Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE
CConformal geometry of surfaces in the Lagrangian–Grassmannianand second order PDE
Dennis TheNovember 6, 2018
Abstract
Of all real Lagrangian–Grassmannians LG ( n, n ), only LG (2 ,
4) admits a distinguished (Lorentzian) conformal structureand hence is identified with the indefinite M¨obius space S , . Using Cartan’s method of moving frames, we study hyperbolic(timelike) surfaces in LG (2 ,
4) modulo the conformal symplectic group
CSp (4 , R ). This CSp (4 , R )-invariant classification isalso a contact-invariant classification of (in general, highly non-linear) second order scalar hyperbolic PDE in the plane. Via LG (2 , Contents
CSp (4 , R )-invariants for surfaces . . . . . . . . . . . . . . . 6 LG (2 , LG (2 ,
4) and
CSp (4 , R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 LG (2 ,
4) as a quadric hypersurface in RP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 The invariant conformal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Elliptic, parabolic, and hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Spheres in LG (2 ,
4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 The Maurer–Cartan form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.7 Moving frames for surfaces in LG (2 ,
4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 a r X i v : . [ m a t h . DG ] S e p Hyperbolic 2-generic surfaces 23
A.1 Null parametrization and differential syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.2 1-adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.3 Monge–Amp`ere invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.4 2-adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.5 3-adaptation for singly-ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.6 3-adaptation for generic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
In this article we have two goals in mind: (i) investigate the local differential geometry of hyperbolic (timelike)surfaces in the real Lagrangian–Grassmannian LG (2 ,
4) modulo the conformal symplectic group
CSp (4 , R ), and (ii)investigate the local contact geometry of (in general, highly non-linear) scalar hyperbolic PDE in the plane. Let usdescribe each in turn and clarify their connection to each other.Recall that LG (2 ,
4) is the set of isotropic 2-planes with respect to a given symplectic form η on R . As a manifold, LG (2 ,
4) is 3-dimensional and admits a transitive action by the symplectic group Sp (4 , R ). For our purposes, η willbe only defined up to scale, hence we use instead CSp (4 , R ). Our basic question then is: Given two (embedded)surfaces M, ˜ M of LG (2 , , does there exist an element g ∈ CSp (4 , R ) such that ˜ M = g · M ? We will address thisquestion in the small and seek differential invariants which determine if a neighbourhood of a point in M is equivalentto a neighbourhood of a point in ˜ M . Most of our study will assume M, ˜ M come equipped with parametrizations i : U → M , ˜ i : U → ˜ M so that equivalence means ˜ i = g · i on U . Ultimately however, we will be interested in the unparametrized equivalence problem, whereby i : U → M and ˜ i : ˜ U → ˜ M are equivalent if there is g ∈ CSp (4 , R ) anda diffeomorphism ϕ : U → ˜ U such that ˜ i ◦ ϕ = g · i on U . While the general study of submanifolds in homogeneousspaces is classical [Jen77], our study of surfaces in LG (2 ,
4) modulo
CSp (4 , R ) has not appeared in the literature.A feature of LG (2 ,
4) absent in higher dimensional Lagrangian–Grassmannians is that LG (2 ,
4) is endowed witha canonical (up to sign)
CSp (4 , R )-invariant Lorentzian conformal structure [ µ ], or equivalently a unique CSp (4 , R )-invariant cone field C . This is a manifestation of the well-known isomorphism Sp (4 , R ) ∼ = Spin (2 , Spin (2 , SO + (2 , SO (2 , B and C Dynkin diagrams, or corresponding (real) Satake diagrams. Withrespect to [ µ ], LG (2 ,
4) is conformally flat and is diffeomorphic to the indefinite M¨obius space S , .The study of the M¨obius space (conformal sphere) S n of definite signature has a long history. We highlightonly the work of Akivis & Goldberg [AG96] which contains an extensive bibliography of the literature on conformalgeometry and which was a source of inspiration for our work here. Using moving frames, Akivis & Goldberg carry outa unified study of submanifolds in conformal spaces. However, their study of the indefinite signature case (Section3.3 in [AG96]) is very brief – e.g. the timelike case occupies only half of p.104 in [AG96]. Four-dimensional conformalstructures in all signatures are studied, but it seems that the three-dimensional indefinite case S , ∼ = LG (2 ,
4) wasnot substantially addressed in their work and has not appeared anywhere in the literature. Thus, one of our goals isto fill in this gap. It should be noted that hypersurface theory in S n for n ≥ S .For n ≥
4, conformal rigidity of a generic hypersurface is determined by the first fundamental form and trace-freesecond fundamental form (c.f. Theorem 2.3.1 in [AG96]); for n = 3, third order invariants come into play [SS80]. Inthe indefinite case, a similar phenomenon occurs. We also remark that LG (2 ,
4) can be identified with the Lie quadric Q in Lie sphere geometry [Cec08], whose elements correspond to oriented spheres and points in R . However, hereas well, no study of surfaces in Q has appeared in the literature.In spirit, our study is similar to the classical theory of surfaces in Euclidean space R modulo the Euclidean group E (3) consisting of rotations, translations and reflections. The mean and Gaussian curvatures feature prominently inthis theory. We derive here analogous local invariants for hyperbolic surfaces in LG (2 ,
4) modulo
CSp (4 , R ). Unlike2he Euclidean case, any hyperbolic surface is locally conformally flat (see Lemma A.1), so has no intrinsic geometry.Thus, our question is an extrinsic one solely concerned with their embedding into the ambient space LG (2 , F = 0 can be realized geometrically as a (7-dimensional) hypersurfacein the second jet-space J = J ( R , R ). This ambient jet space is equipped with a canonical contact system C and thegeometric theory of such PDE is concerned with the study of such hypersurfaces modulo contact transformations, i.e.those diffeomorphisms preserving C . One has the well-known contact-invariant classification into equations of elliptic,parabolic, and hyperbolic type [Gar67]. Less known is the more refined subclassification of hyperbolic PDE into thoseof Monge–Amp`ere (MA), Goursat, and generic types (also called class 6-6, 6-7, and 7-7), based on properties of theso-called Monge subsystems [GK93]. All MA PDE are of the form a ( z xx z yy − ( z xy ) ) + bz xx + cz xy + dz yy + e = 0(the coefficients are functions of x, y, z, z x , z y ) and have been well-studied, while relatively little progress has beenmade on the latter classes, which consist of non-linear equations. For a recent study of generic hyperbolic PDE, see[The08], [The10]. We also highlight the fact that (nonlinear) hyperbolic PDE of the type mentioned above arise ashydrodynamic reductions of certain integrable PDE in three independent variables [Smi09]. We give an equivalentdefinition of these three hyperbolic classes in terms of the 2nd order CSp (4 , R )-invariant classification of hyperbolicsurfaces in LG (2 , J | ξ , for ξ ∈ J , of the projection map π : J → J is diffeomorphic to LG (2 , F = 0 with each fibre J | ξ is a two-dimensional surface; the hyperbolicitycondition is a first-order condition on this surface.3. Regarding C as a rank 4 distribution on J endowed with a (conformal) symplectic form η , any contact trans-formation of J fixing ξ ∈ J acts on the fibre J | ξ = LG ( C ξ , [ η ]) by an element of CSp ( C ξ , [ η ]).Ours is a fibrewise study of π : J → J and CSp (4 , R )-invariants of surfaces in LG (2 ,
4) yield contact invariants fora PDE.
This fibrewise study adds nothing new for MA equations, but new contact invariants are obtained for Goursatand generic equations.
Our work here is principally a contribution to the study of fully non-linear hyperbolic PDE.Added impetus for this work comes from the recent study of curves in general Lagrangian–Grassmannians [Zel05],[ZL07], integrable PDE in three independent variables (hypersurfaces in LG (3 , GL (2 , R )geometry [Smi09], symplectic MA equations in four independent variables (hypersurfaces in LG (4 , LG realization of the fibres of π : J → J dates back atleast to work of Yamaguchi [Yam82], we feel this viewpoint is not well-known and has not been sufficiently explored.Let us give an overview of the contents of our paper. In Section 2, we review the construction of the second jetspace J via the Lagrange–Grassmann bundle. We prove in Theorem 2.7 that CSp (4 , R )-invariants yield contactinvariants for PDE. This is easily seen to hold for J ( R n , R ), so more generally: submanifold theory in LG ( n, n ) modulo CSp (2 n, R ) has implications for the contact-invariant study of (systems of ) scalar PDE in n independentvariables . In Section 3, we delve into the geometry of LG (2 , Sp (4 , R ) ∼ = Spin (2 , LG (2 ,
4) as a quadric hypersurface
Q ⊂ RP , and its canonical CSp (4 , R )-invariant conformal structure and cone field C . To any [ z ] ∈ P V there corresponds a basic surface S [ z ] = P ( z ⊥ ) ∩ Q which we call a “sphere” of indefinite, definite, or degenerate type (locally, a hyperboloid of onesheet, two sheets, or a cone respectively).In Section 4, we use Cartan’s method of moving frames to extract invariants of hyperbolic surfaces M ⊂ LG (2 , doubly-ruled or singly-ruled by null geodesics, or generic surfaces. A doubly-ruled hyperbolic surface is shown to be(an open subset of) an indefinite sphere. In local coordinates, spheres take the same form as a MA equation. Werecover the well-known theorem on contact-invariance of the hyperbolic MA PDE class via the simple argument:1. A Monge–Amp`ere PDE intersects any fibre of π : J → J as an indefinite sphere.2. A contact transformation of J maps indefinite spheres in any fibre to indefinite spheres in any other fibre.If M is given a null parametrization, explicit parametrizations of moving frames are given in Appendix A, leadingin particular to two relative invariants I , I which we call Monge–Amp`ere invariants because of their connection tocorresponding contact invariants for hyperbolic PDE. In the PDE setting, the MA invariants were first calculated byVranceanu [Vra40] for equations of the form z xx = f ( x, y, z, z x , z y , z xy , z yy ). Much later, Jur´aˇs [Jur97] calculated theseinvariants for general F ( x, y, z, z x , z y , z xx , z xy , z yy ) = 0. A third calculation appeared in [The08] which simplifiedthe expression of these invariants. However, all three calculations were significantly involved and did not appeal to3he inherent geometry of surfaces in LG (2 , I , I based on 2-adapted moving frames isconceptually simple and geometrically motivated: their vanishing characterizes indefinite spheres.In Sections 5 & 6, we study the geometry of singly-ruled and generic surfaces. For such surfaces M , pairs ofthird order objects called cone congruences can be geometrically associated to M . The construction of 3-adaptedmoving frames leads to the key notion of the conjugate manifold M (cid:48) , whose dimension dim( M (cid:48) ) is an invariant. If M (cid:48) is also a surface, both M , M (cid:48) are envelopes for the central sphere congruence of M . For a PDE, a correspondingfibrewise construction leads to the notion of its conjugate PDE . No such notion exists for MA equations since fibrewisethese are second order objects (spheres), while the conjugate manifold is a third order construction. In the generic(2-elliptic) case, curvature lines exist which lead to the Lorentzian analogues of contact spheres, canal surfaces, andDupin cyclides. As a preview, our classification of hyperbolic surfaces / PDE is given in Figure 1. Examples whichwe will discuss in this paper are given in Figure 2. (See Definition 2.10 for the notion of a CSI PDE.) M hyperbolic I = I = 0 (cid:121) (cid:121) (cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114) I = 0 or I = 0 (cid:15) (cid:15) I I (cid:54) = 0 (cid:38) (cid:38) (cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76)(cid:76) dim( M (cid:48) ) = 0 dim( M (cid:48) ) = 1 M (cid:48) null curveDoubly-ruled bynull geodesics:2-isotropic Singly-ruled bynull geodesics:2-parabolic (cid:122) (cid:122) (cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116) (cid:15) (cid:15) (cid:36) (cid:36) (cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74)(cid:74) Generic surfaces:2-hyp: (cid:15) = − (cid:15) = 1 (cid:47) (cid:47) (cid:79) (cid:79) (cid:57) (cid:57) (cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116)(cid:116) dim( M (cid:48) ) = 2 M (cid:48) hyperbolic (cid:15) (cid:15) (cid:34) (cid:34) (cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69)(cid:69) (cid:47) (cid:47) M (cid:48) M (cid:48) ) = 0( δ = δ = 0) dim( M (cid:48) ) = 1 M (cid:48) null geodesic( δ = ± δ = 0) dim( M (cid:48) ) = 2 M (cid:48) hyp. 2-par.( δ = ± M (cid:48) M (cid:48) CSp (4 , R )-equivalence; also, hyperbolic PDE up to contact-equivalenceThe last two entries in Table 2 form the complete list (up to contact equivalence) of maximally symmetric hyperbolicPDE of generic type [The08], each having 9-dimensional contact symmetry algebra. In the case of the latter family,this 9-dimensional Lie algebra is independent of the parameter c and has been shown to be isomorphic to a parabolicsubalgebra of the non-compact real form of the 14-dimensional exceptional simple Lie algebra g [The10]. We shouldalso note that while all examples in Figure 2 are of the form F ( z xx , z xy , z yy ) = 0, our classification applies equallywell to those PDE which have x, y, z, z x , z y dependency as well: our classification is fibrewise.In Section 7, we make some concluding remarks and comment on future directions for research. A scalar second order PDE in the plane can be realized as a hypersurface in J = J ( R , R ). We review the geometricconstruction of the jet spaces J k = J k ( R , R ) for k ≤
2. This section generalizes to more independent variables thantwo, but we restrict to two for simplicity and since this is our main focus in this paper.
Identify the zeroth order jet space J with R , regarded as a trivial bundle with fibre R over the base R . Definethe first order jet space J to be the Grassmann bundle Gr (2 , T R ), with projection π : J → J . Any ξ ∈ J is a2-plane in T ¯ ξ R , where ¯ ξ = π ( ξ ). There is a canonical linear Pfaffian system ( I , J ) on J given by I| ξ = ( π ) ∗ ( ξ ⊥ ) , J | ξ = ( π ) ∗ ( T ∗ ¯ ξ R ) . M ⊂ LG (2 ,
4) CSI? a ( z xx z yy − ( z xy ) ) + bz xx + cz xy + dz yy + e = 0( a, b, c, d, e functions of x, y, z, z x , z y ) indefinite sphere (cid:88) F ( z xy , z yy ) = 0 indefinite sphere or singly-ruled , dim( M (cid:48) ) ≤ × z xy = ( z yy ) singly-ruled , dim( M (cid:48) ) = 0 (cid:88) z xy = exp( z yy ) singly-ruled , dim( M (cid:48) ) = 1 , δ = +1 (cid:88) z xy = ln( z yy ) singly-ruled , dim( M (cid:48) ) = 1 , δ = − (cid:88) F ( z xx , z yy ) = 0 indefinite sphere or 2-elliptic × z xx = exp( z yy ) 2-elliptic , dim( M (cid:48) ) = 2 (cid:88) z xx = ( z yy ) , dim( M (cid:48) ) = 2 (cid:88) z xx z yy = − , dim( M (cid:48) ) = 2 (cid:88) z xx ( z yy ) + 1 = 0 2-elliptic , dim( M (cid:48) ) = 0 (cid:88) (3 z xx − z xy z yy + 2( z yy ) ) (2 z xy − ( z yy ) ) = c c > , c < , c = 0and z xy > ( z yy ) ; dim( M (cid:48) ) = 0 (cid:88) (For 0 < c ≤
4, the PDE is not hyperbolic.)Figure 2: Examples of hyperbolic PDEThe canonical system C (also called the contact distribution) on J is defined by C ξ = ( π ) ∗− ( ξ ) ⊂ T ξ J .Let us examine this in local coordinates. Let ξ ∈ J , ¯ ξ = π ( ξ ). Pick local coordinates ( x, y, z ) on an openset ¯ U ⊂ R about ¯ ξ such that dx ∧ dy | ξ (cid:54) = 0. By continuity, dx ∧ dy | ξ (cid:54) = 0 for all ξ in some neighbourhood U ⊂ ( π ) − ( ¯ U ) about ξ . Since ξ is 2-dimensional and dx | ξ , dy | ξ are linearly independent, then on U , dz | ξ = p ( ξ ) dx | ξ + q ( ξ ) dy | ξ . This defines local coordinates p = p ( ξ ) , q = q ( ξ ). Hence, ξ = span { D x , D y } ∈ Gr (2 , T ¯ ξ R ), where D x = ∂∂x (cid:12)(cid:12)(cid:12)(cid:12) ¯ ξ + p ( ξ ) ∂∂z (cid:12)(cid:12)(cid:12)(cid:12) ¯ ξ , D y = ∂∂y (cid:12)(cid:12)(cid:12)(cid:12) ¯ ξ + q ( ξ ) ∂∂z (cid:12)(cid:12)(cid:12)(cid:12) ¯ ξ . Pulling back J coordinates to J , ( x, y, z, p, q ) are local coordinates on J , I is generated by the contact 1-form σ = dz − pdx − qdy , and J = { dz, dx, dy } is associated with the independence condition dx ∧ dy (cid:54) = 0. Finally, C = { σ = 0 } = span (cid:26) ∂∂x + p ∂∂z , ∂∂y + q ∂∂z , ∂∂p , ∂∂q (cid:27) . (2.1)The exterior derivative η = dσ = dx ∧ dp + dy ∧ dq restricts to be a non-degenerate (conformal) symplectic formon the distribution C . (Only the conformal class of η is relevant since σ is only well-defined up to scale.) Withrespect to the basis given in (2.1), η is represented by (cid:18) I − I (cid:19) , where I is the 2 × J as the Lagrange–Grassmann bundle LG ( C, [ η ]) over J , with projection π : J → J , i.e. J = (cid:91) ξ ∈ J J | ξ := (cid:91) ξ ∈ J LG ( C ξ , [ η ]) . Namely, any ˜ ξ ∈ J is a Lagrangian (i.e. 2-dimensional isotropic) subspace of ( C ξ , [ η ]), where ξ = π ( ˜ ξ ). Since( C ξ , [ η ]) is isomorphic to R with its canonical (conformal) symplectic form, it is clear that the fibres of π are alldiffeomorphic to LG (2 , J is given by ˜ C ˜ ξ = ( π ) ∗− ( ˜ ξ ) ⊂ T ˜ ξ J .5iven ˜ ξ ∈ J , pick adapted local coordinates ( x, y, z, p, q ) (as above) on a neighbourhood U of ξ = π ( ˜ ξ ) ∈ J with dx ∧ dy | ˜ ξ (cid:54) = 0. Define ˜ U = { ˜ ξ ∈ ( π ) − ( U ) : dx ∧ dy | ˜ ξ (cid:54) = 0 } . For any ˜ ξ ∈ ˜ U , define r ( ˜ ξ ) , s ( ˜ ξ ) , s ( ˜ ξ ) , t ( ˜ ξ ) by dp | ˜ ξ = r ( ˜ ξ ) dx | ˜ ξ + s ( ˜ ξ ) dy | ˜ ξ , dq | ˜ ξ = s ( ˜ ξ ) dx | ˜ ξ + t ( ˜ ξ ) dy | ˜ ξ . Since ˜ ξ is Lagrangian, then on ˜ ξ we have 0 = η = dσ = dx ∧ dp + dy ∧ dq = ( s − s ) dx ∧ dy so that s = s . Letting s := s = s , we have ˜ ξ = span { ˜ D x , ˜ D y } ∈ LG ( C ξ , [ η ]) ⊂ Gr (2 , T J ), where˜ D x = ∂∂x (cid:12)(cid:12)(cid:12)(cid:12) ξ + p ( ξ ) ∂∂z (cid:12)(cid:12)(cid:12)(cid:12) ξ + r ( ˜ ξ ) ∂∂p (cid:12)(cid:12)(cid:12)(cid:12) ξ + s ( ˜ ξ ) ∂∂q (cid:12)(cid:12)(cid:12)(cid:12) ξ , ˜ D y = ∂∂y (cid:12)(cid:12)(cid:12)(cid:12) ξ + q ( ξ ) ∂∂z (cid:12)(cid:12)(cid:12)(cid:12) ξ + s ( ˜ ξ ) ∂∂p (cid:12)(cid:12)(cid:12)(cid:12) ξ + t ( ˜ ξ ) ∂∂q (cid:12)(cid:12)(cid:12)(cid:12) ξ . (2.2)Pulling back the J coordinates ( x, y, z, p, q ) to J , we have a local coordinate system ( x, y, z, p, q, r, s, t ) on J . Wecall such coordinates standard . The canonical system ˜ C is given by ˜ C = { σ = σ = σ = 0 } , where σ = dz − pdx − qdy, σ = dp − rdx − sdy, σ = dq − sdx − tdy. We see that for sections R → J , ( x, y ) → ( x, y, z, p, q, r, s, t ), on which σ = σ = σ = 0, we have p = z x , q = z y , r = z xx , s = z xy , t = z yy . Definition 2.1.
A (local) contact transformation of J [ or J ] is a (local) diffeomorphism preserving the canonicalsystem C [ or ˜ C ] under pushforward. Any contact transformation φ of J prolongs to a contact transformation pr( φ ) of J , i.e. φ ∗ acts on C , inducinga map of Lagrangian–Grassmannians pr( φ ) : J | ξ → J | ξ (cid:48) . In particular, pr( φ ) preserves fibres of π : J → J .Conversely, by Backl¨und’s theorem [Olv95], if ˜ φ on J is contact, then ˜ φ = pr( φ ) for some φ contact on J . Example 2.2.
It is well-known that (for scalar-valued maps) there are contact transformations of J that arenot the prolongation of diffeomorphisms of J . An example of this is the Legendre transformation (¯ x, ¯ y, ¯ z, ¯ p, ¯ q ) =( − p, − q, z − px − qy, x, y ) , which satisfies ¯ σ = d ¯ z − ¯ pd ¯ x − ¯ qd ¯ y = d ( z − px − qy ) + xdp + ydq = σ . Since φ is contact, then φ ∗ σ = λσ , for some function λ on J , which implies for η = dσ that φ ∗ η = φ ∗ dσ = d ( φ ∗ σ ) = dλ ∧ σ + λη . Restricting to C = { σ = 0 } , we see that η is preserved up to the overall factor of λ , i.e. φ ∗ : ( C ξ , [ η ]) → ( C ξ (cid:48) , [ η ]) is a conformal symplecticomorphism. If ξ (cid:48) = φ ( ξ ) = ξ , then φ ∗ ∈ CSp ( C ξ , [ η ]) and thisacts on J | ξ . Conversely, one may ask if, given ξ ∈ J , any element of CSp ( C ξ , [ η ]) is realized by a local contacttransformation of J . This question has an affirmative answer, but we postpone the proof until Section 3.1. Theseobservations motivate our study of surfaces in LG (2 ,
4) modulo
CSp (4 , R ) (as opposed to a smaller subgroup). CSp (4 , R ) -invariants for surfaces Consider a second order scalar PDE in the plane F = 0, regarded as a hypersurface Σ in J . We use the usualnon-degeneracy assumption that Σ is transverse to π : J → J and that π | Σ : Σ → J is a submersion. Given ξ ∈ J , consider the fibre J | ξ = ( π ) − ( ξ ) = LG ( C ξ , [ η ]) and Σ | ξ = Σ ∩ J | ξ , which is a 2-dimensional surface. Wenow discuss the transfer of differential invariants from the standard LG (2 ,
4) setting to the PDE setting. For now,simply regard LG (2 ,
4) as the set of isotropic 2-planes on the standard ( R , [ η ]) with standard basis and CSp (4 , R )consists of linear transformations of R which preserve η up to scale.Let U be a 2-dimensional connected manifold. Let J n ( U, LG (2 , n -th order jet space of maps s : U → LG (2 , g ∈ CSp (4 , R ) prolongs to an action g ( n ) on J n ( U, LG (2 , Definition 2.3. An n -th order differential invariant for CSp (4 , R ) is a function κ : J n ( U, LG (2 , → R such that κ ( g ( n ) · p ) = κ ( p ) , for any g ∈ CSp (4 , R ) and p ∈ J n ( U, LG (2 , . For short, we call κ a CSp (4 , R ) -invariant. Example 2.4.
We will endow LG (2 , with natural coordinates ( r, s, t ) and the conformal structure [ drdt − ds ] .Locally, J ( U, LG (2 , is given by ( u, v, r, s, t, r u , r v , s u , s v , t u , t v ) . The sign of the determinant of (cid:18) r u t u − s u r u t v + r v t u − s u s v r u t v + r v t u − s u s v r v t v − s v (cid:19) is a first-order CSp (4 , R ) -invariant, distinguishing hyperbolic (timelike), parabolic (null), elliptic (spacelike) surfaces. efinition 2.5. Let ( W, [ η ]) be a 4-dimensional real vector space endowed with a conformal symplectic form. A conformal symplectic (c.s.) basis b = { e i } i =1 is a basis of W such that the matrix ( η ( e i , e j )) i,j =1 is a multiple of (cid:18) I − I (cid:19) . The first two vectors e , e of b span a Lagrangian subspace, i.e. an element of LG ( W, [ η ]) . Lemma 2.6. A CSp (4 , R ) -invariant κ : J n ( U, LG (2 , → R induces a CSp ( C ξ , [ η ]) -invariant κ ξ : J n ( U, J | ξ ) → R .Proof. Fix any c.s. basis b of ( C ξ , [ η ]). This defines an isomorphism (conformal symplectomorphism) ψ b : C ξ → R by mapping b onto the standard basis of R , hence induces ˜ ψ b : J | ξ ∼ = → LG (2 ,
4) and identifies (cid:98) ψ b : CSp ( C ξ , [ η ]) ∼ = → CSp (4 , R ). Note that for g ∈ CSp ( C ξ , [ η ]) and g b = (cid:98) ψ b ( g ) ∈ CSp (4 , R ), we have: ψ g · b ◦ g = ψ b , ψ b ◦ g = g b ◦ ψ b , (2.3)with similar identities for ˜ ψ b . Given a CSp (4 , R )-invariant κ , define κ ξ : J n ( U, J | ξ ) → R by κ ξ ( j nx s ) := κ ( j nx ( ˜ ψ b ◦ s )),where s : U → J | ξ is a local map and j nx s is the n -th jet prolongation of s evaluated at x ∈ U . We claim:1. κ ξ is well-defined: Let b (cid:48) be another c.s. basis, so b (cid:48) = g · b for some g ∈ CSp ( C ξ , [ η ]). By (2.3), κ ( j nx ( ˜ ψ b (cid:48) ◦ s )) = κ ( j nx ( ˜ ψ b ◦ g − ◦ s )) = κ (( g b − ) ( n ) · j nx ( ˜ ψ b ◦ s )) = κ ( j nx ( ˜ ψ b ◦ s )).2. κ ξ is CSp ( C ξ , [ η ])-invariant: Fix g ∈ CSp ( C ξ , [ η ]). By (2.3), κ ξ ( g ( n ) · j nx s ) = κ ξ ( j nx ( g · s )) = κ ( j nx ( ˜ ψ b ◦ ( g · s ))) = κ ( j nx ( g b · ( ˜ ψ b ◦ s ))) = κ ( g b ( n ) · j nx ( ˜ ψ b ◦ s )) = κ ( j nx ( ˜ ψ b ◦ s )) = κ ξ ( j nx s ).The function κ ξ accounts for “vertical” derivatives, i.e. derivatives only along the fibre J | ξ . Consider the bundle J n := ˙ (cid:83) ξ ∈ J J n ( U, J | ξ ) over J . Given a contact transformation ˜ φ : J → J and a map s : U → J | ξ , we have˜ φ ◦ s : U → J | φ ( ξ ) , hence we define Φ ( n ) : J n → J n by Φ ( n ) ( j nx s ) := j nx ( ˜ φ ◦ s ). Define K : J n → R by K ( p ) = κ ξ ( p ),where p ∈ J n ( U, J | ξ ). We say that K is contact-invariant if K ◦ Φ ( n ) = K for any contact transformation ˜ φ : J → J . Theorem 2.7.
Any
CSp (4 , R ) -invariant κ : J n ( U, LG (2 , → R induces a contact-invariant function K : J n → R .In particular, for any constant c ∈ R , K = c defines a contact-invariant class of PDE.Proof. To show K is contact-invariant, we show κ φ ( ξ ) (Φ ( n ) ( j nx s )) = κ ξ ( j nx s ) for any contact ˜ φ = pr( φ ) : J → J and s : U → J | ξ . If φ ( ξ ) = ξ , then ˜ φ ∈ CSp ( C ξ , [ η ]), so by CSp ( C ξ , [ η ])-invariance, κ ξ ◦ Φ ( n ) = κ ξ . If φ ( ξ ) (cid:54) = ξ , chooseany c.s. basis b = { e i } i =1 of ( C ξ , [ η ]). Since φ ∗ is a conformal symplectomorphism, then b (cid:48) = { e (cid:48) i = φ ∗ ( e i ) } i =1 isalso a c.s. basis. We have κ φ ( ξ ) (Φ ( n ) ( j nx s )) = κ φ ( ξ ) ( j nx ( ˜ φ ◦ s )) = κ ( j nx ( ˜ ψ b (cid:48) ◦ ˜ φ ◦ s )) = κ ( j nx ( ˜ ψ b ◦ s )) = κ ξ ( j nx s ). Remark 2.8.
Theorem 2.7 clearly generalizes to J ( R n , R ) : Submanifold theory in LG ( n, n ) modulo CSp (2 n, R ) has implications for the contact-invariant study of systems of scalar PDE in n independent variables. For any surface M ⊂ LG (2 ,
4) (as for a PDE) there is no distinguished choice of (local) parametrization i : U → M and hence invariants that we seek should be independent of reparametrization. Thus, we are ultimately interestedin the unparametrized equivalence problem: Given i : U → M and ˜ i : ˜ U → ˜ M , does there exist a diffeomorphism ϕ : U → ˜ U and an element g ∈ CSp (4 , R ) such that ˜ i ◦ ϕ = g · i on U ? We will study this problem in three steps:1. Apply Cartan’s method of moving frames to find invariants for the parametrized equivalence problem.2. Give a parametric description of the invariants of the parametrized problem. There is a natural choice ofcoordinates ( r, s, t ) on LG (2 ,
4) which correspond to the 2nd derivative coordinates in the PDE setting.3. Investigate how the above invariants change under reparametrization. Unchanged properties will be invariantsfor the unparametrized equivalence problem.Generally, the simplest surfaces to describe are those with constant
CSp (4 , R )-invariants κ . Surfaces in LG (2 , all constant symplectic ( CSp (4 , R )) invariants (CSI) have a natural geometric meaning: Proposition 2.9 (Homogeneity theorem, p.42 of [Jen77]) . For any smooth embedding i : U → G/H , M = i ( U ) isan open submanifold of a homogeneous submanifold of G/H iff M has no non-constant G -invariants. Definition 2.10.
Let Σ ⊂ J be a PDE. If for any (well-defined) CSp (4 , R ) -invariant κ of surfaces in LG (2 , thecorresponding contact-invariant K is constant on Σ , then we call Σ a constant symplectic invariant (CSI) PDE. Several examples of hyperbolic CSI PDE are given in Figure 2.7
Conformal geometry of LG (2 , In Section 2, our main result was that
CSp (4 , R )-invariants for surfaces in LG (2 ,
4) induce contact-invariants forPDE. With this in mind, we describe the ambient geometry of LG (2 ,
4) with a view to preparing for our later studyof hyperbolic surfaces via the method of moving frames. LG (2 , and CSp (4 , R ) On R , take the standard basis { e i } i =1 with dual basis { η i } i =1 . Let η = 2( η ∧ η + η ∧ η ) be the standardsymplectic form represented by J = (cid:18) I − I (cid:19) , so η ( x, y ) = x T Jy . We define: Sp (4 , R ) = (cid:8) X ∈ GL (4 , R ) : X T JX = J (cid:9) = (cid:26)(cid:18) A BC D (cid:19) : A T C = C T A, B T D = D T B, A T D − C T B = I (cid:27) , sp (4 , R ) = (cid:8) X ∈ M at × ( R ) : X T J + JX = 0 (cid:9) = (cid:26)(cid:18) a bc − a T (cid:19) : b, c symmetric (cid:27) , which are 10-dimensional. LG (2 ,
4) is the set of η -isotropic 2-planes in R . This admits a transitive Sp (4 , R )-action,hence LG (2 ,
4) = Sp (4 , R ) /P up to a choice of basepoint o . Choosing o = span { e , e } , we see dim( P ) = 7 since P = (cid:26)(cid:18) A B A T ) − (cid:19) : A ∈ GL (2 , R ) , A − B symmetric (cid:27) , p = (cid:26)(cid:18) a b − a T (cid:19) : b symmetric (cid:27) . Hence, dim( LG (2 , Sp (4 , R )-action is not effective: the global isotropy subgroup is K = {± I } ∼ = Z .In the PDE context (Section 2.1), only the conformal class [ η ] is relevant. This does not affect the definition of LG (2 ,
4) since a 2-plane is η -isotropic iff it is ( λη )-isotropic. We consider instead the conformal symplectic group CSp (4 , R ) = (cid:8) X ∈ GL (4 , R ) : X T JX = ρJ, for some ρ ∈ R × (cid:9) ∼ = Sp (4 , R ) (cid:111) ϕ R × , where R × is embedded into GL (4 , R ) as ρ (cid:55)→ diag( ρI , I ) and note that ϕ : R × → Aut ( Sp (4 , R )) by ϕ ( ρ ) (cid:18) A BC D (cid:19) = (cid:18) ρI I (cid:19) (cid:18) A BC D (cid:19) (cid:18) ρ − I I (cid:19) = (cid:18) A ρBρ − C D (cid:19) . Since diag ( √ ρ − I , √ ρI ) · diag ( ρI , I ) = √ ρI acts trivially, then ρ ∈ R × and diag ( √ ρI , √ ρ − I ) ∈ P have thesame action on LG (2 , R × is redundant and it suffices to consider (cid:99) Sp (4 , R ) = Sp (4 , R ) (cid:111) ϕ Z , where Z is generated by ρ = −
1, i.e. diag ( − I , I ). The stabilizer in (cid:99) Sp (4 , R ) of o is (cid:98) P = P (cid:111) Z , and global isotropy is K .Let us describe a collection of charts on general LG ( n, n ). For more details, see [PT03]. A Lagrangian decomposi-tion of ( R n , η ) is a pair ( L , L ) of transversal Lagrangian subspaces in LG ( n, n ), i.e. R n = L ⊕ L and L ∩ L = 0.Define (cid:116) ( L ) = { L ∈ LG ( n, n ) : L transverse to L } and B sym ( L ) = { symmetric bilinear forms on L } . Associ-ated to any Lagrangian decomposition ( L , L ) is a chart ψ L ,L = ρ L ,L ◦ φ L ,L : (cid:116) ( L ) → B sym ( L ), where themaps φ L ,L : (cid:116) ( L ) → Lin ( L , L ) and ρ L ,L : L → L ∗ are defined by φ L ,L ( L ) = T L , ρ L ,L ( v ) = η ( · , v ) | L (3.1)Here, T L is uniquely defined by L = graph ( T L ) = { v + T L ( v ) : v ∈ L } . A priori the image of ψ L ,L is in B ( L ), butin fact ψ L ,L is a bijection onto B sym ( L ). Moreover, the collection of charts as ( L , L ) ranges over all Lagrangiandecompositions yields a differentiable atlas for LG ( n, n ) [PT03].Let L = o = span { e , e } and L = span { e , e } . We describe the chart ψ = ψ L ,L on LG (2 , L = span { ˜ e = e + re + se , ˜ e = e + se + te } ∈ (cid:116) ( L ) ψ (cid:55)−→ (cid:18) r ss t (cid:19) . (3.2)8he right side of (3.2) is the matrix for: (i) T L ∈ Lin ( L , L ) in the bases { e , e } , { e , e } , or (ii) ψ ( L ) ∈ B sym ( L ) inthe basis { e , e } . Equivalently, exponentiate X = (cid:18) r ss t (cid:19) on sp (4 , R ) / p to obtain (cid:18) I X I (cid:19) /P on LG (2 ,
4) = Sp (4 , R ) /P , and the first two columns yield the left side of (3.2). From (2.1) and (2.2), the fibrewise identification e = ∂ x + p∂ z , e = ∂ y + q∂ z , e = ∂ p , e = ∂ q (3.3)implies that these ( r, s, t ) coordinates in LG (2 ,
4) correspond to the natural ( r, s, t ) coordinates in the J setting.Hereafter, whenever we write ( r, s, t ), we mean such coordinates in one setting or the other.Now let us see the (cid:99) Sp (4 , R )-action in the chart ψ . The element ρ = − ∈ Z acts as ( r, s, t ) (cid:55)→ ( − r, − s, − t ). Forelements in Sp (4 , R ) sufficiently close to the identity, (cid:18) A BC D (cid:19) · (cid:18) I X I (cid:19) /P = (cid:18) A + BX BC + DX D (cid:19) /P = (cid:18) I X I (cid:19) /P, (3.4)so we have the M¨obius-like transformation ˜ X = ( C + DX )( A + BX ) − . Table 1 displays the infinitesimal generators.Rotations Inversions Translations a − r∂ r − s∂ s b − r ∂ r − rs∂ s − s ∂ t c ∂ r a − r∂ s − s∂ t b − s ∂ r − st∂ s − t ∂ t c ∂ t a − s∂ r − t∂ s b − rs∂ r − ( s + rt ) ∂ s − st∂ t c ∂ s a − s∂ s − t∂ t Table 1: Infinitesimal generators for sp (4 , R )-action on LG (2 , Proposition 3.1.
Given ξ ∈ J , any element of CSp ( C ξ , [ η ]) is realized by a local contact transformation of J .Proof. Fix ξ ∈ J . Choose coordinates ( x, y, z, p, q ) on J as in Section 2.1 such that ξ = ( x, y, z, p, q ) = (0 , , , , r, s, t ) coordinates in the LG (2 ,
4) and J | ξ settings. Next, we realize the vector fieldsin Table 1 as the ( r, s, t ) components of a prolongation X (2) of a contact symmetry X . X X (2) ( r, s, t ) components of X (2)12 x ∂ z x ∂ z + x∂ p + ∂ r ∂ r xy∂ z xy∂ z + y∂ p + x∂ q + ∂ s ∂ s y ∂ z y ∂ z + y∂ q + ∂ t ∂ t x∂ x x∂ x − p∂ p − r∂ r − s∂ s − r∂ r − s∂ s y∂ x y∂ x − p∂ q − r∂ s − s∂ t − r∂ s − s∂ t x∂ y x∂ y − q∂ p − s∂ r − t∂ s − s∂ r − t∂ s y∂ y y∂ y − q∂ q − s∂ s − t∂ t − s∂ s − t∂ t p∂ x + p ∂ z p∂ x + p ∂ z − r ∂ r − rs∂ s − s ∂ t − r ∂ r − rs∂ s − s ∂ t q∂ x + p∂ y + pq∂ z q∂ x + p∂ y + pq∂ z − rs∂ r − ( s + rt ) ∂ s − st∂ t − rs∂ r − ( s + rt ) ∂ s − st∂ t q∂ y + q ∂ z q∂ y + q ∂ z − s ∂ r − st∂ s − t ∂ t − s ∂ r − st∂ s − t ∂ t Since the components of all X (1) := ( π ) ∗ X (2) above are homogeneous in ( x, y, z, p, q ), then restricted to ξ =(0 , , , , π , i.e. they preserve J | ξ = LG ( C ξ , [ η ]). Thus, anyinfinitesimal generator of the CSp ( C ξ , [ η ])-action on J | ξ is realizable by an infinitesimal contact transformation.But Sp (4 , R ) ∼ = Sp ( C ξ , η ) is connected so this is true in terms of regular (finite) transformations. Lastly, if ρ ∈ R × ,the contact transformation ( x, y, z, p, q ) (cid:55)→ ( ρx, ρy, ρz, p, q ) preserves J | ξ and induces η (cid:55)→ ρη .Because of Proposition 3.1, we can easily recognize various transformations of LG (2 ,
4) from their correspondingjet transformations. For example, in the jet space setting we know the scalings ( x, y, z ) → ( λ x, λ y, µz ) induces( r, s, t ) → ( µλ r, µλ λ s, µλ t ). Hence, the latter is also a transformation of LG (2 , .2 LG (2 , as a quadric hypersurface in RP By the Pl¨ucker embedding, the full Grassmannian Gr (2 ,
4) embeds as the decomposable elements in P ( (cid:86) R ).The induced (cid:99) Sp (4 , R )-action on (cid:86) R preserves V := (cid:86) R = { z ∈ (cid:86) R : η ( z ) = 0 } , which is a 5-dimensionalirreducible subspace. The restriction of the Pl¨ucker embedding to LG (2 ,
4) has image the quadric hypersurface Q = { [ z ] ∈ P V : (cid:104) z, z (cid:105) = 0 } , where (cid:104) z , z (cid:105) = 3( η ∧ η )( z ∧ z ) is a non-degenerate symmetric bilinear form. Notethat 3( η ∧ η )( e ∧ e ∧ e ∧ e ) = − η ∧ η ∧ η ∧ η )( e , e , e , e ) = − det( η i ( e j )) = −
1. On V , take the basis B : e ∧ e ,e ∧ e ,e ∧ e − e ∧ e ,e ∧ e ,e ∧ e ⇒ (cid:104)· , ·(cid:105) B = −
10 0 0 1 00 0 − − , (3.5)so (cid:104)· , ·(cid:105) has signature (2 ,
3) = (+ + − − − ). Since LG (2 ,
4) is (cid:99) Sp (4 , R )-invariant, so are Q and (cid:104)· , ·(cid:105) . Fixing the basis B , there is a homomorphism Φ : (cid:99) Sp (4 , R ) → O (2 , φ = Φ ∗ : sp (4 , R ) → so (2 ,
3) given by φ a a b b a a b b c c − a − a c c − a − a = a + a b b b c a − a − a b c − a − a − b c − a a − a b c − c c − ( a + a ) (3.6)is an isomorphism. Since Sp (4 , R ) is connected and O (2 ,
3) has four connected components, then Φ yields a 2-to-1 covering map Sp (4 , R ) → SO + (2 , Sp (4 , R ) ∼ = Spin (2 , ρ = − ∈ Z acts on B as thematrix diag (1 , − , − , − ,
1) which has determinant −
1. Hence, Φ : (cid:99) Sp (4 , R ) → O + (2 ,
3) is surjective with kernel K = {± I } . The effective symmetry group of Q is O + (2 ,
3) with stabilizer P + = Φ( P ) at [ e ∧ e ]. Remark 3.2.
As groups, O (2 , /SO + (2 , ∼ = Z × Z . Let V + , V − be the maximal positive and negative definitesubspaces of V . Here, O + (2 , ⊂ O (2 , preserves the orientation of V − , e.g. − I ∈ O − (2 , . We can write the condition (cid:104) z, z (cid:105) = 0 defining Q in coordinates as | x | −| y | := x + x − y − y − y = 0, where[ z ] = [( x, y )] ∈ Q and we may assume x ∈ S , y ∈ S . Since [( x, y )] = [( − x, − y )], then LG (2 , ∼ = ( S × S ) / Z .With respect to the chart ψ in Section 3.1, the Lagrangian subspace L = span { ˜ e , ˜ e } in (3.2) satisfies˜ e ∧ ˜ e = e ∧ e + re ∧ e + s ( e ∧ e + e ∧ e ) + te ∧ e + ( rt − s ) e ∧ e , so with respect to B , L = [1 , r, s, t, rt − s ] ∈ Q . Thus, ψ : [1 , r, s, t, rt − s ] ∈ Q (cid:55)−→ ( r, s, t ) ∈ R . Points of LG (2 , ψ − are { [0 , a, b, c, d ] : ac − b = 0 } , which is precisely the degenerate sphere S [ Z ] (see Section 3.5),where Z = (0 , , , , ψ (cid:48) centered at Z , i.e. span { e , e } by taking ( e (cid:48) , e (cid:48) , e (cid:48) , e (cid:48) ) = ( e , e , − e , − e )and constructing similar coordinates [1 , r (cid:48) , s (cid:48) , t (cid:48) , r (cid:48) t (cid:48) − ( s (cid:48) ) ] B (cid:48) = as above with respect to the new basis B (cid:48) . Relating B (cid:48) to B , we see [1 , r (cid:48) , s (cid:48) , t (cid:48) , r (cid:48) t (cid:48) − ( s (cid:48) ) ] B (cid:48) = [ rt − s , − t, s, − r, B (cid:48) . Thus, the coordinate change formula ψ (cid:48) ◦ ψ − is r (cid:48) = − trt − s , s (cid:48) = srt − s , t (cid:48) = − rrt − s . (3.7)In fact, at least three coordinate charts are needed to cover LG (2 , Q not covered by ψ − or ( ψ (cid:48) ) − is Q ∩ { [0 , a, b, c,
0] : ac − b = 0 } . In the PDE setting, the Legendre transformation (see Example 2.2)prolongs to give (3.7) on the J coordinates ( r, s, t ) . There is a distinguished (up to sign) Lorentzian conformal structure [ µ ], equivalently a cone field C = { µ = 0 } , on LG (2 , µ ] is an equivalence class of metrics: µ ∼ µ (cid:48) iff µ (cid:48) = λ µ , where λ is nonvanishing. Forconformal structures, all (cid:99) Sp (4 , R )-invariant ones on LG (2 ,
4) correspond to Ad ( (cid:98) P )-invariant ones on sp (4 , R ) / p = (cid:26)(cid:18) c (cid:19) : c symmetric (cid:27) / p . The element ρ = − Z factor of (cid:98) P acts as c (cid:55)→ − c . Via the adjoint action, (cid:18) A B A T ) − (cid:19) · (cid:18) c (cid:19) / p = (cid:18) A T ) − cA − (cid:19) / p . (3.8)Note det( c ) transforms as det( c ) (cid:55)→ det( A ) − det( c ). The conformal class of its polarization is Ad ( (cid:98) P )-invariantand corresponds to an (cid:99) Sp (4 , R )-invariant conformal structure [ µ ] on LG (2 , ± µ ] are the only two suchstructures. Thus, LG (2 ,
4) is endowed with a canonical
CSp (4 , R )-invariant cone field C = { µ = 0 } .For a second description, let (cid:98) Q = { z ∈ V : (cid:104) z, z (cid:105) = 0 } = cone ( Q ). For any [ z ] ∈ Q and ρ ∈ R × , the affine tangentspace (cid:98) T [ z ] Q is the tangent space to (cid:98) Q at any ρz , translated to the origin, i.e. (cid:98) T [ z ] Q = T ρz (cid:98) Q = T z (cid:98) Q = { w : (cid:104) z, w (cid:105) = 0 } = z ⊥ , dim( z ⊥ ) = 4 . The canonical surjection π : V → P V restricts to π : (cid:98) Q → Q and induces vector space isomorphisms ( π ∗ ) ρz : z ⊥ /(cid:96) z → T [ z ] Q , where (cid:96) z = span { z } . These satisfy ( π ∗ ) ρz ( ρw ) = ( π ∗ ) z ( w ), ∀ w ∈ z ⊥ . The form induced by (cid:104)· , ·(cid:105) on z ⊥ /(cid:96) z isnon-degenerate and has signature (1 , T [ z ] Q via ( π ∗ ) z or ( π ∗ ) ρz and these differ by a factor of ρ .This conformal class on T [ z ] Q induces a conformal structure on all of Q . It is (cid:99) Sp (4 , R )-invariant because so is (cid:104)· , ·(cid:105) . Remark 3.3.
Although ( π ∗ ) ρz depends on ρ ∈ R × , there is a well-defined mapping of subspaces since ( π ∗ ) ρz = ρ ( π ∗ ) z ,e.g. if M ⊂ Q is a submanifold, then T [ z ] M is identified with a subspace of z ⊥ /(cid:96) z . Alternatively, there is the canonicalisomorphism z ⊥ /(cid:96) z ∼ = → (cid:96) z ⊗ T [ z ] Q , [ w ] (cid:55)→ ρz ⊗ ( π ∗ ) ρz ( w ) = z ⊗ ( π ∗ ) z ( w ) , or equivalently, (cid:96) ∗ z ⊗ ( z ⊥ /(cid:96) z ) ∼ = T [ z ] Q . We have a third description in terms of the local coordinates ( r, s, t ) in the chart ψ . Taking c = (cid:18) r ss t (cid:19) in(3.8), the invariant conformal structure [ µ ] has representative µ = dr (cid:12) dt − ds (cid:12) ds = drdt − ds . Each vector field Z in Table 1 preserves [ µ ] under Lie derivation, i.e. L Z µ = λ ( Z ) µ for some scalar function λ ( Z ). Given a tangentvector v = v ∂∂r + v ∂∂s + v ∂∂t , µ ( v, v ) = v v − v and this is positive / zero / negative iff v lies inside [on, outside]the null cone. This is well-defined on the conformal class [ µ ]. Definition 3.4.
Let M ⊂ LG (2 , be a surface. We say M is:1. elliptic (spacelike) if for any x ∈ M , T x M ∩ C x is a single point; equivalently, N x M is timelike.2. parabolic (null) if for any x ∈ M , T x M ∩ C x consists of a single line; equivalently, N x M is null.3. hyperbolic (timelike) if for any x ∈ M , T x M ∩ C x consists of two distinct lines; equivalently, N x M is spacelike. In the chart ψ , suppose M has equation F ( r, s, t ) = 0. On M , 0 = dF = F r dr + F s ds + F t dt . Assuming( F r , F s , F t ) (cid:54) = 0 on M , then T x M for any x ∈ M is spanned by the (linearly dependent) vectors F r ∂ s − F s ∂ r , F t ∂ r − F r ∂ t , F t ∂ s − F s ∂ t . The µ -orthogonal complement of T x M yields the normal space N x M spanned by n = F t ∂∂r − F s ∂∂s + F r ∂∂t and we have µ ( n, n ) = F r F t − F s , whose sign is well-defined on [ µ ]. Hence, M is elliptic /parabolic / hyperbolic iff at each point of M , F r F t − F s is positive / zero / negative. Example 3.5.
Let c (cid:54) = 0 be a constant. The surface rt = c is elliptic if c > and hyperbolic if c < . Using therescaling ( r, s, t ) → ( λr, λs, λt ) which is a CSp (4 , R ) transformation, all are equivalent to rt = 1 or rt = − . The canonical cone field C on LG (2 ,
4) transfers to each fibre J | ξ in the PDE setting. Alternatively, the sign of µ ( n, n ) is a discrete CSp (4 , R )-invariant on M , which by Theorem 2.7 yields a contact-invariant for PDE. Surprisingly,even the following definition has not appeared in the literature. Definition 3.6.
Σ = { F = 0 } ⊂ J is elliptic / parabolic / hyperbolic at ˜ ξ ∈ J if for ξ = π ( ˜ ξ ) , T ˜ ξ (Σ | ξ ) intersectsthe canonical cone C ˜ ξ ⊂ T ˜ ξ ( J | ξ ) in one point / along one line / along two distinct lines. Equivalently, the normalline to T ˜ ξ (Σ | ξ ) ⊂ T ˜ ξ ( J | ξ ) lies inside / on / outside C ˜ ξ . LG perspective, “elliptic / parabolic / hyperbolic” is really a first order consequence of theexistence of the cone field C , which itself is a manifestation of the special isomorphism Sp (4 , R ) ∼ = Spin (2 , J coordinates ( x, y, z, p, q, r, s, t ), the sign of F r F t − F s evaluated pointwise on Σ determines the PDEtype. We recover the usual invariant distinguishing elliptic, parabolic, hyperbolic PDE [Gar67]. LG (2 , Using (cid:104)· , ·(cid:105) , there is a bijective (polar) correspondence between points and hyperplanes in P V . Definition 3.7.
For any [ z ] ∈ P V , we refer to S [ z ] = P ( z ⊥ ) ∩ Q as a sphere . Remark 3.8.
We caution the reader that spheres as we have defined above are topologically different from the usualspheres in Euclidean geometry. Rather, a sphere above is the intersection of a hyperplane with Q . This terminologyis borrowed from classical conformal geometry in definite signature. Orthogonality characterizes incidence with S [ z ] : if [ w ] ∈ Q , then [ w ] ∈ S [ z ] iff (cid:104) w, z (cid:105) = 0. Hence, spheres are givenby linear equations. Let H + = { [ z ] ∈ P V : (cid:104) z, z (cid:105) > } and H − = { [ z ] ∈ P V : (cid:104) z, z (cid:105) < } . Then P V = H + ˙ ∪ H − ˙ ∪ Q .The spheres determined by elements in H + , H − , Q are referred to as definite, indefinite, degenerate respectively.Note that any [ z ] ∈ Q plays a dual role as: (i) a point in Q , and (ii) a sphere in Q with vertex (singularity) at [ z ]. Lemma 3.9. Sp (4 , R ) acts transitively on each of H + , H − , Q .2. Let (cid:104) z, z (cid:105) < . The stabilizer in Sp (4 , R ) of [ z ] acts transitively on S [ z ] .Proof. Exercise for the reader.Using Lemma 3.9, and by examining points in H + , H − , Q , the corresponding spheres are topologically RP ,( S × S ) / Z , and a pinched torus respectively. However, this will not play an essential role in the sequel. Morerelevant for us is the local picture. Let us describe general spheres in the chart ψ . With respect to the basis B inSection 3.2, fix z = ( z , z , z , z , z ) ∈ V and note (cid:104) z, z (cid:105) = 2( z z − z − z z ). Then ψ ( S [ z ] ) = { ( r, s, t ) : − z + rz − sz + tz − z ( rt − s ) = 0 } . (3.9)If z = 0, then ψ ( S [ z ] ) is a plane, while if z (cid:54) = 0, then ψ ( S [ z ] ) = (cid:26) ( r, s, t ) : (cid:16) r − z z (cid:17) (cid:16) t − z z (cid:17) − (cid:16) s − z z (cid:17) = (cid:104) z,z (cid:105) z (cid:27) is a hyperboloid of 2-sheets, a hyperboloid of 1-sheet, or a cone iff (cid:104) z, z (cid:105) is positive, negative, or zero, respectively. Proposition 3.10.
Let [ z ] ∈ H − . The indefinite sphere S [ z ] is doubly-ruled by null geodesics. roof. For any [ w ] ∈ S [ z ] , (cid:98) T [ w ] S [ z ] = w ⊥ ∩ z ⊥ has signature (+ −
0) since (cid:104) z, z (cid:105) < (cid:104)· , ·(cid:105) has signature (2 , { u , u , w } of (cid:98) T [ w ] S [ z ] . Let u be either u or u . Then u correponds to a nulldirection in the tangent space T [ w ] S [ z ] . Also, (cid:104) u, u (cid:105) = (cid:104) u, w (cid:105) = (cid:104) u, z (cid:105) = 0. For any a, b ∈ R (not both zero), (cid:104) au + bw, au + bw (cid:105) = 0, so [ au + bw ] ∈ Q and (cid:104) au + bw, z (cid:105) = 0, so [ au + bw ] is a null projective line contained in S [ z ] ,which is a null line in the chart ψ . Since the conformal structure [ µ ] is flat, null lines are null geodesics.For our later study of hyperbolic surfaces in LG (2 , B H : e ∧ e ,e ∧ e ,e ∧ e ,e ∧ e − e ∧ e ,e ∧ e ⇒ (cid:104)· , ·(cid:105) B H = −
10 0 1 0 00 1 0 0 00 0 0 − − , (3.10)which is B (3.5) with third and fourth basis vectors interchanged. Any basis v = ( v , ..., v ) of V with scalar products (cid:104) v i , v j (cid:105) given by the matrix in (3.10) will be called a hyperbolic frame . Any such frame has a natural geometricinterpretation as a 5-tuple of spheres: ([ v ] , [ v ] , [ v ] , [ v ]) forms a null diamond inscribed in the indefinite sphere S [ v ] . The picture given in Figure 4 is a sample configuration, but should not be misleading: e.g. one or more of thepoints [ v i ], i = 1 , , ,
4, may lie at on the sphere at infinity. (If [ v ] is, then the hyperboloid degenerates to a plane.)Figure 4: Geometric interpretation of a hyperbolic frameLet P x ∈ LG (2 ,
4) correspond to [ x ] ∈ Q ⊂ P V under the Pl¨ucker map. The following is obvious: Lemma 3.11.
Let [ x ] , [ y ] ∈ Q . Then (cid:104) x, y (cid:105) (cid:54) = 0 iff R = P x ⊕ P y is a Lagrangian decomposition. Thus, if [ x ] (cid:54) = [ y ] and (cid:104) x, y (cid:105) = 0 , then dim( P x ∩ P y ) = 1 . Thus, given a basis v of V as above, we can interpret it in the symplectic setting. Corresponding to v , we have • we have a pair of Lagrangian decompositions R = P [ v ] ⊕ P [ v ] and R = P [ v ] ⊕ P [ v ] . • P [ v ] = ( P [ v ] ∩ P [ v ] ) ⊕ ( P [ v ] ∩ P [ v ] ) and P [ v ] = ( P [ v ] ∩ P [ v ] ) ⊕ ( P [ v ] ∩ P [ v ] ).Finally, inversion with respect to an indefinite or definite sphere S [ z ] is defined in a natural manner: Let [ w ] ∈ Q .Its inversion [ ˜ w ] with respect to S [ z ] is ˜ w = w − (cid:104) w,z (cid:105)(cid:104) z,z (cid:105) z . If [ ˜ w ] = [ w ], we must have ˜ w = w and (cid:104) w, z (cid:105) = 0, i.e.[ w ] ∈ S [ z ] . Inversion with respect to a degenerate sphere is undefined.13 .6 The Maurer–Cartan form Recall the effective symmetry group of Q = LG (2 ,
4) is O + (2 , { e i } i =1 on R and the hyperbolic frame B H on V , c.f. (3.10), the map φ = (Φ ∗ ) e arising from Φ : (cid:99) Sp (4 , R ) → O + (2 ,
3) is φ a a b b a a b b c c − a − a c c − a − a = a + a b b b c a − a − a b c a − a − a b c − a − a − b c c − c − ( a + a ) , Denoting the components of the (left-invariant) MC form ω = g − dg on O + (2 ,
3) as ω = α + α β β β θ α − α − α β θ α − α − α β θ − α − α − β θ θ − θ − ( α + α ) , (3.11)the MC structure equations dω + ω ∧ ω = 0 are0 = dα + α ∧ α + β ∧ θ + β ∧ θ , dα + ( α − α ) ∧ α + β ∧ θ + β ∧ θ , dα + α ∧ α + β ∧ θ + β ∧ θ , dα + α ∧ ( α − α ) + β ∧ θ + β ∧ θ , dβ + 2 α ∧ β + 2 α ∧ β , dθ + 2 θ ∧ α + 2 θ ∧ α , dβ + 2 α ∧ β + 2 α ∧ β , dθ + 2 θ ∧ α + 2 θ ∧ α , dβ + ( α + α ) ∧ β + α ∧ β + α ∧ β , dθ + θ ∧ α + θ ∧ ( α + α ) + θ ∧ α . (3.12)With respect to O + (2 , → O + (2 , /P + = Q , the forms θ , θ , θ are semi-basic and ˜ µ = θ (cid:12) θ − θ (cid:12) θ has P + -invariant conformal class [˜ µ ]. Hence, [˜ µ ] is the pullback by the projection of a conformal class [ µ ] on LG (2 , LG (2 , LG (2 , Identify O + (2 ,
3) with its orbit through a chosen basis v (0) of V . (Later, we will take v (0) = B H to study hyperbolicsurfaces.) For any v = ( v , v , v , v , v ) ∈ O + (2 , O + (2 , → Q , v (cid:55)→ [ v ]. Given any (embedded) surface i : M (cid:44) → Q ⊂ P V , the zeroth order frame bundle F ( M ) → M is the pullback of O + (2 , → Q by i , and the diagram F ( M ) ˜ i (cid:47) (cid:47) p (cid:15) (cid:15) O + (2 , (cid:15) (cid:15) M i (cid:47) (cid:47) Q commutes. A moving frame v for M is a local section of F ( M ) → M , i.e. we always have [ v ] ∈ M . Let ˜ v = ˜ i ◦ v . Remark 3.12.
Notationally, we do not distinguish a moving frame v from a particular frame v ∈ F ( M ) . Oneshould infer the former if derivatives, pullbacks, etc. are taken and the latter when defining adapted frame bundles. Since v = v (0) g for some (local) map g : M → O + (2 , d v = v (0) dg = v g − dg = v ω , or in components, d v j = ω ij v i , where the row / column index of ω are on top / bottomrespectively. Here, ω is the MC form of O + (2 ,
3) given in (3.11), or more precisely, its pullback to M by ˜ v . Followingthe usual practice with moving frames, we will not notationally distinguish the MC form with its pullback. Theintegrability conditions for ω are the MC structure equations dω + ω ∧ ω = 0 given in (3.12).14he structure group H of F ( M ) → M is P + = Φ( P ) ⊂ O + (2 , v , v (cid:48) satisfy v (cid:48) = v h for some (local) map h : M → H . The change of frame formula is given by˜ v (cid:48)∗ ω = h − (˜ v ∗ ω ) h + h − dh. (3.13)More generally, let G be a Lie group with Lie algebra g and Maurer–Cartan form ω G . The following theorems[Jen77] provide the theoretical basis for Cartan’s method of moving frames. Theorem 3.13.
Let f , f : M → G be two smooth maps of a connected manifold M into a Lie group G . Then f ( x ) = g · f ( x ) for all x ∈ M and for a fixed g ∈ G iff f ∗ ω G = f ∗ ω G . Theorem 3.14.
Let M be a smooth manifold, ψ ∈ Ω ( M, g ) a g -valued 1-form. Then for any x ∈ M , there existsan open neighbourhood U of x ∈ M and a function f : U → G such that f ∗ ω G = ψ iff ω satisfies dψ + ψ ∧ ψ = 0 . Our application of the method of moving frames will proceed by finding geometrically adapted sections of F ( M ) → M , or equivalently by composing with the map F ( M ) → O + (2 , M to G = O + (2 , F k ( M ) → M with structure group H k ⊂ H . We begin our study of hyperbolic surfaces in Q = LG (2 ,
4) via moving frames. While the main text of this articlecontains an account of our moving frames study in an abstract form, we encourage the reader to concurrently readAppendix A which contains parallel calculations in parametric form. For the remainder of this paper, i : U → Q isa smooth embedded hyperbolic surface and M = i ( U ). In defining F ( M ) as in Section 3.7, we choose the initialbasis v (0) = B H , c.f. (3.10). Thus, any v ∈ F ( M ) is a hyperbolic frame. Let [ v ] ∈ M . The affine tangent space (cid:98) T [ v ] M is the tangent space (translated to the origin) to the cone (cid:99) M ⊂ V over M at any (nonzero) point of (cid:96) v = span { v } . We have v ∈ (cid:98) T [ v ] M ⊂ v ⊥ = span { v , v , v , v } for any v ∈ F ( M ).By hyperbolicity of M , there is: (i) a complementary normal line to T [ v ] M in T [ v ] Q , and (ii) two distinguished nulllines in T [ v ] M . These data lift to (cid:98) T [ v ] M to give the flags of subspaces: (cid:96) v ⊂ n i | v ⊂ (cid:98) T [ v ] M ⊂ v ⊥ ⊂ V, dim( n i | v ) = 2 , n | v ∩ n | v = (cid:96) v (cid:96) v ⊂ N | v ⊂ v ⊥ ⊂ V, dim( N | v ) = 2 , N | v ∩ (cid:98) T [ v ] M = (cid:96) v . Define F ( M ) = { v ∈ F ( M ) : ( v , v ) ∈ ( n | v × n | v ) ∪ ( n | v × n | v ) , v ∈ N | v } and H = H (cid:48) (cid:111) ( Z × Z ), where H (cid:48) = X ( r , r , s , s , s ) = r s s s s s − s r r r s r r s r r − s r r : r r > , s i ∈ R and Z × Z has generators R = diag (cid:18) − , (cid:18) − − (cid:19) , − , − (cid:19) and R = diag (1 , − , − , − , v ∈ F ( M ), (cid:98) T [ v ] M = span { v , v , v } . Comparing with (3.11), any 1-adapted moving frame v has θ = 0 and { θ , θ } is a local coframing on T ∗ M . Explicitly, the 1-adapted moving frame structure equations d v j = ω ij v i are: d v = ( α + α ) v + θ v + θ v d v = β v + ( α − α ) v − α v + θ v d v = β v + ( α − α ) v − α v + θ v (4.1) d v = 2 β v − α v − α v d v = β v + β v − β v − ( α + α ) v The components of ω : (i) can be calculated in terms of the moving frame, e.g. θ = (cid:104) d v , v (cid:105) = −(cid:104) d v , v (cid:105) , and (ii)satisfy the MC equations (3.12). 15 .2 2-adaptation: central sphere congruence By the dθ MC equation (3.12), 0 = θ ∧ α + θ ∧ α . Hence, Cartan’s lemma implies there are second orderfunctions λ ij on M so that: α = λ θ + λ θ , α = λ θ + λ θ . (4.2)Since (cid:104) d v , v (cid:105) = 2 α = 2 λ θ , (cid:104) d v , v (cid:105) = 2 α = 2 λ θ , then λ ij are the components of the second fundamentalform II of ( M, µ ). However, the conformal change (cid:98) µ = e u µ , induces (cid:99) II = II − µ [grad µ ( u )] ⊥ , so II is not well-definedon the conformal class [ µ ]. Letting H = tr µ ( II ) be the mean curvature, the trace-frace second fundamental form trfr µ ( II ) = II − Hµ is well-defined on [ µ ], c.f. [HJ03]. Indeed, µ ij = (cid:18) (cid:19) are the components of (cid:104)· , ·(cid:105) withrespect to the basis { v , v } mod v , and trfr µ ( II ) has components a ij = λ ij − Hµ ij , where H = µ k(cid:96) λ k(cid:96) = λ (with µ ij the inverse of µ ij ). Thus, µ ij a ij = 0 and a ij only has diagonal components diag ( λ , λ ). The eigenvaluesof a ik = a ij µ jk are the roots of 0 = det ( a ij − sµ ij ) = λ λ − s . Definition 4.1.
Let v be any 1-adapted moving frame on M . Let v c = v + 2 λ v .1. The map γ : M → H − , p (cid:55)→ [ v c | p ] is called the conformal Gauss map .2. Pointwise, S [ v c ] = P ( v ⊥ c ) ∩ Q is the central tangent sphere (CTS). The collection of CTS over M , i.e. the imageof γ , is called the central sphere congruence (CSG) on M .3. The 2-adapted frame bundle of M is F ( M ) = { v ∈ F ( M ) : [ v ] = [ v c ] } with structure group H = H (cid:48) (cid:111) ( Z × Z ) with H (cid:48) = { X ( r , r , s , s , ∈ H (cid:48) } . ¯ α ¯ α ¯ θ ¯ θ ¯ λ ¯ λ ¯ λ ¯ v c H (cid:48) r ( α − s θ ) r ( α − s θ ) r r θ r r θ λ − s r r r λ
11 1 r r λ v c R α α θ θ λ λ λ − v c R α α − θ − θ − λ − λ − λ − v c (4.3)From (4.3), S [ v c ] is indeed H -invariant, i.e. the CSG is well-defined, and moreover λ , λ are relative invariants.Since (cid:104) v , v (cid:105) = −
2, then if v is 2-adapted, we must have v = ± v c and λ = 0. Each S [ v ] is incident with M at[ v ] and since (cid:104) d v , v (cid:105) = 0, then T [ v ] M is also tangent to S [ v ] . Equivalently, M is an envelope for its CSG.Define ( D v )( w , w ) = w d ( w d v ), where w , w are vector fields on M . We can compute trfr µ ( II ) via (cid:104) D v ( w , w ) , v (cid:105) = (cid:104) w (( α + α )( w ) d v + θ ( w ) d v + θ ( w ) d v ) , v (cid:105) = 2( α ( w ) θ ( w ) + α ( w ) θ ( w ))= 2( λ θ ( w ) θ ( w ) + λ θ ( w ) θ ( w )) (4.4)Thus, (cid:104) D v ( · , · ) , v (cid:105) is the symmetric tensor 2( λ θ + λ θ ). Since v may be rescaled and v can change sign,then (cid:104) D v ( · , · ) , v (cid:105) is only well-defined up to scale. A similar calculation as above yields (cid:104) D v ( · , · ) , v + 2 s v (cid:105) = 2( λ θ + λ θ − s ( θ ⊗ θ + θ ⊗ θ )) = 2( θ , θ ) (cid:18) λ − s − s λ (cid:19) (cid:18) θ θ (cid:19) (4.5)The directions along which (4.5) vanishes determine directions of second-order tangency of C n with M . Definition 4.2.
Asymptotic directions are those tangent vectors w for which (cid:104) D v ( w, w ) , v (cid:105) = 0 . Principaldirections are eigendirections of a ij = µ ik a kj . Integral curves of principal directions are called curvature lines . Definition 4.3.
Define (cid:15) = sgn( λ λ ) . M is if it is 2-hyperbolic ( (cid:15) = − or 2-elliptic ( (cid:15) = 1) . Along asymptotic directions, S [ v c ] has second order contact with M . Let us observe that from (4.1), we have θ = 0 ⇒ d v ≡ , d v ≡ − α v = − λ θ v mod { v , v } θ = 0 ⇒ d v ≡ , d v ≡ − α v = − λ θ v mod { v , v } Recalling that λ , λ are relative invariants, then by (4.4), we have a geometric interpretation for λ , λ :16erminology Invariant characterization Geometric interpretation2-isotropic λ = λ = 0 M is doubly-ruled by null geodesics2-parabolic Exactly one of λ or λ is zero M is singly-ruled by a null geodesic2-hyperbolic λ λ < M generic; carries an asymptotic net;no curvature lines2-elliptic λ λ > M generic; no asymptotic directions;carries a net of curvature linesTable 2: Second order CSp (4 , R )-invariant classification of (un)parametrized hyperbolic surfaces • λ = 0 iff any curve satisfying θ = θ = 0 is a null geodesic • λ = 0 iff any curve satisfying θ = θ = 0 is a null geodesicA priori, Table 2 is a classification of parametrized surfaces, but we show in Appendix A.2 that it is also aclassification of unparametrized surfaces. If M is 2-isotropic, any 2-adapted moving frame satisfies α = α = 0.The dα , dα equations reduce to 0 = β ∧ θ = β ∧ θ , which implies β = 0 and hence by (4.1), d v = 0. Thus,the CTS is constant for all of M . Since (cid:104) v , v (cid:105) = 0, then [ v ] ∈ S [ v ] , i.e. M is an open subset of S [ v ] . Proposition 4.4.
A hyperbolic surface M ⊂ Q is 2-isotropic iff it is an open subset of an indefinite sphere.Proof. The first direction has been established. For the converse, work in the chart ψ . M = S [0 , , , , has localparametrization [1 , u, , v, uv ]. The moving frame v = (1 , u, , v, uv ), v = (0 , , , , v = (0 , , , , v =(0 , , , , v = (0 , , , ,
1) is 2-adapted with structure equations d v = θ v + θ v and d v i = 0 for i = 1 , , , θ = du , θ = dv , so λ = λ = 0. Using Lemma 3.9, any indefinite sphere is 2-isotropic.Since Sp (4 , R ) acts transitively on H − , then there are no invariants for such surfaces. Indeed, any such is a CSIsurface with s = 0 having 6-dimensional symmetry algebra (as a subalgebra of sp (4 , R )): ∂ r , ∂ t , − r ∂ r − rs∂ s − s ∂ t , − s ∂ r − st∂ s − t ∂ t , − r∂ r − s∂ s , − s∂ s − t∂ t . In (3.9), spheres were given in local coordinates ( r, s, t ). Fibrewise a MA PDE is a sphere.
Theorem 4.5.
The class of hyperbolic MA PDE is contact-invariant.Proof.
Being hyperbolic and 2-isotropic are
CSp (4 , R )-invariant conditions. By Theorem 2.7, we are done. (Thisargument is equivalent to that given in the Introduction.)Thus, we have reproven a classical fact from a completely different (and arguably simpler) point of view. Contact-invariance of the elliptic and parabolic MA PDE classes are similarly established. Remark 4.6.
Although indefinite spheres admit no
CSp (4 , R ) -invariants, hyperbolic MA PDE certainly do admitcontact invariants, e.g. the wave ( z xy = 0) and Liouville equation ( z xy = exp( z )) are contact-inequivalent [GK93]. A parametric description of the relative invariants λ , λ appearing in Table 2 is given in Appendix A. Computationsare made in local coordinates ( r, s, t ) of the chart ψ and with respect to a null parametrization ( u, v ) on a hyperbolicsurface M with respect to [ µ ], where µ = dr (cid:12) dt − ds (cid:12) ds . The null parameter condition is r u t u − s u = r v t v − s v = 0.The functions λ , λ are multiples of the Monge–Amp`ere (relative) invariants I , I , given by I = det r uu s uu t uu r u s u t u r v s v t v , I = det r vv s vv t vv r u s u t u r v s v t v . (4.6)17 second description of the MA invariants appears in Appendix A.3. If M is given implicitly by F ( r, s, t ) = 0 andnull parameters ( u, v ) on M , we have I = r u s u t u T F rr F rs F rt F sr F ss F st F tr F ts F tt r u s u t u , I = r v s v t v T F rr F rs F rt F sr F ss F st F tr F ts F tt r v s v t v . (4.7) Remark 4.7.
Equations (4.6) and (4.7) differ by an overall scaling, but this does not affect the classification result.
Proposition 4.8.
Let M ⊂ Q be given by F ( r, s, t ) = 0 . The following hold:1. F ( r, s ) = 0 is hyperbolic iff F s (cid:54) = 0 . If so, then it is 2-isotropic or 2-parabolic.2. F ( r, t ) = 0 is hyperbolic iff F r F t < . If so, then it is 2-isotropic or 2-elliptic.Proof. From Section 3.4, M is hyperbolic iff F r F t − F s <
0. We calculate I , I and use Table 2 to classify M .Equation Local null basis of T M
Hess ( F ) Invariants F ( r, s ) = 0 n = ∂∂t n = F r ∂∂s − F s ∂∂r F rr F rs F sr F ss
00 0 0 I = 0 I = F rr F s − F rs F r F s + F ss F r F ( r, t ) = 0 n = F r ∂∂t − F t ∂∂r + √− F r F t ∂∂s n = F r ∂∂t − F t ∂∂r − √− F r F t ∂∂s F rr F rt F tr F tt I = I = F rr F t − F rt F r F t + F tt F r In the case F ( r, t ) = 0 is 2-generic, we have I = I (cid:54) = 0, so (cid:15) = sgn( I I ) = sgn(( I ) ) = +1, so it is 2-elliptic.The MA invariants I , I are necessarily also relative contact-invariants for hyperbolic PDE, where we mustinterpret the null parameters u, v as functions of ξ = ( x, y, z, p, q ) ∈ J , i.e. dependent on the fibre J | ξ . As discussedin the Introduction, the MA invariants for hyperbolic PDE have been calculated several times in the literature byVranceanu [Vra40], Jur´aˇs [Jur97], and The [The08]. The calculations in these papers were quite involved. Ourcomputation above of I , I based on a 2-adapted lift for a hyperbolic surface M ⊂ LG (2 ,
4) is geometrically simple.Let us compare I , I above with the MA invariants calculated in [The08], denoted here I ( T )1 , I ( T )2 : I ( T )1 = det F r F s F t λ + F t (cid:16) F t λ + (cid:17) r (cid:16) F t λ + (cid:17) s (cid:16) F t λ + (cid:17) t , I ( T )2 = det F r λ + F r F s F t (cid:16) F r λ + (cid:17) r (cid:16) F r λ + (cid:17) s (cid:16) F r λ + (cid:17) t where λ + = F s + √− ∆, where ∆ = F r F t − F s <
0, and (without loss of generality) it is assumed that F s ≥ ξ . Evaluated on Σ = { F = 0 } , hyperbolic PDE are classified by: MA ( I ( T )1 = I ( T )2 = 0), Goursat( I ( T )1 = 0 or I ( T )2 = 0, but not both), generic ( (cid:15) ( T ) = sgn( I ( T )1 I ( T )2 ) = ± I ( T )1 , I ( T )2 depend onlyon second derivatives in ( r, s, t ). Hence, fibrewise, for any ξ ∈ J , they must be CSp ( C ξ , [ η ])-invariants of surfacesΣ | ξ ⊂ J | ξ and must be a function of the second order CSp (4 , R )-invariants I , I for surfaces in LG (2 , I ( T )1 = I ( T )2 = 0 and I = I = 0. The PDE z xy = ( z yy ) is in theGoursat class, while s = t regarded as a surface in LG (2 ,
4) has I (cid:54) = 0, I = 0. The PDE 3 z xx ( z yy ) + 1 = 0 is inthe generic hyperbolic class with (cid:15) ( T ) = 1; the surface 3 rt + 1 = 0 in LG (2 ,
4) is generic, and from Proposition 4.8,it is 2-elliptic. Thus, both invariant descriptions coincide.
Theorem 4.9.
A hyperbolic PDE Σ ⊂ J is: Monge–Amp`ereGoursat2-elliptic2-hyperbolic iff for any ξ ∈ J , Σ | ξ ⊂ J | ξ is an indefinite spheresingly-ruled by null geodesics2-elliptic ( (cid:15) = +1) ( (cid:15) = − and these PDE classes are contact-invariant and mutually contact-inequivalent. Remark 4.10.
The statement for hyperbolic 2-elliptic / 2-hyperbolic in the above theorem is a definition. .4 3-adaptation: cone congruences and the conjugate manifold In the generic and singly-ruled cases, there are additional geometric objects canonically associated with a third orderneighbourhood of M . Suppose that v is a 2-adapted moving frame, so λ = 0. The dα , dα equations yield0 = ( dλ + λ (3 α − α )) ∧ θ + β ∧ θ , 0 = ( dλ + λ (3 α − α )) ∧ θ + β ∧ θ , and hence by Cartan’slemma, there exist third order functions λ ijk on M such that dλ + λ (3 α − α ) = λ θ + λ θ , (4.8) β = λ θ + λ θ , (4.9) dλ + λ (3 α − α ) = λ θ + λ θ (4.10)On a generic surface M (so λ λ (cid:54) = 0), define v n = v − λ λ v , v n = v − λ λ v , v n (cid:48) = v + λ λ v , v n (cid:48) = v + λ λ v . (4.11)¯ α + ¯ α ¯ α − ¯ α ¯ β H (cid:48) α + α − r s θ − s r θ + dr r α − α − r s θ + s r θ + dr r r β + s r r α + r s r α R α + α − ( α − α ) β R α + α α − α − β (4.12)¯ λ ¯ λ ¯ λ ¯ λ ¯ v n ¯ v n ¯ v n (cid:48) ¯ v n (cid:48) H (cid:48) r ( r λ − s λ ) r r ( λ + s r λ ) r r λ + s λ r r λ − r s λ r r r v n r v n r v n (cid:48) r v n (cid:48) R λ λ λ λ − v n − v n − v n (cid:48) − v n (cid:48) R λ λ λ λ − v n − v n − v n (cid:48) − v n (cid:48) (4.13)The two cone pairs {S [ v n ] , S [ v n ] } and {S [ v n (cid:48) ] , S [ v n (cid:48) ] } are geometrically associated to M . The former normalizes¯ λ = ¯ λ = 0 and β = 0, while the latter normalizes ¯ λ = ¯ λ = 0, c.f. (4.9). We choose the former to defineour 3-adaptation, c.f. Remark 4.14. Definition 4.11.
Let M be a hyperbolic 2-generic. The 3-adapted frame bundle and its structure group are F ( M ) = { v ∈ F ( M ) : { [ v ] , [ v ] } = { [ v n ] , [ v n ] }} , H = H (cid:48) (cid:111) ( Z × Z ) , where H (cid:48) = (cid:110) X ( r , r ) := diag (cid:16) r , r , r , , r (cid:17) : r r > (cid:111) , and R , R generate the Z × Z factor. If M is singly-ruled, then using R , we may assume λ (cid:54) = 0 and λ = 0 (hence, λ = λ = 0). In this case, v n and v n (cid:48) are not well-defined. We refer to S [ v n ] as the primary cone congruence and S [ v n (cid:48) ] the secondary conecongruence . The change of frame ( r , r , s , s ) = (cid:16) , , λ λ , − λ λ (cid:17) normalizes β = 0 and λ = λ = 0. Hence, dλ + λ (3 α − α ) = 0, i.e. 3 α − α = − d (ln( λ )) is exact. Definition 4.12.
Let M be hyperbolic singly-ruled. The 3-adapted frame bundle and its structure group are ¯ F ( M ) = { v ∈ F ( M ) : [ v ] = [ v n (cid:48) ] , [ v ] = [ v n ] } , ¯ H = H (cid:48) (cid:111) Z , where H (cid:48) is as in Definition 4.11, and R generates the Z factor. In both the singly-ruled and generic cases, the residual structure groups ¯ H and H preserve [ v ]. Hence, for any3-adapted frame, the null diamond ([ v ] , [ v ] , [ v ] , [ v ]) inscribed on S [ v ] is geometrically associated to M . Definition 4.13.
For any 3-adapted frame v , we call [ v ] ∈ Q the conjugate point . For a 3-adapted moving frame v , we call the image of [ v ] (regarded as a map M → Q ) the conjugate manifold M (cid:48) ⊂ Q of M . Given a 3-adapted v for M , we have M (cid:48) tangent to S [ v ] at [ v ] since by (4.1), (cid:104) d v , v (cid:105) = 2 β = 0. Thus, if M (cid:48) is also a surface, it is a second envelope for CSG of M . 19 emark 4.14. The requirement β = 0 for the second envelope motivates our choice of defining the 3-adaptationvia {S [ v n ] , S [ v n ] } instead of {S [ v n (cid:48) ] , S [ v n (cid:48) ] } . We later express dim( M (cid:48) ) in terms of the invariants of M . We immediately caution the reader on several points.In general: (i) M (cid:48) may have singularities, (ii) the CTS of M , M (cid:48) may not agree pointwise, (iii) ( M (cid:48) ) (cid:48) (cid:54) = M , (iv) M (cid:48) may not have the same type as M (even if dim( M (cid:48) ) = 2).Thus far, we can canonically assign to any singly-ruled or generic surface M a geometric moving system of spheres S [ v ] , ..., S [ v ] . The residual scaling freedom in individual frame vectors represented in the 3-adapted structure groups¯ H or H will subsequently be reduced by normalizing coefficients in the MC structure equations. After reducing asmuch as possible, the residual structure functions will be candidates for invariants for our equivalence problem, butwe must still investigate their transformation under ¯ H or H . For a hyperbolic singly-ruled surface M , we constructed in Definition 4.12 the 3-adapted frame bundle ¯ F ( M )with structure group ¯ H . Any 3-adapted moving frame satisfies λ (cid:54) = 0, λ = 0, β = α = 0, α = λ θ ,3 α − α = − d (ln | λ | ), hence 3 dα = dα . The integral curves of θ = θ = 0 are null geodesics. The dα , dα , dβ equations (3.12) imply 3 β ∧ θ = − dα = − dα = β ∧ θ and α ∧ β = 0. By Cartan’slemma, there exist fourth order functions Λ , Λ such that β = Λ θ − Λ θ , β = Λ θ . The dβ , dβ equations (3.12) yield 0 = − d Λ + 4Λ α ) ∧ θ + ( d Λ + 2Λ ( α + α )) ∧ θ and 0 = ( d Λ +2Λ ( α + α )) ∧ θ . By Cartan’s lemma, there exist fifth order functions Λ , Λ such that d Λ + 4Λ α = Λ θ − Λ θ , d Λ + 2Λ ( α + α ) = Λ θ . (5.1)The MC equations (3.12) for 3-adapted moving frames reduce to0 = 3 dα + Λ θ ∧ θ , dα + Λ θ ∧ θ , dθ + 2 θ ∧ α , dθ + 2 θ ∧ α . (5.2)Under an ¯ H -frame change, α , α , θ , θ transform according to (4.12)-(4.13) (with s = s = 0). We also have:¯ β ¯ β ¯ λ ¯Λ ¯Λ ¯Λ ¯Λ X ( r , r ) ∈ H (cid:48) r r β r r β r r λ r r Λ
11 1 r Λ r r Λ r r Λ R − β − β − λ Λ Λ − Λ − Λ (5.3) Definition 5.1.
Let δ = sgn(Λ ) ∈ {− , , } and δ = sgn(Λ ) ∈ {− , , } . From (5.3), δ , δ are CSp (4 , R )-invariants and we assume they are locally constant. Setting r = λ ( r ) , wenormalize ¯ λ = 1, hence 3¯ α = ¯ α and ¯ α = ¯ θ . • δ (cid:54) = 0: Normalize ¯Λ = δ by setting r = sgn( λ ) (cid:112) | Λ | , r = sgn( λ ) | Λ | / | λ | − / . The structuregroup is reduced to the identity. The residual functions are ζ = Λ | λ || Λ | / and ζ = δ Λ | λ | / | Λ | / . Under¯ H , ζ and ( ζ ) are invariant so these are the fundamental invariants. • δ = 0, δ = ±
1: Normalize ¯Λ = δ by setting r = √ | Λ | λ , r = | Λ | λ . The structure group is reduced to theidentity. The residual function is ζ = δ Λ | Λ | / , whose square is ¯ H -invariant. The fundamental invariant is ζ . • δ = δ = 0: The residual group ¯ H r = { diag ( r , r, r , , r ) : r > } induces ¯ α = α + dr r by (3.13). Since dα = 0 by (5.2), then α is locally exact by Poincar´e’s lemma, and we normalize ¯ α = 0. There are noresidual invariant functions. There is a residual change of frame, v (cid:48) = v h for h : U → ¯ H r locally constant.20lassification Invariants Normalized moving frame structure equations Integrability conditionsdim( M (cid:48) ) = 2 : δ = ± ζ , ( ζ ) d v = 2 ζ θ v + θ v + θ v d v = ( ζ θ − δ θ ) v + ζ θ v − θ v + θ v d v = δ θ v − ζ θ v + θ v d v = − θ v d v = δ θ v + ( ζ θ − δ θ ) v − ζ θ v dθ dθ − ζ θ ∧ θ dζ − δ θ ) ∧ θ dim( M (cid:48) ) = 1 : δ = 0 ,δ = ± ζ d v = 2 ζθ v + θ v + θ v d v = δ θ v + ζθ v − θ v + θ v d v = − ζθ v + θ v d v = − θ v d v = δ θ v − ζθ v dθ dθ − ζθ ∧ θ dζ ∧ θ dim( M (cid:48) ) = 0 : δ = δ = 0 − d v = θ v + θ v d v = − θ v + θ v d v = θ v d v = − θ v d v = 0 0 = dθ = dθ Table 3: Invariant sub-classes and structure equations for surfaces singly-ruled by null geodesicsDropping bars, we summarize the results in Table 3. By Theorem 3.13, our solution to the parametrized equiva-lence problem for hyperbolic singly-ruled surfaces is:
Theorem 5.2 (Invariants and equivalence for hyperbolic singly-ruled surfaces) . Let U be connected and i, ˜ i : U → LG (2 , hyperbolic singly-ruled surfaces. If M = i ( U ) and ˜ M = ˜ i ( U ) are CSp (4 , R ) -equivalent, then δ , δ agree for M, ˜ M on U , and1. δ (cid:54) = 0 : ζ and ( ζ ) agree for M, ˜ M on U .2. δ = 0 , δ (cid:54) = 0 : ζ agrees for M, ˜ M on U .3. δ = δ = 0 : no additional conditions.Conversely, if the above hold, then M, ˜ M are CSp (4 , R ) -equivalent. In Appendix A.5, we show that ζ , ( ζ ) , ζ are invariant under reparametrizations. Hence, these solve thecorresponding unparametrized equivalence problem. By Theorem 3.14, we have: Theorem 5.3 (Bonnet theorem for hyperbolic singly-ruled surfaces) . The integrability conditions in Table 3 are theonly local obstructions to the existence of a hyperbolic singly-ruled surface with prescribed invariants.
The δ , δ invariants for M have an interpretation in terms of the conjugate manifold M (cid:48) . Let v be a 3-adaptedmoving frame. From the v equation (4.1), T [ v ] M (cid:48) ⊂ T [ v ] Q ∼ = v ⊥ /(cid:96) v is spanned by the coefficients of θ and θ , i.e.Λ v + Λ v and Λ v mod v . (5.4) Proposition 5.4.
Let M be a singly-ruled hyperbolic surface. Then dim( M (cid:48) ) = rank (cid:18) Λ Λ (cid:19) . We have dim( M (cid:48) ) = 0 iff δ = δ = 0. If dim( M (cid:48) ) = 1, then δ = 0 and δ (cid:54) = 0. From Table 3, we see d v , d v ≡ { v , v } . Hence, the null curve M (cid:48) is in fact a null geodesic. Suppose dim( M (cid:48) ) = 2, i.e. δ (cid:54) = 0. The conformalstructure on M (cid:48) is represented with respect to the basis (5.4) by (multiples of) the matrix (cid:18) Λ Λ Λ (cid:19) , so21 (cid:48) is hyperbolic. Let us further classify M (cid:48) . Given a 3-adapted v on M , ˜ v = (˜ v , ˜ v , ˜ v , ˜ v , ˜ v ) = ( v , v , v , − v , v )is 2-adapted on M (cid:48) , so the CSGs of M , M (cid:48) agree, with˜ θ = β , ˜ θ = β , ˜ β = θ , ˜ β = θ , ˜ θ = 0 , ˜ β = 0 , ˜ α = − α , ˜ α = − α , ˜ α = − α , ˜ α = 0 , ˜ λ = − λ Λ , ˜ λ = λ Λ Λ . Hence, M (cid:48) is singly-ruled. In general, ˜ v is not 3-adapted unless ˜ λ = 0. By the discussion preceding Definition4.12, we consider v (cid:48) = ( v , v + s v , v , − v , v + s v ), where s = ˜ λ λ = − Λ ) . This is 3-adapted for M (cid:48) , and β (cid:48) = ˜ β = θ = 1Λ β = 1Λ ˜ θ = 1Λ θ (cid:48) ⇒ Λ (cid:48) = 1Λ ⇒ δ (cid:48) = δ ⇒ dim(( M (cid:48) ) (cid:48) ) = 2 . Note S [ v ] , S [ v (cid:48) ] always agree. As observed earlier, ζ is nonconstant, so Λ and s cannot vanish identically. Thus,in general S [ v ] (cid:54) = S [ v (cid:48) ] and ( M (cid:48) ) (cid:48) (cid:54) = M . The functions Λ , Λ , Λ , Λ are calculated in Appendix A.5 and it is shown that δ , δ , ζ , ( ζ ) , ζ are invariantsof unparametrized surfaces. If M is hyperbolic singly-ruled CSI, then dim( M (cid:48) ) = 0 or dim( M (cid:48) ) = 1 with ζ constant. Proposition 5.5.
Let M be a hyperbolic singly-ruled surface such that dim( M (cid:48) ) = 2 . Then M is not homogeneous.Proof. From Table 3, the integrability condition (3 dζ − δ θ ) ∧ θ = 0 implies ζ cannot be constant. By Theorem2.9, all such surfaces are not homogeneous. Example 5.6.
Consider ( r, s, t ) = ( uu + v − ln( u ) , u u + v − u, u u + v − u ) . This is a null parametrization with I = u (3 v − u )4( u + v ) , I = 0 , ν = u ( u + v ) , Λ = − v − u ) . Thus, δ = − and dim( M (cid:48) ) = 2 . Proposition 5.7.
All hyperbolic singly-ruled surfaces M of the form F ( s, t ) = 0 have dim( M (cid:48) ) ≤ .Proof. From Proposition 4.8, M is hyperbolic if F s (cid:54) = 0. By the implicit function theorem, we may assume that F isgiven locally by F ( s, t ) = s − f ( t ). This has null parametrization ( r, s, t ) = ( g ( u ) + v, f ( u ) , u ), where g (cid:48) ( u ) = f (cid:48) ( u ) .We compute ν = 1, I = f (cid:48)(cid:48) ( u ) (cid:54) = 0, I = I = 0, hence from (A.13), Λ = 0, so δ = 0 and dim( M (cid:48) ) ≤ M of the form s = f ( t ) with f (cid:48)(cid:48) ( t ) (cid:54) = 0 satisfy:Λ = 13 (cid:18) f (cid:48)(cid:48)(cid:48)(cid:48) ( u ) f (cid:48)(cid:48) ( u ) (cid:19) − (cid:18) f (cid:48)(cid:48)(cid:48) ( u ) f (cid:48)(cid:48) ( u ) (cid:19) , Λ = Λ (cid:18) ln | Λ || f (cid:48)(cid:48) ( u ) | / (cid:19) u , ζ = 2 δ Λ | Λ | / and we obtain from (A.12) the 3-adapted moving frame v : v = x , v = x u + f (cid:48)(cid:48)(cid:48) ( u )3 f (cid:48)(cid:48) ( u ) x , v = x v , v = 2 ι N , v = Z + f (cid:48)(cid:48)(cid:48) ( u )3 f (cid:48)(cid:48) ( u ) x v (5.5)where x = (1 , r, s, t, rt − s ), N = (cid:0) , − f (cid:48) ( u ) , − , , f ( u ) − uf (cid:48) ( u ) (cid:1) , Z = (0 , , , , ι = ± v differsfrom B H by an element of O + (2 , v , v , v is zero, hence they lie on the sphereat infinity S [ Z ] . Consequently, we cannot picture the conjugate manifold M (cid:48) or the normalizing cones in the ( r, s, t )coordinate chart ψ . Since x v = (0 , , , , u ), then v ∈ span { Z , (0 , , , , } which is an isotropic subspace andwhose projectivization is a null geodesic in Q .From (5.5), if f (cid:48)(cid:48)(cid:48) ( u ) = 0, then v = Z is constant, so s = t has dim( M (cid:48) ) = 0. Less obvious is if f ( u ) = u , then v = (0 , − u , , , v ] is constant. From Table 3, there are no CSp (4 , R )-invariants if dim( M (cid:48) ) = 0. Hence, Theorem 5.8.
Any hyperbolic singly-ruled surface M with dim( M (cid:48) ) = 0 is locally CSp (4 , R ) -equivalent to either ofthe surfaces s = t ( t > or s = t ( t > . r, s, t ) Λ Λ δ ζ s + 1 = √ − t ( u + v − tanh( u ) , − u ) , tanh( u )) 1 0 +1 01 − s = √ t + 1 ( − u + v + tan( u ) , − sec( u ) , tan( u )) − − s = ln( t ) ( − u + v, ln( u ) , u ) u u +1 8 s = e t ( e u + v, e u , u ) −
19 427 − s = √ t ( ln( u ) + v, √ u, u ) u s = t n ( n (cid:54) = 2 , − , ) ( n n − u n − + v, u n , u ) n − n u n − n − n +1)27 u ∗ n − | n − || n +1 | ∗ δ = +1 if − < n < δ = − n < − n > . Theorem 5.9.
Let M ⊂ LG (2 , be a hyperbolic singly-ruled surface with dim( M (cid:48) ) = 1 . If ζ is constant and if:1. δ = +1 , then M is locally CSp (4 , R ) -equivalent to s = t n , where − < n < satisfies ζ = n − (2 − n )( n +1) . If ζ = 0[ ζ = 8] then M is also locally CSp (4 , R ) -equivalent to s + 1 = √ − t [ s = ln( t )] .2. δ = − , then M is locally CSp (4 , R ) -equivalent to s = t n , where n < − or n > satisfies ζ = n − | n − || n +1 | > .If ζ = 0 [ ζ = 64] , then M is also locally CSp (4 , R ) -equivalent to − s = √ t [ s = e t ] . Remark 5.10.
At present, we do not know which singly-ruled CSI surfaces have invariants δ = − and < ζ ≤ . For M hyperbolic 2-generic, we defined (Definition 4.11) the 3-adapted frame bundle ( F ( M ) , H ). Any 3-adapted moving frame v satisfies λ λ (cid:54) = 0, λ = 0 and α = λ θ , α = λ θ , β = 0 with dλ + λ (3 α − α ) = λ θ , dλ + λ (3 α − α ) = λ θ . (6.1)The dβ equation (3.12) becomes λ θ ∧ β + λ θ ∧ β = 0, so by Cartan’s lemma, β = b λ θ + b θ , β = b θ + b λ θ , (6.2)where b ij are fourth order functions on M . The remaining MC equations (3.12) for 3-adapted moving frames are0 = dα + ( λ λ − b ) θ ∧ θ , dβ + 2 α ∧ β , dθ + 2 θ ∧ α , (6.3)0 = dα + ( λ λ − b ) θ ∧ θ , dβ + 2 α ∧ β , dθ + 2 θ ∧ α . (6.4)With κ , κ , τ defined below in (6.7), (6.8), an H -frame change induces:¯ α + ¯ α ¯ α − ¯ α ¯ θ ¯ θ ¯ β ¯ β X ( r , r ) ∈ H (cid:48) α + α + dr r α − α + dr r r r θ r r θ r r β r r β R α + α − ( α − α ) θ θ β β R α + α α − α − θ − θ − β − β (6.5)¯ λ ¯ λ ¯ λ ¯ λ ¯ b ¯ b ¯ b ¯ κ ¯ κ ¯ τX ( r , r ) ∈ H (cid:48) r r λ λ r r r r λ λ r r b r b r b r κ κ τR λ λ λ λ b b b κ κ (cid:15)τR − λ − λ λ λ − b b b − κ − κ τ (6.6)Setting r = sgn( λ ) (cid:112) | λ λ | , r = sgn( λ ) (cid:16) | λ || λ | (cid:17) / , we can normalize ¯ α = ¯ θ , ¯ α = (cid:15) ¯ θ , where (cid:15) =sgn( λ λ ) = ±
1. We refer to the corresponding 3-adapted moving frame as normalized . The structure group isreduced to: (i) Z , generated by R , if (cid:15) = 1 ( M (cid:15) = − M roposition 6.1. Any hyperbolic 2-generic surface M ⊂ LG (2 , admits at most a 2-dimensional symmetry group. Equations (6.1) reduce to 3¯ α − ¯ α = 4 κ ¯ θ , 3¯ α − ¯ α = 4 κ ¯ θ , or equivalently¯ α = 32 κ ¯ θ + 12 κ ¯ θ , ¯ α = 12 κ ¯ θ + 32 κ ¯ θ , where κ = (cid:15) λ = sgn( λ ) λ | λ | / | λ | / , κ = 14 ¯ λ = sgn( λ ) λ | λ | / | λ | / . (6.7)From (6.3)-(6.4), we have d ¯ θ = − κ ¯ θ ∧ ¯ θ , d ¯ θ = κ ¯ θ ∧ ¯ θ . Write df = f , ¯ θ + f , ¯ θ and similarly for iteratedcoframe derivatives. The dα , dα equations in (6.3), (6.4) imply4( κ , + κ κ )¯ θ ∧ ¯ θ = d (3¯ α − ¯ α ) = (4 (cid:15) − b − ¯ b )¯ θ ∧ ¯ θ , − κ , + κ κ )¯ θ ∧ ¯ θ = d (3¯ α − ¯ α ) = − (4 (cid:15) − b − ¯ b )¯ θ ∧ ¯ θ , which implies τ := ¯ b = sgn( λ ) b (cid:112) | λ λ | , ¯ b = 12 κ , − κ , − κ κ + (cid:15), ¯ b = 12 κ , − κ , − κ κ + (cid:15). (6.8)The dβ , dβ equations in (6.3), (6.4) yield the integrability conditions: (cid:15)τ , = ¯ b , + 4 κ ¯ b − (cid:15)κ τ, τ , = ¯ b , + 4 κ ¯ b − κ τ. (6.9)The τ terms in κ τ , − κ τ , cancel, hence the integrability condition 0 = d τ = − τ , + τ , − κ τ , + κ τ , becomes0 = 2( κ , − κ , ) τ − (cid:15) (¯ b , + 4 κ , ¯ b + 4 κ ¯ b , ) + ¯ b , + 4 κ , ¯ b + 4 κ ¯ b , − κ τ , + 3 κ τ , = − κ , − κ , ) τ + T ( κ , κ , (cid:15) ) (6.10)where T = T ( κ , κ , (cid:15) ) is a third order differential function of κ , κ given by T = ¯ b , + 4 κ , ¯ b + 7 κ ¯ b , + 12 κ ¯ b − (cid:15) (¯ b , + 4 κ , ¯ b + 7 κ ¯ b , + 12 κ ¯ b ) . We have κ , = κ , iff ¯ b = ¯ b iff b = b which is H -invariant. From (6.10), we have: Proposition 6.2. If κ , (cid:54) = κ , , then τ is a third order function of κ , κ . Definition 6.3.
We refer to κ H = ( κ + κ ) as the conformal mean-squared curvature , κ G = κ κ as the conformalGaussian curvature , and τ as the conformal torsion . Note that κ H , κ G recover κ , κ up to sign and swap. By Theorem 3.13, our solution to the parametrizedequivalence problem for hyperbolic 2-generic surfaces is: Theorem 6.4 (Invariants and equivalence for hyperbolic generic surfaces) . Let U be connected and i, ˜ i : U → LG (2 , hyperbolic 2-generic surfaces. If M = i ( U ) and ˜ M = ˜ i ( U ) are CSp (4 , R ) -equivalent, then for M, M (cid:48) on U : (i) (cid:15) , κ H , κ G agree, (ii) τ agree if (cid:15) = 1 ; τ agree if (cid:15) = − . Conversely, if the above hold, then M, ˜ M are CSp (4 , R ) -equivalent. In Appendix A.6, we show that { ( κ ) , ( κ ) } and τ (if (cid:15) = 1) or τ (if (cid:15) = −
1) are invariant under reparametriza-tions. Hence, these solve the corresponding unparametrized equivalence problem. Dropping bars over the objectsassociated with the normalized moving frame, we have the results in Table 6.1. By Theorem 3.14, we have:
Theorem 6.5 (Bonnet theorem for hyperbolic generic surfaces) . The integrability conditions in Table 6.1 are theonly local obstructions to the existence of a hyperbolic 2-generic surface with prescribed invariants. d v = 2( κ θ + κ θ ) v + θ v + θ v d v = ( τ θ + b θ ) v − ( κ θ − κ θ ) v − θ v + θ v d v = ( b θ + (cid:15)τ θ ) v + ( κ θ − κ θ ) v − (cid:15)θ v + θ v d v = − (cid:15)θ v − θ v d v = ( b θ + (cid:15)τ θ ) v + ( τ θ + b θ ) v − κ θ + κ θ ) v dθ = − κ θ ∧ θ ,dθ = κ θ ∧ θ ,τ , = (cid:15) ( b , + 4 κ b ) − κ ττ , = b , + 4 κ b − κ τ − κ , − κ , ) τ + T ( κ , κ , (cid:15) )Definitions Fundamental invariants b = κ , − κ , − κ κ + (cid:15), dκ = κ , θ + κ , θ b = κ , − κ , − κ κ + (cid:15), dκ = κ , θ + κ , θ T = b , + 4 κ , b + 7 κ b , + 12 κ b − (cid:15) ( b , + 4 κ , b + 7 κ b , + 12 κ b ) . (cid:15) = ± τ (if (cid:15) = 1) , τ (if (cid:15) = − κ G = κ κ κ H = ( κ + κ ) Table 4: Invariant description of hyperbolic 2-generic surfaces
We express dim( M (cid:48) ) in terms of κ , κ , τ . Let v be any 3-adapted moving frame for M . Since β = 0, then by the d v equation in (4.1), the tangent space T [ v ] M (cid:48) ⊂ T [ v ] Q ∼ = v ⊥ /(cid:96) v is spanned by the coefficients of θ and θ , i.e. w = b v + b λ v and w = b λ v + b v mod v . (6.11) Proposition 6.6. dim( M (cid:48) ) = rank (cid:18) (cid:15)τ ¯ b ¯ b τ (cid:19) , where ¯ b , ¯ b , τ were defined in (6.8) .Proof. dim( M (cid:48) ) = rank (cid:18) b λ b b b λ (cid:19) . Since dim( M (cid:48) ) is geometric, use the normalized moving frame. Corollary 6.7. dim( M (cid:48) ) = 0 iff τ = 0 and κ , = κ , = (cid:15) − κ κ . Example 6.8.
While every singly-ruled hyperbolic surface with dim( M (cid:48) ) = 0 is homogeneous, the analogous assertionin the generic hyperbolic case is false: The integrability conditions reduce to: κ , = κ , = (cid:15) − κ κ , dθ = − κ θ ∧ θ , dθ = κ θ ∧ θ . (6.12) Since { θ } and { θ } are both Frobenius, introduce coordinates u, v such that θ = f du , θ = gdv , where f, g arenonvanishing functions of u, v . The θ , θ equations in (6.12) yield κ = f v fg , κ = g u fg . The remaining equation yields f (cid:18) f v f g (cid:19) u = 1 g (cid:18) g u f g (cid:19) v = (cid:15) − f v g u f g ⇐⇒ F uv = G uv = (cid:15)e F + G , where F = ln | f | , G = ln | g | . Thus, G = F + φ ( u ) + ψ ( v ) . Reparametrizing ˜ u = ˜ u ( u ) and ˜ v = ˜ v ( v ) , we can without loss of generality assume that G = F . We should solve F uv = (cid:15)e F which is almost the Liouville equation z xy = e z . By means of Backl¨und transfor-mations, the latter has the well-known general solution z = ln (cid:16) X (cid:48) ( x ) Y (cid:48) ( y )( X ( x )+ Y ( y )) (cid:17) , which we use to get the general solution F = ln (cid:16) U (cid:48) ( u ) V (cid:48) ( v )( U ( u )+ V ( v )) (cid:17) of the former in the case (cid:15) = 1 . Set ( U ( u ) , V ( v )) = ( u , v ) , so F = G = ln (cid:16) u v ( u + v ) (cid:17) .Hence, f = g = 3 (cid:113) u v ( u + v ) and κ = v − u u v , κ = u − v uv . Since all integrability conditions are satisfied, a surface M locally exists with κ , κ as given and τ = 0 . Since κ , κ are nonconstant, M is nonhomogeneous. If dim( M (cid:48) ) = 1, then τ = 0 and exactly one of b or b is zero. In this case, M (cid:48) is a null curve. Ifdim( M (cid:48) ) = 2, then b λ λ − b b (cid:54) = 0. The conformal structure on M (cid:48) , expressed in terms of (6.11), is (cid:18) b b λ b b + b λ λ b b + b λ λ b b λ (cid:19) . Its determinant is − ( b λ λ − b b ) <
0, so M (cid:48) is hyperbolic.Now classify M (cid:48) at 2nd order (c.f. Table 2). For any 3-adapted moving frame v on M , v (cid:48) = ( v (cid:48) , v (cid:48) , v (cid:48) , v (cid:48) , v (cid:48) ) =25 v , v , v , v , v ) is at least 1-adapted on M (cid:48) . Writing out the structure equations for v (cid:48) and comparing with (4.1),the corresponding MC forms for the adaptation to M (cid:48) are related to those for the adaptation to M by: θ (cid:48) = 0 ,β (cid:48) = 0 , α (cid:48) = − α ,α (cid:48) = − α , α (cid:48) = α ,α (cid:48) = α , θ (cid:48) = β ,θ (cid:48) = β , β (cid:48) = θ ,β (cid:48) = θ . Inverting (6.2), we have (cid:18) θ θ (cid:19) = 1 b λ λ − b b (cid:18) b λ − b − b b λ (cid:19) (cid:18) β β (cid:19) . Writing α (cid:48) = λ (cid:48) θ (cid:48) + λ (cid:48) θ (cid:48) = λ b λ λ − b b ( − b θ (cid:48) + b λ θ (cid:48) ) and α (cid:48) = λ (cid:48) θ (cid:48) + λ (cid:48) θ (cid:48) = λ θ = λ b λ λ − b b ( b λ θ (cid:48) − b θ (cid:48) ), we obtain λ (cid:48) = − λ b b λ λ − b b , λ (cid:48) = b λ λ b λ λ − b b , λ (cid:48) = − λ b b λ λ − b b or λ (cid:48) = r r (cid:18) − (cid:15) ¯ b (cid:15)τ − ¯ b ¯ b (cid:19) , λ (cid:48) = 1 r (cid:18) (cid:15)τ(cid:15)τ − ¯ b ¯ b (cid:19) , λ (cid:48) = 1 r r (cid:18) − ¯ b (cid:15)τ − ¯ b ¯ b (cid:19) where r , r leading to the normalized frame were defined after (6.6). The CSG for M (cid:48) is given by v c (cid:48) = v (cid:48) +2 λ (cid:48) v (cid:48) = v + 2 λ (cid:48) v . Thus, v (cid:48) is 2-adapted iff τ = 0. Since β (cid:48) = 0, v (cid:48) is moreover 3-adapted for M (cid:48) . Hence, ( M (cid:48) ) (cid:48) = M . Theorem 6.9. If M is hyperbolic 2-generic surface and dim( M (cid:48) ) = 2 , then M (cid:48) is also hyperbolic. Moreover, ( M (cid:48) ) (cid:48) = M iff τ = 0 iff the central tangent spheres to M and M (cid:48) agree at conjugate points. From Section 4.2, we see that changing v (cid:48) to v c (cid:48) does not affect λ (cid:48) , λ (cid:48) . Hence, ( λ (cid:48) , λ (cid:48) ) classify M (cid:48) at 2ndorder. Taking into the differential order of b ij , this is a 4th order classification of M .2nd order classification of M (cid:48) Invariant characterizationindefinite sphere τ (cid:54) = 0 and ¯ b = ¯ b = 0singly-ruled τ (cid:54) = 0 and exactly one of ¯ b or ¯ b is zero2-hyperbolic (cid:15)τ (cid:54) = ¯ b ¯ b , (cid:15) ¯ b ¯ b < (cid:15)τ (cid:54) = ¯ b ¯ b , (cid:15) ¯ b ¯ b > M when M (cid:48) is a hyperbolic surface Remark 6.10.
Examples of 2-generic CSI surfaces M with M (cid:48) M (cid:48) an indefinite sphere or M (cid:48) singly-ruled) are non-vacuous. In the 2-elliptic case, λ λ > v , which we moreover assume is 3-adapted. Inthis case M has no asymptotic directions, but carries a net of curvature lines. From a ij = diag ( λ , λ ) and µ ij = (cid:18) (cid:19) , the eigenvalues and eigendirections of a ik = µ ij a jk = (cid:18) λ λ (cid:19) are s ± = ±√ λ λ and v ± = λ v + s ± v . From the d v structure equation, these eigendirections (modulo v ) are respectively given by thevanishing of θ ± = λ θ + s ∓ θ . The contact spheres are B ± = v + 2 s ± v . Under the action of H , we have that[ ¯ B ± ] = [ B ± ] or [ B ∓ ]. Thus, the pair ([ B + ] , [ B − ]) is geometric. Differentiating, dB ± = d v + 2( ds ± ) v + 2 s ± d v = − α v − α v + d ( λ λ ) s ± v + 2 s ± [( α + α ) v + θ v + θ v ]= − λ θ v − λ θ v + 1 s ± [ λ dλ + λ dλ ] v + 2 s ± [( α + α ) v + θ v + θ v ]= − λ θ ± v ∓ + 1 s ± [ λ λ θ + λ λ θ ] v = − λ θ ± v ∓ + 12 s ± (cid:20) λ λ λ ( θ + + θ − ) + λ λ s − ( θ + − θ − ) (cid:21) v
26s the 1-form coefficients of v and v ± in dB ± are linearly independent, the congruence determined by each of [ B ± ]depends on 2 parameters. However, along the curvature directions determined by θ + and θ − , we have θ + = 0 : dB + = a + s + θ − v ; θ − = 0 : dB − = a − s − θ + v , where a ± = λ λ λ ± λ λ s + . We have a + = 0 ⇒ dB + = (cid:18) − λ v − + λ λ λ s + v (cid:19) θ + ⇒ dB + = 0 along θ + = 0 ,a − = 0 ⇒ dB − = (cid:18) − λ v + + λ λ λ s − v (cid:19) θ − ⇒ dB − = 0 along θ − = 0 . If a + = 0 or a − = 0, then B + or B − respectively depend on only 1 parameter. Since we always have [ v ] ∈ S [ B ± ] ,then M is the envelope of a 1-parameter family of (indefinite) spheres and M is called a canal surface . Definition 6.11. If M is the envelope of two 1-parameter families of spheres, then M is called a Dupin cyclide . We see from above that M is a Dupin cyclide iff a + = a − = 0. This condition is equivalent to λ = λ = 0,or κ = κ = 0. By the integrability conditions, τ , = τ , = 0, so τ is constant. By Theorem 2.9, we have: Theorem 6.12.
A hyperbolic surface M ⊂ LG (2 , is a Dupin cyclide iff M is 2-elliptic with κ = κ = 0 and τ ∈ R constant. Moreover, distinct τ ∈ R correspond to CSp (4 , R ) -inequivalent classes of Dupin cyclides. Any Dupincyclide M is a CSI surface with dim( M (cid:48) ) (cid:54) = 0 : (1) τ = ± iff dim( M (cid:48) ) = 1 , and (2) τ (cid:54) = ± iff dim( M (cid:48) ) = 2 . Remark 6.13.
It is well-known that all Dupin cyclides in Euclidean signature are inversions of the standard torus.Since inversions generate the conformal group, then there is only a single equivalence class in that setting. InLorentzian signature, there are infinitely many non-equivalent Dupin cyclides.
Example 6.14.
In the next section, we show for rt = − that κ = κ = 0 and τ = 2 , so this is a Dupin cyclide M with dim( M (cid:48) ) = 2 . ( r, s, t ) = ( − e − ( u + v ) , u − v, e u + v ) is a null parametrization. We have ν = 4 and I = I = I = − ,which implies λ λ = I I ν = . From (A.2) , we have N = (cid:0) , − e − ( u + v ) , , e u + v , − (cid:1) . From (A.15) , a 3-adaptedmoving frame is v = x = (1 , − e − ( u + v ) , u − v, e u + v , − − ( u − v ) ) v = x u = (0 , e − ( u + v ) , , e u + v , − u − v )) , v = 14 x v = 14 (0 , e − ( u + v ) , − , e u + v , u − v )) , v = ι N − x ) = − ι (cid:16) , e − ( u + v ) , u − v, − e u + v , − ( u − v ) (cid:17) , v = Z + 18 N − x = − (cid:16) , e − ( u + v ) , u − v, − e u + v , − − ( u − v ) (cid:17) . The points [ v ] , [ v ] are located on the sphere at infinity S [ Z ] in the coordinate chart ψ . From [ v ] , the conjugate surface M (cid:48) has equation rt = − , which has κ = κ = 0 , τ = 2 , so is again a Dupin cyclide and M (cid:48) is CSp (4 , R ) -equivalentto M . The contact spheres are S [ B ] , S [ B ] , where [ B ] = [0 , e − ( u + v ) , , − e u + v , ⇒ S [ B ] : re u + v − te − ( u + v ) + 2 = 0 , [ B ] = [1 , , u − v, , − ( u − v ) ] ⇒ S [ B ] : rt − s + 2 s ( u − v ) + 1 − ( u − v ) = 0 . ore generally, one can show that the conjugate surface to rt = − k is rt = − k . Hence, for M above, ( M (cid:48) ) (cid:48) (cid:54) = M . Let M be 2-generic CSI, i.e. κ , κ , τ ∈ R are constant, so ¯ b = ¯ b = (cid:15) − κ κ . The remaining integrabilityconditions (6.9)-(6.10) reduce to0 = 2 κ ¯ b − (cid:15)κ τ, κ ¯ b − κ τ, T = 12( κ − (cid:15)κ )¯ b . Solving these equations and using Proposition 6.6 and the 2nd order classification of M (cid:48) in Section 6.2, we have:dim( M (cid:48) ) (cid:15) κ κ τ Inequivalent cases Sub-classification of M (cid:48) ± (cid:54) = 0 (cid:15)κ < κ ≤ − ± − (cid:54) = ± − ∈ R τ ≥ (cid:54) = 0 κ − κ ) (cid:54) = 0 0 < κ (cid:54) = 1 2-elliptic2 1 (cid:54) = 0 − κ − κ ) κ > Remark 6.15.
Comparing with Table 4 in [The08] we see that ( (cid:15), m, n, B ) corresponds to ( (cid:15), κ , − κ , τ ) here. Example 6.16.
Let (cid:15) = ± and m ∈ R nonzero. Consider the surfaces M = M ± (cid:15),m given by r = −
13 ( (cid:15)mu + v ) , s = −
12 ( (cid:15)m u − v ) , t = − ( (cid:15)m u + v ) . (6.14) Here, M + (cid:15),m corresponds to u + mv > and M − (cid:15),m corresponds to u − mv < . Let x = (1 , r, s, t, rt − s ) . Then (cid:104) x u , x u (cid:105) = (cid:104) x v , x v (cid:105) = 0 and ν = (cid:15)m ( u + mv ) . We calculate I = − m ( mv + u ) , I = − (cid:15)m ( mv + u ) , I = 0 , hence I I = (cid:15)m ( mv + u ) , so sgn( I I ) = (cid:15) (cid:54) = 0 , hence such surfaces are 2-elliptic if (cid:15) = 1 and 2-hyperbolic if (cid:15) = − .Define ς = sgn( mv + u ) . Using (A.16) , we find that κ = − ς(cid:15) | m | , κ = − ς | m | , τ = 0 ⇒ κ κ = (cid:15) ⇒ dim( M (cid:48) ) = 0 . From (A.15) , the conjugate point is located at [ v ] = [ Z ] = [0 , , , , , i.e. the center of the sphere at infinity. Note ( κ ) = m , ( κ ) = m are independent of c and sgn( m ) . By Theorem 6.4, M + (cid:15),m is CSp (4 , R ) -equivalent to any of M − (cid:15),m , M + (cid:15), − m , M + (cid:15), m . A minimal list of inequivalent surfaces is given by all M + (cid:15),m such that (cid:15) = ± , m ∈ (0 , .Under the substitution ( r, s, t ) = ( z xx , z xy , z yy ) , these surfaces correspond to the class of maximally symmetrichyperbolic PDE of generic type and were studied in [The08], [The10]. Define α = 1 − (cid:15)m . The PDE given by (6.14)28 as shown in [The08] to be contact-equivalent to α = 0 : 3 rt + 1 = 0; α (cid:54) = 0 : (3 r − st + 2 t ) (2 s − t ) = c ( m ) := (1 + (cid:15)m ) (cid:16) (cid:15)m (cid:17) . (6.15) Examining the calculation, we see that the contact transformations used resulted in vertical transformations on thefibres of π : J → J . On these fibres J | ξ , CSp ( C ξ , [ η ]) ∼ = CSp (4 , R ) transformations were used. Thus, the surfaces (6.14) and (6.15) are CSp (4 , R ) -equivalent. We note that c ( m ) = c ( − m ) = c ( m ) . Thus, (cid:15) = ± , m ∈ (0 , sufficesand this is the parameter range specified in [The08]. From Proposition 4.8, let us examine hyperbolic 2-generic surfaces M given by F ( r, t ) = 0 with F r F t <
0. Bythe implicit function theorem, we may assume that F is given locally by F ( r, t ) = r − f ( t ) = 0 with f (cid:48) ( t ) >
0. ByLemma A.1, a null parametrization exists, which we write as( r, s, t ) = ( f ( t ( u, v )) , s ( u, v ) , t ( u, v )) : f (cid:48) ( t )( t u ) = ( s u ) , f (cid:48) ( t )( t v ) = ( s v ) , Γ = det (cid:18) s u t u s v t v (cid:19) (cid:54) = 0 . None of s u , s v , t u , t v can vanish anywhere. Let F ( t ) be an antiderivative of (cid:112) f (cid:48) ( t ). Using the CSp (4 , R ) transfor-mation ( r, s, t ) → ( r, − s, t ) (induced from ( x, y, z ) → ( − x, y, z )) and interchanging u, v if necessary, we may assume (cid:112) f (cid:48) ( t ) t u = s u , (cid:112) f (cid:48) ( t ) t v = − s v . (If instead (cid:112) f (cid:48) ( t ) t v = s v , then F ( t ) = s and so ( r, s, t ) = ( f ( t ) , F ( t ) , t ), whichis a curve.) From the first equation, we have F ( t ) = s + g ( v ). Differentiation yields (cid:112) f (cid:48) ( t ) t v = s v + g (cid:48) ( v ) andsubstitution in the second yields − s v = g (cid:48) ( v ), so s uv = 0. Write s = s ( u ) − s ( v ). Since s u , s v (cid:54) = 0, then we canreparametrize so s = u − v . Hence, t u = t v = 1 (cid:112) f (cid:48) ( t ) , Γ = t v + t u = 2 (cid:112) f (cid:48) ( t ) (cid:54) = 0Integration yields F ( t ) = u + v , so that t = F − ( u + v ). This makes sense since F (cid:48) ( t ) (cid:54) = 0, so F is 1-1. Thus, ourparametrization is ( r, s, t ) = ( f ( F − ( u + v )) , u − v, F − ( u + v )). We haveWe have ν = 4 and I = I = I = f (cid:48)(cid:48) ( t )( f (cid:48) ( t )) / . Since I I > M is 2-elliptic, c.f. Prop 4.8. From (A.16): κ = κ = sgn( f (cid:48)(cid:48) ( t )) 4 f (cid:48) ( t ) ( f (cid:48)(cid:48) ( t )) [ Sf ]( t ) , τ = 2 − f (cid:48) ( t ) f (cid:48)(cid:48) ( t ) (cid:18) [ Sf ]( t ) f (cid:48) ( t ) (cid:19) (cid:48) (6.16)where [ Sf ]( t ) = f (cid:48)(cid:48)(cid:48) ( t ) f (cid:48) ( t ) − f (cid:48)(cid:48) ( t )) f (cid:48) ( t ) is the Schwarzian derivative of f . Example 6.17.
The following are examples of hyperbolic 2-elliptic CSI surfaces M with dim( M (cid:48) ) = 2 , c.f. (6.13) :(i) r = t with t > ; κ = κ = − , τ = − , and (ii) r = e t : κ = κ = − , τ = − . After some calculations,one can show that the conjugate surface M (cid:48) is: (i) r = t , and (ii) r = 9 e t − . The Schwarzian vanishes on linear fractional transformations, hence f ( t ) = − t satisfies [ Sf ]( t ) = 0 and κ = κ = 0, τ = 2 from (6.16). Hence, M = { rt = − } is the simplest example of a Dupin cyclide. It has dim( M (cid:48) ) = 2. As described in Section 2.3, our
CSp (4 , R )-invariant study of hyperbolic surfaces in LG (2 ,
4) yields contact-invariantinformation for hyperbolic PDE in the plane. Our view of PDE from this perspective has led to a new and simpleproof of contact-invariance of the Monge–Amp`ere PDE, a geometric interpretation of of elliptic, parabolic, hyperbolicPDE, a reinterpretation of class 6-6, 6-7, 7-7 hyperbolic PDE as the 2nd order classification of hyperbolic surfaces in LG (2 , LG (2 , CSp (4 , R )-invariant class of hyperbolic surfaces, find coframe structure equations and investigate thegeometry of the corresponding classes of hyperbolic PDE.3. Study submanifold theory in general Lagrangian–Grassmannians LG ( n, n ) modulo CSp (2 n, R ). This studywill be significantly more difficult than in LG (2 ,
4) as there is no longer a connection to conformal geometry.The first-order distinguished structure on the tangent spaces of LG ( n, n ) is now a degree n cone instead of aquadratic cone as in the conformal case. Acknowledgements
We thank Igor Zelenko for pointing out that the fibres of π : J ( R n , R ) → J ( R n , R ) are diffeomorphic to LG ( n, n ). This was the initial seed that grew into this work. We also thank Niky Kamran and Abraham Smith forfruitful discussions and encouragement. We gratefully acknowledge financial support from the National Sciences andEngineering Research Council of Canada in the form of an NSERC Postdoctoral Fellowship. A Parametric description of moving frames
A.1 Null parametrization and differential syzygies
Given a smooth hyperbolic surface M ⊂ LG (2 ,
4) with basepoint o , we may (using the CSp (4 , R )-action) assume o = { e , e } . Let ( r, s, t ) be standard coordinates about o corresponding to [1 , r, s, t, rt − s ] ∈ Q , c.f. Section 3,expressed with respect to the basis B (3.5) on which (cid:104)· , ·(cid:105) has matrix (3.5). Let ( r ( u, v ) , s ( u, v ) , t ( u, v )) be a localparametrization i : U → M and let x = x ( u, v ) = (1 , r, s, t, rt − s ). Since (cid:104) x , x (cid:105) = 0, then (cid:104) x , x u (cid:105) = (cid:104) x , x v (cid:105) = 0,where x u = (0 , r u , s u , t u , r u t + rt u − ss u ), x v = (0 , r v , s v , t v , r v t + rt v − ss v ). In order to substantially simplify ourlater formulas we will assume that u, v are null coordinates , the local existence of which is guaranteed: Lemma A.1 (Existence of null coordinates [Wei95]) . Let ( S, µ ) be a smooth surface with a Lorentzian metric µ .Given any p ∈ S , there exist (smooth) coordinates u, v such that ∂ u , ∂ v are null vectors and µ = f dudv with f > . The u, v coordinate lines are null curves with respect to [ µ ], where µ = dr (cid:12) dt − ds (cid:12) ds . Analytically, (cid:104) x u , x u (cid:105) = 2( r u t u − s u ) = 0 , (cid:104) x v , x v (cid:105) = 2( r v t v − s v ) = 0 . (A.1)Since [ µ ] on M is non-degenerate, then ν := (cid:104) x u , x v (cid:105) = r u t v + r v t u − s u s v (cid:54) = 0 . Define Z = (0 , , , ,
1) and N = , det (cid:18) r u s u r v s v (cid:19) ,
12 det (cid:18) r u t u r v t v (cid:19) , det (cid:18) s u t u s v t v (cid:19) , det r s tr u s u t u r v s v t v , (A.2)which is like a Lorentzian cross-product. We have N ∈ { x , x u , x v , Z } ⊥ . Using (A.1), we find ν N := (cid:113) − (cid:104) N , N (cid:105) = ν so that (cid:104) N , N (cid:105) = − ν N ) . Define I = (cid:104) N , x uu (cid:105) , I = (cid:104) N , x vv (cid:105) , I = (cid:104) N , x uv (cid:105) , which can also be written I = det r uu s uu t uu r u s u t u r v s v t v , I = det r vv s vv t vv r u s u t u r v s v t v , I = det r uv s uv t uv r u s u t u r v s v t v . (A.3)The functions I , I are the MA invariants, as demonstrated in (A.6). These functions satisfy some non-trivialdifferential syzygies. First, note that with respect to the basis { x , x u , x v , N , Z } , x uu = ν u ν x u − I ν N , x uv = − I ν N + ν Z , x vv = ν v ν x v − I ν NN u = − I ν x u − I ν x v + ν u ν N , N v = − I ν x u − I ν x v + ν v ν N (cid:104) x uu , x (cid:105) = (cid:104) x vv , x (cid:105) = (cid:104) x uu , x u (cid:105) = (cid:104) x vv , x v (cid:105) = (cid:104) x uv , x u (cid:105) = (cid:104) x uv , x v (cid:105) = 0 , (cid:104) x uu , x v (cid:105) = ν u , (cid:104) x vv , x u (cid:105) = ν v , (cid:104) x uv , x (cid:105) = − ν, (cid:104) N u , N (cid:105) = − νν u , (cid:104) N v , N (cid:105) = − νν v , (cid:104) N u , x (cid:105) = (cid:104) N v , x (cid:105) = 0 , (cid:104) N u , x u (cid:105) = − I , (cid:104) N u , x v (cid:105) = (cid:104) N v , x u (cid:105) = − I , (cid:104) N v , x v (cid:105) = − I . which are differential consequences of (A.1). We examine integrability conditions. After some simplification,0 = x vvu − x uvv = (cid:18) (ln | ν | ) uv + 2( I I − I ) ν (cid:19) x v + 2 (cid:18)(cid:18) I ν (cid:19) v − ν (cid:18) I ν (cid:19) u (cid:19) N x uuv − x uvu = (cid:18) (ln | ν | ) uv + 2( I I − I ) ν (cid:19) x u + 2 (cid:18)(cid:18) I ν (cid:19) u − ν (cid:18) I ν (cid:19) v (cid:19) N N uv − N vu = (cid:18)(cid:18) I ν (cid:19) u − ν (cid:18) I ν (cid:19) v (cid:19) x u + (cid:18) ν (cid:18) I ν (cid:19) u − (cid:18) I ν (cid:19) v (cid:19) x v Thus, we obtain the differential syzygies(ln | ν | ) uv = 2( I − I I ) ν , (cid:18) I ν (cid:19) u = 1 ν (cid:18) I ν (cid:19) v , (cid:18) I ν (cid:19) v = 1 ν (cid:18) I ν (cid:19) u . (A.4) Remark A.2.
The identities (A.4) are extremely complicated, yet appear deceivingly simple. The latter two identitiesin full yields two degree 5 (differential) polynomial in the 15 variables r u , r v , ..., t uu , t uv , t vv (since the third orderderivative terms cancel), each having 62 terms. Using MAPLE’s Groebner package, we have verified that thesedifferential polynomials vanish on the ideal generated by the relations (A.1) and their differential consequences.
A.2 1-adaptation
We assume a regular parametrization, so x u , x v never vanish. Take our initial 1-adapted moving frame v to be v = x , v = x u , v = 1 ν x v , v = 2 ν ι N , v = Z , (A.5)where ι = ± v differs from B H by an element of O + (2 , U is connected,so ι is constant. Using (A.5), (4.1), θ = (cid:104) d v , v (cid:105) , θ = (cid:104) d v , v (cid:105) and α = (cid:104) d v , v (cid:105) , α = (cid:104) d v , v (cid:105) are (cid:26) θ = duθ = νdv , (cid:26) α = λ θ + λ θ α = λ θ + λ θ , where λ = ιI ν , λ = ιI ν , λ = ιI ν . (A.6)Note that under null reparametrizations, we have(˜ u, ˜ v ) ˜ ι ˜ ν ˜ N ˜ I ˜ I ˜ I ˜ λ ˜ λ ˜ λ ˜ λ ( f ( u ) , g ( v )) sgn( f u ) ι f u g v ν f u g v N I ( f u ) g v I f u ( g v ) I ( f u ) ( g v ) sgn( f u )( f u ) λ sgn( f u )( f u ) λ λ λ ( v, u ) sgn( ν ) ι ν − N − I − I − I − sgn( ν ) ν λ − sgn( ν ) ν λ λ λ In particular, note that ˜ α = | f u | α under (˜ u, ˜ v ) = ( f ( u ) , g ( v )) and ˜ α = − sgn( ν ) να under (˜ u, ˜ v ) = ( v, u ). This isto be expected that the MC forms are not necessarily invariant under reparametrizations since our choice of 1-adaptedframe in (A.5) is not. For example, if (˜ u, ˜ v ) = ( f ( u ) , g ( v )), then (˜ v , ˜ v , ˜ v , ˜ v , ˜ v ) = ( v , f u v , f u v , sgn( f u ) v , v ). A.3 Monge–Amp`ere invariants
Let us give another set of formulas for the MA invariants I , I , given in (A.3). Let F ( r, s, t ) = 0 be an implicitdescription of a hyperbolic surface M endowed with null coordinates ( u, v ). Recall that I = (cid:104) N , x uu (cid:105) , I = (cid:104) N , x vv (cid:105) ,where we can take N = (0 , − F t , F s , − F r , − rF r − sF s − tF t ). (This may differ from (A.2) by an overall scaling, butthis will not affect the classification result.) Thus, I and I have the simple expressions I = − ( F r r uu + F s s uu + F t t uu ) , I = − ( F r r vv + F s s vv + F t t vv ) . (A.7)31valuated on the surface. There is another natural way to express these invariants. Let n = r u s u t u , n = r v s v t v , Hess ( F ) = F rr F rs F rt F sr F ss F st F tr F ts F tt . Differentiating F = 0, we obtain F r r u + F s s u + F t t u = 0, F r r v + F s s v + F t t v = 0. Differentiating again, one findsthat 0 = dF (( n ) u )) + n T Hess ( F ) n and 0 = dF (( n ) v )) + n T Hess ( F ) n . Combining this with (A.7), we obtain I = n T Hess ( F ) n , I = n T Hess ( F ) n . (A.8) A.4 2-adaptation
Let s = λ = ιI ν . The CTS is given by v c = v + 2 s v . Hence, a 2-adapted lift is v = x , v = x u , v = 1 ν x v , v = v c = 2 ιν (cid:18) N + I ν x (cid:19) , v = Z − I ν N − I ν x . (A.9)For (A.9), we calculate θ = du , θ = νdv as before, and λ = 0, while λ , λ are as in (A.6). Moreover, α + α = 0 , α − α = ν u ν du, α = λ du, α = λ νdv,dλ + λ (3 α − α ) = λ θ + λ θ , dλ + λ (3 α − α ) = λ θ + λ θ ,λ = ν (cid:18) ιI ν (cid:19) u , λ = 1 ν (cid:18) ιI ν (cid:19) v , λ = 1 ν (cid:18) ιI ν (cid:19) u , λ = 1 ν (cid:18) ιI ν (cid:19) v ,β = 2 I I ν du + I ν dv, β = I ν du + 2 I I ν dv,β = d (cid:18) ιI ν (cid:19) = λ θ + λ θ ⇒ λ = (cid:18) ιI ν (cid:19) u , λ = 1 ν (cid:18) ιI ν (cid:19) v . Consequently, from the two expressions for λ , λ we recover again the latter two identities in (A.4). A.5 3-adaptation for singly-ruled surfaces
Assume M is singly-ruled, I (cid:54) = 0, I = 0. Hence, v is the parameter along the null ruling and (A.4) becomes(ln | ν | ) uv = 2 I ν , (cid:18) I ν (cid:19) u = 1 ν (cid:18) I ν (cid:19) v , (cid:18) I ν (cid:19) v = 0 . (A.10)Let s = λ λ = (cid:0) ln (cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:1) u , s = − λ λ = − ν (cid:0) ln (cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:1) v . The PNC is v n = ν x v + s x and the SNC is v n (cid:48) = x u + s x .Hence, a 3-adapted lift for a singly-ruled hyperbolic surface is v = x , v = x u + 13 (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u x , v = 1 ν (cid:18) x v − (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v x (cid:19) , v = 2 ιν (cid:18) N + I ν x (cid:19) , (A.11) v = Z − I ν N − ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v x u + 13 ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u x v − (cid:18) ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v + I ν (cid:19) x (A.12)For this framing, α = λ du , α = β = 0, and λ = λ = 0, so 3 α − α = − d (ln | λ | ) is exact. Moreover, α = − (cid:18) ln | I || ν | / (cid:19) u du, α = ν u ν du + (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v dv, β = Λ θ − Λ θ , β = Λ θ , where β = −(cid:104) d v , v (cid:105) and β = −(cid:104) d v , v (cid:105) , henceΛ = 2 I I ν + ν (cid:18) ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u (cid:19) u − (cid:18)(cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u (cid:19) , Λ = 1 ν (cid:18) ln | ν | / | I | (cid:19) uv (A.13)32he absence of a θ term in β is equivalent to (cid:0) ν (cid:0) I ν (cid:1) v (cid:1) v = 0, a differential consequence of (A.10). From (5.1),Λ = Λ (cid:18) ln | Λ | ν | I | / (cid:19) u , Λ = Λ (cid:18) ln | Λ | ν | I | / (cid:19) u = − ν Λ ,v , and the absence of the θ term in the d Λ equation (5.1) yields the syzygy(ln | Λ | ) v + 2 (cid:18) ln | I || ν | (cid:19) v = 0 ⇒ (cid:18) Λ I ν (cid:19) v = 0Under the null reparametrization ˜ u = f ( u ), ˜ v = g ( v ), we have˜Λ = Λ ( f u ) , ˜Λ = Λ , ˜Λ = Λ f u , ˜Λ = Λ ( f u ) , ˜ ζ = ζ , ˜ ζ = sgn( f u ) ζ , ˜ ζ = sgn( f u ) ζ, where ζ = Λ | λ || Λ | / , ζ = δ Λ | λ | / | Λ | / , and ζ = δ Λ | Λ | / . Thus, δ = sgn(Λ ), δ = sgn(Λ ), ζ , ( ζ ) , ζ areinvariants of unparametrized surfaces. A.6 3-adaptation for generic surfaces
Let us assume that M is 2-generic. Let s = − λ λ = − (cid:0) ln (cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:1) u , s = − λ λ = − ν (cid:0) ln (cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:1) v . The normalizingcones are given by v n = x u + s x and v n = ν x v + s x . Hence, a 3-adapted lift is v = x , v = x u − (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u x , v = 1 ν (cid:18) x v − (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v x (cid:19) , v = 2 ιν (cid:18) N + I ν x (cid:19) , (A.14) v = Z − I ν N − ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v x u − ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u x v + (cid:18) ν (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u − I ν (cid:19) x . (A.15)For this framing, α = λ du , α = λ νdv (c.f. (A.6)), β = 0, and λ = λ = 0. Moreover, α = − ν u ν du + (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) u du, β = (cid:18) I I ν − ν I (cid:18) I ν (cid:19) uv (cid:19) du + (cid:18) I ν + (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) uv (cid:19) dvα = ν u ν du + (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) v dv, β = 1 ν (cid:18) I ν − (cid:18) ln (cid:12)(cid:12)(cid:12)(cid:12) I ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) uv (cid:19) du + 1 ν (cid:18) I I ν − ν I (cid:18) I ν (cid:19) uv (cid:19) dvλ = λ (cid:18) ln | I || I | | ν | (cid:19) u , λ = λ ν (cid:18) ln | I | | I || ν | (cid:19) v . For β , we used the differential identity (A.4). From (6.2), and since θ = du , θ = νdv , we have b λ = 2 I I ν − ν I (cid:18) I ν (cid:19) uv , b λ = 1 ν (cid:18) I I ν − ν I (cid:18) I ν (cid:19) uv (cid:19) ⇒ b = ι (cid:18) I ν − ν I I (cid:18) I ν (cid:19) uv (cid:19) . From (6.7) and (6.8), we have κ = ν | I || I | (cid:18) | I | / | I | / | ν | / (cid:19) v , κ = | ν | | I || I | (cid:18) | I | / | I | / | ν | / (cid:19) u , τ = sgn( I ν ) ν (cid:112) | I I | (cid:18) I ν − ν I I (cid:18) I ν (cid:19) uv (cid:19) (A.16)Taking into account the formulas from Section A.2, we calculate the effect of null reparametrizations:(˜ u, ˜ v ) ˜ κ ˜ κ ˜ τ ( f ( u ) , g ( v )) : sgn( f u ) κ sgn( f u ) κ τ ( v, u ) : sgn( ν ) κ sgn( ν ) κ (cid:15)τ eferences [AABMP10] D. Alekseevsky, R. Alonso-Blanco, G. Manno, and R. Pugliese. Contact geometry of multidimensionalMonge-Amp`ere equations: characteristics, intermediate integrals and solutions. arXiv:1003.5177v1 ,2010.[AG96] M. A. Akivis and V. V. Goldberg. Conformal Differential Geometry and its Generalizations . Pure andApplied Mathematics (New York). John Wiley & Sons, Inc., 1996.[Car10] E. Cartan. Les syst`emes de Pfaff `a cinq variables et les ´equations aux d´eriv´ees partielles du secondordre.
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