IINTRODUCTION TO EXTERIOR DIFFERENTIAL SYSTEMS
BENJAMIN MCKAY
C o n t e n t s
1. Introduction, 12. Expressing differential equations using differential forms, 13. The Cartan–Kähler theorem, 24. Example: Lagrangian submanifolds, 45. The last character, 46. Example: harmonic functions, 57. Generality of integral manifolds, 58. Example: triply orthogonal webs, 79. Example: isometric immersion, 810. Prolongation, 911. Back to isometric immersion, 1012. For further information, 1113. Linearization, 1114. The characteristic variety, 13References, 13 1.
I n t ro d u c t i o n
We assume that the reader is familiar with elementary differential geometry onmanifolds and with differential forms. These lectures explain how to apply theCartan–Kähler theorem to problems in differential geometry. We want to decide ifthere are submanifolds of a given dimension inside a given manifold on which givendifferential forms vanish. The Cartan–Kähler theorem gives a linear algebra test: ifthe test passes, such submanifolds exist. I will not give a proof or give the mostgeneral statement of the theorem, as it is difficult to state precisely.For a proof of the Cartan–Kähler theorem, see [2], which we will follow veryclosely, and also the canonical reference work [1] and the canonical textbook [6].The last two also give proof of the Cartan–Kuranishi theorem, which we will onlybriefly mention.2.
E x p r e s s i n g d i f f e r e n t i a l e q u at i o n s u s i n g d i f f e r e n t i a lf o r m s
Take a differential equation of second order 0 = f ( x, u, u x , u xx ). To write it as afirst order system, add a new variable p to represent u x , and a new equation: u x = p, f ( x, u, p, p x ) . It is easy to generalize this to any number of variables and equations of any order:reduce any system of differential equation to a first order system.
Date : June 30, 2017.Thanks to Daniel Piker for the use of his images of triply orthogonal webs. a r X i v : . [ m a t h . DG ] J un B. MCKAY
To express a first order differential equation 0 = f ( x, u, u x ), add a variable p torepresent the derivative u x , let ϑ = du − p dx on the manifold M = { ( x, u, p ) | f ( x, u, p ) } (assuming it is a manifold). A submanifold of M of suitable dimension on which0 = ϑ and 0 = dx is locally the graph of a solution. It is easy to generalize this toany number of variables and any number of equations of any order.3. T h e C a rta n – K ä h l e r t h e o r e m An integral manifold of a collection of differential forms is a submanifold onwhich the forms vanish. An exterior differential system is an ideal I ⊂ Ω ∗ of smoothdifferential forms on a manifold M , closed under exterior derivative, which splitsinto a direct sum I = I ⊕ I ⊕ · · · ⊕ I n of forms of various degrees: I p .. = I ∩ Ω p . Any collection of differential forms has thesame integral manifolds as the exterior differential system it generates. An exteriordifferential system is analytic if it is locally generated by real analytic differentialforms.Some trivial examples: the exterior differential system generated by(1) 0,(2) Ω ∗ ,(3) the pullbacks of all forms via a submersion,(4) dx ∧ dy + dx ∧ dy in R ,(5) dy − z dx on R . Problem 1.
What are the integral manifolds of our trivial examples?The elements of I are 0-forms, i.e. functions. All I -integral manifolds lie in thezero locus of these functions. Replace our manifold M by that zero locus (whichmight not be a manifold, a technical detail we will ignore); henceforth we add tothe definition of exterior differential system the requirement that I = 0.An integral element at a point m ∈ M of an exterior differential system I is alinear subspace E ⊂ T m M on which all forms in I vanish. Every tangent spaceof an integral manifold is an integral element, but some integral elements of someexterior differential systems don’t arise as tangent spaces of integral manifolds. Problem 2.
What are the integral elements of our trivial examples?The polar equations of an integral element E are the linear functions w ∈ T m M ϑ ( w, e , e , . . . , e k )where ϑ ∈ I k +1 and e , e , . . . , e k ∈ E . They vanish on a vector w just when thespan of { w } ∪ E is an integral element. If an integral element E is contained inanother one, E ⊂ F , then all polar equations of E occur among those of F : largerintegral elements have more (or at least the same) polar equations. Problem 3.
What are the polar equations of the integral elements of our trivialexamples?A partial flag E • is a sequence of nested linear subspaces E ⊂ E ⊂ E ⊂ · · · ⊂ E p XTERIOR DIFFERENTIAL SYSTEMS 3 in a vector space. The increments of a partial flag are the integers measuring howthe dimensions increase: dim E , dim E − dim E , dim E − dim E , ...dim E p − dim E p − . A flag is a partial flag E ⊂ E ⊂ E ⊂ · · · ⊂ E p for which dim E i = i . Danger: most authors require that a flag have subspaces ofall dimensions; we don’t : we only require that the subspaces have all dimensions0 , , , . . . , p up to some dimension p . In particular, the increments of any flag are0 , , , . . . , E • of integral elements form a partial flag in thecotangent space. The characters s , s , . . . , s p of E • are the increments of its polarequations, i.e. the numbers of linearly independent polar equations added at eachincrement in the flag. Problem 4.
What are the characters of the integral flags of our trivial examples?The rank p Grassmann bundle of a manifold M is the set of all p -dimensionallinear subspaces of tangent spaces of M . Problem 5.
Recall how charts are defined on the Grassmann bundle. Prove thatthe Grassmann bundle is a fiber bundle over the underlying manifold.The integral elements of an exterior differential system form a subset of theGrassmann bundle. Let us inquire whether this subset is a submanifold of theGrassmann bundle; if so, let us predict its dimension. We say that a flag of integralelements predicts the dimension dim M + s + 2 s + · · · + ps p ; an integral element predicts the dimension predicted by the generic flag inside it. Theorem 1 (Cartan’s bound) . Every integral element predicts the dimension of asubmanifold of the Grassmann bundle containing all nearby integral elements.
An integral element E correctly predicts dimension if the integral elements near E form a manifold of dimension predicted by E . An integral element which correctlypredicts dimension is involutive . Theorem 2 (Cartan–Kähler) . There is an integral manifold tangent to everyinvolutive integral element of any analytic exterior differential system.
If an integral element is involutive, then all nearby integral elements are too, asthe nonzero polar equations will remain nonzero. An exterior differential system is involutive if its generic maximal dimensional integral element is involutive.
Problem 6.
The Frobenius theorem in this language: on a manifold M ofdimension p + q , take an exterior differential system I locally generated by q linearlyindependent 1-forms together with all differential forms of degree more than p : I k = Ω k for k > p . Prove that I is involutive if and only if every 2-form in I is asum of terms of the form ξ ∧ ϑ where ϑ is a 1-form in I . Prove that this occurs justwhen the ( n − k )-dimensional I -integral manifolds form the leaves of a foliation F of M . Prove that then I consists precisely of the 1-forms vanishing on the leavesof F . B. MCKAY E x a m p l e : L ag r a n g i a n s u b m a n i f o l d s
Let ϑ .. = dx ∧ dy + dx ∧ dy + · · · + dx n ∧ dy n . Let I be the exterior differential system generated by ϑ on M .. = R n . The integralmanifolds of I are called Lagrangian manifolds . Let us employ the Cartan–Kählertheorem to prove the existence of Lagrangian submanifolds of complex Euclideanspace. Writing spans of vectors in angle brackets,Flag Polar equations Characters E = { } { } s = 0 E = h ∂ x i (cid:10) dy (cid:11) s = 1... ... ... E n = h ∂ x , ∂ x , . . . , ∂ x n i (cid:10) dy , dy , . . . , dy n (cid:11) s n = 1The flag predictsdim M + s + 2 s + · · · + n s n = 2 n + 1 + 2 + · · · + n. The nearby integral elements at a given point of M are parameterized by dy = a dx ,which we plug in to ϑ = 0 to see that a can be any symmetric matrix. So the spaceof integral elements has dimensiondim M + n ( n + 1)2 = 2 n + n ( n + 1)2 , correctly predicted. Therefore the Cartan–Kähler theorem proves the existence ofLagrangian submanifolds of complex Euclidean space, one (at least) through eachpoint, tangent to each subspace dy = a dx , at least for any symmetric matrix a closeto 0. Problem 7.
On a complex manifold M , take a Kähler form ϑ and a holomorphicvolume form Ψ, i.e. closed forms expressed in local complex coordinates as ϑ = √− g µ ¯ ν dz µ ∧ dz ¯ ν , Ψ = f ( z ) dz ∧ dz ∧ · · · ∧ dz n , with f ( z ) a holomorphic function and g µ ¯ ν a positive definite self-adjoint complexmatrix of functions. Prove the existence of special Lagrangian manifolds , i.e. integralmanifolds of the exterior differential system generated by the pair of ϑ and theimaginary part of Ψ. 5. T h e l a s t c h a r ac t e r
In applying the Cartan–Kähler theorem, we are always looking for submanifoldsof a particular dimension p . For simplicity, we can add the hypothesis that ourexterior differential system contains all differential forms of degree p + 1 and higher.In particular, the p -dimensional integral elements are maximal dimensional integralelements. The polar equations of any maximal dimensional integral element E p cut out precisely E p , i.e. there are dim M − p independent polar equations on E p . We encounter s , s , . . . , s p polar equations at each increment, so the numberof independent polar equations is s + s + · · · + s p . Our hypothesis helps us tocalculate s p from the other characters: s + s + · · · + s p − + s p = dim M − p. XTERIOR DIFFERENTIAL SYSTEMS 5
For even greater simplicity, we take this as a definition for the final character s p ,throwing out the previous definition. Now we can ignore any differential forms ofdegree more than p when we test Cartan’s bound.6. E x a m p l e : h a r m o n i c f u n c t i o n s
We will prove the existence of harmonic functions on the plane with given valueand first derivatives at a given point. On M = R x,y,u,u x ,u y , let I be the exteriordifferential system generated by ϑ .. = du − u x dx − u y dy, Θ .. = du x ∧ dy − du y ∧ dx. Note that dϑ = − du x ∧ dx − du y ∧ dy also belongs to I because any exterior differential system is closed under exteriorderivative. An integral surface X ⊂ M on which 0 = dx ∧ dy is locally the graph ofa harmonic function u = u ( x, y ) and its derivatives u x = ∂u∂x , u x = ∂u∂x .Each integral plane E (i.e. integral element of dimension 2) on which dx ∧ dy = 0is given by equations du x = u xx dx + u xy dy,du y = u yx dx + u yy dy, for a unique choice of 4 constants u xx , u xy , u yx , u yy subject to the 2 equations u xy = u yx and 0 = u xx + u yy . Hence integral planes at each point have dimension2. The space of integral planes has dimension dim M + 2 = 5 + 2 = 7.Each vector inside that integral plane has the form v = ( ˙ x, ˙ y, u x ˙ x + u y ˙ y, u xx ˙ x + u xy ˙ y, u yx ˙ x + u yy ˙ y )Each integral line E is the span E = h v i of a nonzero such vector. Compute v (cid:18) dϑ Θ (cid:19) = (cid:18) ˙ xdu x + ˙ ydu y − ( u xx ˙ x + u xy ˙ y ) dx − ( u yx ˙ x + u yy ˙ y ) dy ˙ ydu x − ˙ xdu y + ( u xx ˙ x + u xy ˙ y ) dy − ( u yx ˙ x + u yy ˙ y ) dx (cid:19) . and Flag Polar equations Characters E = { } h ϑ i s = 1 E = h v i h ϑ, v dϑ, v Θ i s = 2Since we are only interested in finding integral surfaces, we compute the finalcharacter from s + s + s = dim M − . The Cartan characters are ( s , s , s ) = (1 , ,
0) with predicted dimension dim M + s + 2 s = 5 + 2 + 2 · G e n e r a l i t y o f i n t e g r a l m a n i f o l d s
The proof of the Cartan–Kähler theorem (which we will not give) constructsintegral manifolds inductively, starting with a point, then building an integral curve,and so on. The choice of the initial data at each inductive stage consists of s constants, s functions of 1 variable, and so on. Different choices of this initial datagive rise to different integral manifolds in the final stage. In this sense, the integralmanifolds depend on s constants, and so on. B. MCKAY
If one describes some family of submanifolds in terms of the integral manifolds ofan exterior differential system, someone else might find a different description of thesame submanifolds in terms of integral manifolds of a different exterior differentialsystem, with different Cartan characters.For example, any smooth function y = f ( x ) of 1 variable is equivalent informationto having a constant f (0) and a function y = f ( x ) of 1 variable.For example, immersed plane curves are the integral curves of I = 0 on M = R .Check that any integral flag E = { } , E = h v i has ( s , s ) = (0 , I generated by ϑ .. = sin φ dx − cos φ dy on M .. = R x,y × S φ . Here ( s , s ) = (1 , C n depend on 1 function of n variables. This countis correct: those which are graphs y = y ( x ) are precisely of the form y = ∂S∂x for some potential function S ( x ), unique up to adding a real constant. On the otherhand, the proof of the Cartan–Kähler theorem builds up each Lagrangian manifoldfrom a choice of one function of one variable, one function of two variables, and soon. Similarly, harmonic functions depend on 2 functions of 1 variable. Summingup: we “trust” the last nonzero Cartan–Kähler s p to tell us the generality of theintegral manifolds: they depend on s p functions of p variables, but we don’t “trust” s , s , . . . , s p − . XTERIOR DIFFERENTIAL SYSTEMS 7 E x a m p l e : t r i p ly o rt h o g o n a l w e b s
Images (a), (b), (c): Daniel Piker, 2015
On a 3-dimensional Riemannian manifold X , a triply orthogonal web is a tripleof foliations whose leaves are pairwise perpendicular. We will see that these exist,locally, depending on 3 functions of 2 variables. Each leaf is perpendicular to aunique smooth unit length 1-form η i , up to ± , which satisfies 0 = η i ∧ dη i , by theFrobenius theorem. Let M be the set of all orthonormal bases of the tangent spacesof X , with obvious bundle map x : M → X , so that each point of M has the form m = ( x, e , e , e ) for some x ∈ X and orthonormal basis e , e , e of T x X . The soldering 1-forms ω , ω , ω on M are defined by v ω i = h e i , x ∗ v i . Note: they are 1-forms on M , not on X . Let ω = ω ω ω . Define cross product α × β on R -valued 1-forms by α × β ( u, v ) = α ( u ) × β ( v ) − α ( v ) × β ( u ) . By the fundamental lemma of Riemannian geometry, there is a unique R -valued1-form γ for which dω = γ × ω . Our triply orthogonal web is precisely a section B. MCKAY X → M of the bundle map M → X on which 0 = ω i ∧ dω i for all i , hence an integral3-manifold of the exterior differential system I on M generated by the 3-forms ω ∧ dω , ω ∧ dω , ω ∧ dω . Using the equations above, I is also generated by γ ∧ ω ∧ ω , γ ∧ ω ∧ ω , γ ∧ ω ∧ ω . The 3-dimensional I -integral manifolds on which0 = ω ∧ ω ∧ ω are locally precisely the triply orthogonal webs. The 3-dimensional integral elementson which 0 = ω ∧ ω ∧ ω are precisely given by γ = p p p p p p ω for any p ij , hence 6 + 6 = 12 dimensions of integral elements. Since the systemis generated by 3-forms, on any integral flag in this integral element, 0 = s = s .Count out ( s , s , s , s ) = (0 , , , E x a m p l e : i s o m e t r i c i m m e r s i o n
Take a surface P with a Riemannian metric. Naturally we are curious if there isan isometric immersion f : P → R , i.e. a smooth map preserving the lengths of allcurves on P . For example, these surfacesare isometric immersions of a piece of this paraboloidMore generally, take a Riemannian manifold P of dimension 3. We ask if thereis an isometric immersion f : P → P .On the orthonormal frame bundle F P , denote the soldering forms as ω = ω + iω .By the fundamental lemma of Riemannian geometry there is a unique 1-form (theconnection 1-form) γ so that dω = iγ ∧ ω and dγ = ( i/ Kω ∧ ¯ ω . As above, on F P there is a soldering 1-form ω and a connection 1-form γ so that d ω = γ × ω andThis ensures that d γ = 12 γ × γ + 12 (cid:16) s − R (cid:17) ω × ω . with Ricci curvature R ij = R ji and scalar curvature s = R ii . XTERIOR DIFFERENTIAL SYSTEMS 9
If there is an isometric immersion f : P → P , then let X .. = X f ⊂ M .. = F P × F P be its adapted frame bundle , i.e. the set of all tuples( p, e , e , p , e , e , e )where p ∈ P with orthonormal frame e , e ∈ T p P and p ∈ P with orthonormalframes e , e , e ∈ T p P , so that f ∗ e = e and f ∗ e = e . Let I be the exteriordifferential system on M generated by the 1-forms ϑ ϑ ϑ .. = ω − ω ω − ω ω . Along X , all of these 1-forms vanish, while the 1-forms ω , ω , γ remain linearlyindependent. Conversely, we will eventually prove that all I -integral manifoldson which ω , ω , γ are linearly independent are locally frame bundles of isometricimmersions. For the moment, we concentrate on asking whether we can apply theCartan–Kähler theorem.Compute: d ϑ ϑ ϑ = − γ − γ − ( γ − γ ) 0 0 γ − γ ∧ ω ω γ mod ϑ , ϑ , ϑ . We count s = 2 , s = 1 , s = 0. Each 3-dimensional integral element has ω = ω ,so is determined by the linear equations giving γ , γ , γ in terms of ω , ω , γ onwhich dϑ = 0: γ γ γ − γ = a bc − a (cid:18) ω ω (cid:19) . Therefore there is a 3-dimensional space of integral elements at each point. But s + 2 s = 4 >
3: no integral element correctly predicts dimension, so we can’tapply the Cartan–Kähler theorem.What to do? On every integral element, we said that γ γ γ − γ = a bc − a (cid:18) ω ω (cid:19) . Make a new manifold M .. = M × R a,b,c , and on M let I be the exterior differentialsystem generated by ϑ ϑ ϑ .. = γ γ γ − γ − a bc − a (cid:18) ω ω (cid:19) . P ro l o n g at i o n
Take an exterior differential system I on a manifold M . What should we doif there are no involutive integral elements? Let M be the set of all pairs ( m, E )consisting of a point m of M and an I -integral element E ⊂ T m M . So M is asubset of the Grassmann bundle over M . Locally on M , take a local basis ω, ϑ, π of the 1-forms, with ϑ a basis for the 1-forms in I . We can write each integralelement on which ω has linearly independent components as the solutions of thelinear equations 0 = ϑ, π = aω for some constants a . On an open subset of M , a is a function valued in some vector space. Pull back the 1-forms ϑ, ω, π to M via the map ( m, E ) ∈ M m ∈ M . On M , let ϑ .. = π − aω . The exteriordifferential system I on M generated by ϑ is the prolongation of I . Inductively,let M (1) .. = M , I (1) .. = I , M ( k +1) .. = (cid:0) M ( k ) (cid:1) , I ( k +1) .. = (cid:0) I ( k ) (cid:1) , if defined. Theorem 3 (Cartan–Kuranishi) . If each M ( k ) is a submanifold of the Grassmannbundle over M ( k − , with finitely many connected components, and if each M ( k ) → M ( k − is a submersion, then all but finitely many I ( k ) are involutive. B ac k t o i s o m e t r i c i m m e r s i o n
Returning to our example of isometric immersion of surfaces, we have prolongationgiven by ϑ ϑ ϑ .. = γ γ γ − γ − a bc − a (cid:18) ω ω (cid:19) . Note that 0 = dϑ , dϑ , dϑ modulo ϑ , ϑ , ϑ , so we can forget about them.Calculate the exterior derivatives: d ϑ ϑ ϑ = − Da DbDc − Da ∧ (cid:18) ω ω (cid:19) + tω ∧ ω mod ϑ , . . . , ϑ . where DaDbDc .. = da + ( b + c ) γ − R ω db − aγ + R ω dc − aγ and the torsion is t .. = s − R − K − a − bc. This torsion clearly has to vanish on any 3-dimensional I -integral element, i.e.every 3-dimensional I -integral element lives over the subset of M on which0 = s − R − K − a − bc. To ensure that this subset is a submanifold, we let M ⊂ M be the set of pointswhere this equation is satisfied and at least one of a, b, c is not zero. Clearly M ⊂ M is a submanifold, on which we find Da, Db, Dc linearly dependent. On M , every3-dimensional I -integral element on which ω , ω , γ are linearly independent has s = 2, s = 0 and 2 dimensions of integral elements at each point. Therefore theexterior differential system is in involution: there is an integral manifold through eachpoint of M , and in particular above every point of the surface. The prolongationexposes the hidden necessary condition for existence of a solution: the relation t = 0between the curvature of the ambient space, that of the surface, and the shapeoperator.We won’t prove the elementary: Lemma 1.
Take any smooth 3-dimensional integral manifold X of the linearPfaffian system constructed above. Suppose that on X , = ω ∧ ω ∧ γ . Every pointof X lies in some open subset X ⊂ X so that X is an open subset of the adaptedframe bundle of an isometric immersion P → P of an open subset P ⊂ P . To sum up:
Theorem 4.
Take any surface P with real analytic Riemannian metric, withchosen point p ∈ P and Gauss curvature K . Take any 3-manifold P with realanalytic Riemannian metric, with chosen point p , and a linear isometric injection F : T p P → T p P . Let ν be a unit normal vector to the image of F . Let R be theRicci tensor on that 3-manifold and s the scalar curvature. Pick a nonzero quadraticform q on the tangent plane T p P so that det q = K + R ( ν, ν ) − s . XTERIOR DIFFERENTIAL SYSTEMS 11
Then there is a real analytic isometric immersion f of some neighborhood of p to P , so that f ( p ) = F and so that f induces shape operator q at p . Fo r f u rt h e r i n f o r m at i o n
For proof of the Cartan–Kähler theorem see [2], which we followed very closely,and also the canonical reference work [1] and the canonical textbook [6]. The lasttwo also give proof of the Cartan–Kuranishi theorem. For more on triply orthogonalwebs in Euclidean space, and orthogonal webs in Euclidean spaces of all dimensions,see [3, 4, 7, 8]. For more on isometric immersions and embeddings see [5].13.
L i n e a r i z at i o n
Take an exterior differential system I on a manifold M , and an integral manifold X ⊂ M . Suppose that the flow of a vector field v on M moves X through a family ofintegral manifolds. So the tangent spaces of X are carried by the flow of v throughintegral elements of I . Equivalently, the pullback by that flow of each form in I also vanishes on X . So L v ϑ vanishes on X for each ϑ ∈ I . Problem 8.
Prove that all vector fields v tangent to X satisfy this equation.More generally, suppose that E ⊂ T m M is an integral element of I . If a vectorfield v on M carries E through a family of integral elements, then 0 = L v ϑ | E for each ϑ ∈ I . Take local coordinates x, y , say x = (cid:0) x , x , . . . , x p (cid:1) and y = (cid:0) y , y , . . . , y q (cid:1) ,where E is the graph of dy = 0. We use a multiindex notation where if I = ( i , i , . . . , i m )then dx I = dx i ∧ dx i ∧ · · · ∧ dx i m . Allow the possibility of m = 0, no indices, for which dx I = 1. Let ( − I mean( − m . Lemma 2.
Take an integral element E ⊂ T m M for an exterior differential system I on a manifold M . Take local coordinates x, y , say x = (cid:0) x , x , . . . , x p (cid:1) and y = (cid:0) y , y , . . . , y q (cid:1) , where E is the graph of dy = 0 . For any ϑ ∈ I , if we write ϑ = c IA dx I then L v ϑ | E = ∂v a ∂x j c Ia dx Ij + v a ∂c I ∂y a dx I . Proof.
Expand out ϑ = c IA ( x, y ) dx I ∧ dy A . and v = v j ∂∂x j + v b ∂∂y b . Note that v c I dx I = ( − J v i c JiK dx JK . Commuting with exterior derivative, L v dx i = ∂v i ∂x j dx j + ∂v i ∂y b dy b , L v dy a = ∂v a ∂x j dx j + ∂v a ∂y b dy b . By the Leibnitz rule, L v ϑ = v i ∂c IA ∂x i dx I ∧ dy A + v a ∂c IA ∂y a dx I ∧ dy A + c JiKA dx J ∧ (cid:18) ∂v i ∂x j dx j + ∂v i ∂y b dy b (cid:19) ∧ dx K ∧ dy A + c IBaC dx I ∧ dy B ∧ (cid:18) ∂v a ∂x j dx j + ∂v a ∂y b dy b (cid:19) ∧ dy C . On E , dy = 0 so L v ϑ | E = v i ∂c I ∂x i dx I + v a ∂c I ∂y a dx I + c JiK dx J ∧ ∂v i ∂x j dx j ∧ dx K + c Ia dx I ∧ ∂v a ∂x j dx j . Write the tangent part of v as v = v i ∂∂x i . Let A be the linear map A : E → E given by A ij = ∂v i ∂x j and apply this by derivation to forms on E ,( Aξ )( v , . . . , v k ) = ξ ( Av , v , . . . , v k ) − ξ ( v , Av , v , . . . , nv k ) + . . . . Then L v ϑ | E = v dϑ | E + v a ∂c I ∂y a dx I + Aϑ | E + c Ia dx I ∧ ∂v a ∂x j dx j . Since 0 = ϑ | E = dϑ | E , we find L v ϑ | E = v a ∂c I ∂y a dx I + c Ia dx I ∧ ∂v a ∂x j dx j . (cid:3) In our coordinates, take any submanifold X which is the graph of some functions y = y ( x ). Suppose that the linear subspace E = ( dy = 0) at ( x, y ) = (0 ,
0) is anintegral element. Pullback the forms from the ideal: ϑ | X = X c IA ( x, y ) dx I ∧ ∂y a ∂x j dx j ∧ · · · ∧ ∂y a ‘ ∂x j ‘ dx j ‘ . The right hand side, as a nonlinear first order differential operator on functions y = y ( x ), has linearization ϑ L v ϑ | E . That linearization is applied to sections ofthe normal bundle T M | X /T X , which in coordinates are just the functions v a . Thelinearized operator at the origin of our coordinates depends only on the integralelement E = T m X , not on the choice of submanifold X ⊂ M tangent to E . Problem 9.
Compute the linearization of u xx = u yy + u zz + u x around u = 0 bysetting up this equation as an exterior differential system.Take sections v of the normal bundle of X which vanish at m . Then thelinearization applied to these sections is L v ϑ | E = c Ia dx I ∧ ∂v a ∂x j dx j = Aϑ | E , where A is the linear map A = ∂v a ∂x j , XTERIOR DIFFERENTIAL SYSTEMS 13 the linearization of v around the origin, and Aϑ as usual means the derivation actionof linear maps A on differential forms ϑ : Aϑ ( u , u , . . . , u k ) = ϑ ( Au , u , . . . , u k ) − ϑ ( u , Au , . . . , u k )+ · · ·± ϑ ( u , u , . . . , Au k )if ϑ is a k -form.The leading order terms of the linearization of an exterior differential systemform the tableau : ϑ m ∈ I m , A ∈ E ∗ ⊗ ( T ∗ M/E ) Aϑ m | E , where I m is the set of all values ϑ m of differential forms in I .14. T h e c h a r ac t e r i s t i c va r i e t y
Lemma 3.
Take an integral manifold X of an exterior differential system I on amanifold M . The characteristic variety Ξ x ⊂ P E of the linearization of the exterior differential system at any integral element E isprecisely the set of hyperplanes in E which lie not only in the integral element E but also in some other integral element different from E .Proof. Take a submanifold X tangent to E . Apply the linearization operator L v ϑ | E as above to sections v of the normal bundle of X . We can also apply this operatorformally to sections of the complexified normal bundle. In particular, we computethat for any smooth function f on X , and real constant λe − iλf L e iλf v ϑ = L v ϑ + iλdf ∧ ( v ϑ ) . In particular, the symbol of the linearization is σ ( ξ ) v = ξ ∧ ( v ϑ ) . The linearization of I is just the sum of the linearizations for any spanning set offorms ϑ ∈ I . The characteristic variety Ξ at a point x ∈ X is therefore preciselythe set of ξ ∈ E ∗ for which there is some section v of the normal bundle of X with v ( x ) = 0 and ξ ∧ ( v ϑ ) | E = 0for every ϑ ∈ I . If we look at the hyperplane (0 = ξ ) ⊂ E , the vector v can beadded to that hyperplane to make an integral element enlarging the hyperplane. Sothe characteristic variety Ξ x of I is precisely the set of hyperplanes in E for whichthere is more than one way to extend that hyperplane to an integral element: youcan extend it to become E or to become this other extension containing v . (cid:3) R e f e r e n c e s
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