Unit Tangent Bundles, CR Leaf Spaces, and Hypercomplex Structures
aa r X i v : . [ m a t h . DG ] F e b UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEXSTRUCTURES
CURTIS PORTER
Abstract.
Unit tangent bundles UM of semi-Riemannian manifolds M are shown to be examples ofdynamical Legendrian contact structures, which were defined in recent work [25] of Sykes-Zelenko togeneralize leaf spaces of 2-nondegenerate CR manifolds. In doing so, Sykes-Zelenko extended the classi-fication in Porter-Zelenko [20] of regular, 2-nondegenerate CR structures to those that can be recoveredfrom their leaf space. The present paper treats dynamical Legendrian contact structures associated with2-nondegenerate CR structures which were called strongly regular in Porter-Zelenko, named L-contactstructures . Closely related to Lie-contact structures, L-contact manifolds have homogeneous modelsgiven by isotropic Grassmannians of complex 2-planes whose algebra of infinitesimal symmetries is oneof so ( p + 2 , q + 2) or so ∗ (2 p + 4) for p ≥ q ≥
0. Each 2-plane in the homogeneous model is a split-quaternionic or quaternionic line, respectively, and more general L-contact structures arise on contactmanifolds with hypercomplex structures, unit tangent bundles being a prime example. The Ricci curva-ture tensor of M is used to define the Ricci-shifted
L-contact structure on UM , whose Nijenhuis tensorvanishes when M is conformally flat. In the language of Sykes-Zelenko (for M analytic), such UM isthe leaf space of a 2-nondegenerate CR manifold which is recoverable from UM , providing a new sourceof examples of 2-nondegenerate CR structures. Contents
1. Introduction 22. Homogeneous Models 42.1. Linear Algebra of (Split) Quaternionic Structures 42.2. Split-Quaternionic Lines as Leaves of a 2-Nondegenerate Levi Foliation 52.3. Quaternionic Lines as Leaves of a 2-Nondegenerate Levi Foliation 73. L-contact Manifolds 83.1. Almost (Hyper)CR Structures 83.2. 2-Nondegenerate CR Manifolds and Their Leaf Spaces 103.3. Adapted Coframings 123.4. Split-Quaternionic Structure Equations 143.5. Quaternionic Structure Equations 164. L-contact Structure on Unit Tangent Bundles 174.1. The Unit Tangent Bundle of a Semi-Riemannian Manifold 174.2. The Orthonormal Frame Bundle 194.3. Quaternionic L-contact in Semi-Riemannian Signature ( p + 1 , p ) 21Appendix A. The Homogeneous Model of § Mathematics Subject Classification. Introduction
A CR manifold U whose Levi form L has k -dimensional kernel is foliated by complex submanifoldsof complex dimension k , hence has a local product structure U ≈ U × C k , where U is the leaf space ofthe Levi foliation. For concreteness, we can take U to be a regular level set of a smooth, non-constantfunction ρ : C n + k +1 → R so that T U = ker d ρ contains a corank-1 distribution given by the kernel D = ker ∂ρ ⊂ T U of the holomorphic differential. The CR structure of U is the splitting C D = H ⊕ H of the complexification of D , where H, H are the intersections with C T U of the holomorphic and anti-holomorphic bundles of the ambient space C n + k +1 . The Levi kernel K ⊂ H consists of null directions for L = ∂∂ρ , so that K ⊕ K is the complexified tangent bundle of the leaves of the Levi foliation (we alwaysassume K has constant rank).The Levi foliation of U introduces an equivalence relation on U – two points being equivalent if they liein the same leaf – and the quotient map Q : U → U sends each point to its equivalence class in the (2 n +1)-dimensional leaf space (since our considerations are local, we can assume without further comment thatthe leaf space is a manifold and the quotient map is smooth). A local trivialization U ≈ U × C k adaptedto the Levi foliation is a straightening ([8, § U ; equivalently, U is straightenable if Q ∗ H ⊂ C T U is a well-defined CR structure on the leaf space. In this article weconsider the opposite extreme, where U is ( § U does not inherit a canonicalCR structure from U , but rather a k -(complex)-dimensional family { Q ∗ H e u ⊂ C T u U | e u ∈ Q ( e u ) = u } (1.0.1)of almost -CR structures at each u ∈ U . In this sense, a 2-nondegenerate CR manifold determines a highlynontrivial fibration over its leaf space.Classification of 2-nondegenerate CR manifolds has been an active research program for the pastdecade, motivating new developments in the method of equivalence (see [20], [25], and references therein).In general, the method of equivalence classifies manifolds carrying some geometric structure by realizingthem as curved versions of a homogeneous model in the spirit of E. Cartan’s espaces g´en´eraliz´es [24,Preface]. Successful application of the method to 2-nondegenerate CR manifolds of arbitrary (odd)dimension in [20] required a regularity assumption on the fibers of Q : U → U . Regularity is a stringentrequirement, as it is generically absent in the moduli space of all possible 2-nondegenerate CR structures,but recent work [25] of Sykes-Zelenko generalizing beyond the regular setting showed that (in the pseudo-convex case, and for arbitrary signature in dimensions 7 and 9) it is also generically true that non-regularsymbols do not admit homogeneous models.A key strategy of Sykes-Zelenko is to shift focus from the 2-nondegenerate CR manifold U to its leafspace U , as follows. U has a contact distribution ∆ = Q ∗ D , and L descends to a symplectic form L : C ∆ × C ∆ → C ( T U/ ∆) so that (1.0.1) and its complex conjugate Q ∗ H define two k -dimensionalcomplex submanifolds of the Lagrangian Grassmannian bundle LaGr ( C ∆) → U of L -Lagrangian (i.e., almost-CR ) splittings of C ∆. A dynamical Legendrian contact structure ([25, Def. 2.3]) on a contactmanifold ( U, ∆) is exactly that: a pair of submanifolds of LaGr ( C ∆) that are appropriately related bycomplex conjugation. Working in the analytic category , Sykes-Zelenko pass to the complexification of U in order to show that a 2-nondegenerate CR structure is recoverable from the dynamical Legendriancontact structure on its leaf space, at least under certain hypotheses ([25, Prop. 2.6]) on Lie derivativesalong K which will always be satisfied for our purposes. Though we work in the smooth (real) category,and we are only interested in regular The geometric PDE underlying dynamical Legendrian contact structures is determined at finite jet order, so analyticityis only employed in [25] to analytically continue local coordinate charts on U , because the complex-conjugation map on C U simplifies several constructions on bundles over U . NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 3
For a semi-Riemannian manifold (
M, g ) (where g has at least one positive eigenvalue), the unit tangentbundle of M is U M = { u ∈ T M | g ( u, u ) = 1 } . (1.0.2)The Levi-Civita connection of g determines a splitting T T M = −−→ T M ⊕ T ↑ M into horizontal and verticalbundles canonically isomorphic to T M , and g is lifted to the Sasaki metric ([22]) ˆ g on T M evaluatingon −−→
T M and its orthogonal complement T ↑ M exactly as g evaluates on T M . Just as one identifies T u S n ∼ = u ⊥ ⊂ R n +1 , the vertical part of T u ( U x M ) is the vertical lift of u ⊥ = ker g ( u, · ) ⊂ T x M . On all of T u ( U M ), θ | u = ˆ g ( −→ u , · ) ∈ Ω ( U M ) is a contact 1-form, where −→ u ∈ −−→ T M is the horizontal lift of u ∈ U M .Thus, we have a contact distribution ∆ ⊂ T U M and symplectic form L : C ∆ × C ∆ → C ,∆ = ker θ, L = − i d θ. It remains to appoint a submanifold of
LaGr ( C ∆ u ) for each u ∈ U M , and to this end we returnto 2-nondegenerate CR structures for inspiration. Among regular structures classified in [20], stronglyregular k = rank C K = 1) are modeled on homogeneous spaces whose Liealgebra g of infinitesimal symmetries is a real form of so ( n + 4 , C ). Specifically, if L has signature ( p, q )for p + q = n , g is one of so ( p + 2 , q + 2) or so ∗ (2 p + 4) (the latter is only possible if q = p ). An essentialobservation for the present work is that the dynamical Legendrian contact structure on the leaf space U of a strongly regular 2-nondegenerate CR structure U is generated by a hyper-CR structure ( § U ;i.e., a pair of endomorphisms J, K : ∆ → ∆ satisfying J = − , J ◦ K = − K ◦ J , and K = ε (where is the identity and ε = ± ε = 1is called split-quaternionic while ε = − rank∆ to be even) is quaternionic .Our primary object of study is therefore a contact manifold ( U, ∆) with a dynamical Legendriancontact structure given by a complex curve in the bundle LaGr ( C ∆) which is generated by a hyper-CRstructure on J, K : ∆ → ∆; in short, we call U an L-contact manifold . For U = U M , where thesignature of the semi-Riemannian metric g is ( p + 1 , q ), the standard L-contact structure is generated bythe split-quaternionic structure
J, K : T T M → T T M, J = (cid:20) − (cid:21) , K = (cid:20) (cid:21) on T T M = −−→ T M ⊕ T ↑ M .
An alternative L-contact structure is given by “shifting” the standard one by the Ricci curvature tensorof M (Definition 4.4 in § Theorem. (4.7, § U M to be the leaf space of a2-nondegenerate CR manifold, it is sufficient that M is conformally flat and real-analytic. L-contact structures of the quaternionic variety are available on
U M when the signature of g is ( p + 1 , p ).These are discussed in § §
2, emphasizing the algebraic consequences of a (split) quaternionic structure. This offersa casual encounter with 2-nondegenerate CR geometry before the formal definitions of §
3. Leaf spacesof 2-nondegenerate CR manifolds motivate the definition of L-contact manifolds in § §§ § § § § M, g ) fibers over
U M , realizing the structureequations of §§ U = U M . In particular, this is sufficient to compare the L-contact structureof
U M to the homogeneous models in §
2. Finally, Appendix A exhibits local hypersurface realizationsof the 2-nondegenerate CR and L-contact models of § Here “L” could plausibly refer to Lagrange, Legendre, Levi, Lie, or just leaf , so the reader can choose their favorite.
CURTIS PORTER a unit tangent bundle in § A.2. § A.1 is a brief account of some similarities between L-contact and Liecontact geometry, which was a principal motivation for this work.2.
Homogeneous Models
Linear Algebra of (Split) Quaternionic Structures.
Denote i = √−
1, and recall the Paulimatrices, σ = (cid:20) (cid:21) , σ = (cid:20) (cid:21) , σ = (cid:20) − ii (cid:21) , σ = (cid:20) − (cid:21) . (2.1.1)The R -algebra A will refer to one of(2.1.2) Quaternions : A = H = R { σ , i σ , − i σ , i σ } = (cid:26) (cid:20) w − zz w (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) w, z ∈ C (cid:27) ;Split-Quaternions : A = (cid:0) H = R { σ , σ , σ , i σ } = (cid:26) (cid:20) w zz w (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) w, z ∈ C (cid:27) . Let V be a real vector space and V = V ⊗ C its complexification, so that an R -basis of V is a C -basisof V . An ε -quaternionic structure on V is given by a linear map K : V → V with K = ε (where is theidentity and ε = ±
1) which is extended to V by conjugate-linearity: K ( cv ) = c K v for c ∈ C and v ∈ V . Thecase ε = 1 is called split-quaternionic and the case ε = − quaternionic . Quaternionic structuresmay exist only when dim R V = dim C V is even. Write the external direct sum V ⊕ V as length-2 rowvectors with V -entries so that the right A -module structure given by matrix multiplication (on the right)is well-defined on the real subspace V = (cid:8)(cid:2) v K v (cid:3) | v ∈ V (cid:9) ∼ = V . The latter is an isomorphism of real vector spaces, by which we can say that the A -span of v ∈ V is v A = (cid:26) (cid:2) v K v (cid:3) (cid:20) w εzz w (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) w, z ∈ C (cid:27) . (2.1.3)If v ∧ K v = 0, v A is the complex 2-plane C { v, K v } ∈ Gr ( V ) in the Grassmannian of complex 2-planes in V . Equivalently, v ∧ K v = 0 ⇒ v A is the complex-projective line P ( v + c K v ) ⊂ PV ( c ∈ C ) through P ( v )in the complex-projectivization of V . We also call (2.1.3) a split-quaternionic line ( ε = 1) or quaternionicline ( ε = − A ∈ GL ( V ) on V ⊕ V is given by the standard action on the first summand and theconjugate A on the second, where conjugation on GL ( V ) is determined by conjugation on V with respectto the real subspace V ⊂ V . Evidently, A ( V ) ⊂ V when A K = K A . G ( V , K ) ⊂ GL ( V ) denotes thecorresponding subgroup of symmetries of V .Let V be equipped with a symmetric, nondegenerate bilinear form b ∈ S V ∗ with respect to which K is symmetric, b ( K v , v ) = b ( v , K v ) , v , v ∈ V. Note that K is b -orthogonal when ε = 1, and when ε = − b has split signature. G ( V, b ) ⊂ GL ( V ) is the real orthogonal group specified by the signature of b . The C -bilinear extensionof b to V is denoted b , so that the complex orthogonal group G ( V , b ) is the complexification of G ( V, b ).Antilinearity of K implies b ( K v , v ) = b ( v , K v ) , v , v ∈ V , (2.1.4)which defines a Hermitian form h ( v , v ) on V . Moreover, (2.1.4) shows that b extends naturally to an A -valued form on V , b A ( (cid:2) v K v (cid:3) , (cid:2) v K v (cid:3) ) = √ ε (cid:20) b ( v , v ) b ( v , K v ) b ( K v , v ) b ( K v , K v ) (cid:21) = (cid:20) √ ε b ( v , v ) √ ε h ( v , v ) √ ε h ( v , v ) ε √ ε b ( v , v ) (cid:21) . NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 5
Thus, the symmetries G ( V , b A ) of the pair ( b , K ) on V may be thought of as the intersection of complex-orthogonal G ( V , b ) and unitary symmetries G ( V , h ) of V . Writing g for the Lie algebra of the symmetrygroup G ( V , b A ), we have in the quaternionic case g = so ∗ (dim V ) and in the split-quaternionic case g = so (sig( b )) where sig( b ) is the signature of b on V .2.2. Split-Quaternionic Lines as Leaves of a 2-Nondegenerate Levi Foliation.
For n ≥
1, let V = R n +4 with b ∈ S V ∗ and split-quaternionic K : V → V represented in the standard basis of columnvectors by b = σ ǫ ij σ , K = σ n
00 0 σ , where ǫ ij = (cid:26) ǫ i = ± ≤ i = j ≤ n i = j , p of the ǫ i ’s are 1 q of the ǫ i ’s are − (cid:27) . (2.2.1)To assemble a basis of V = C n +4 adapted to b and h = K t b , begin with ν, K ν ∈ V spanning a totally null2-plane, add to these mutually orthogonal unit vectors υ i ∈ V – i.e., b ( υ i , υ j ) = h ( υ i , υ j ) = ǫ ij – andname ν ′ , K ν ′ the h -duals of ν, K ν . With respect to the ordered basis ν, K ν, υ i , ν ′ , K ν ′ ∈ V , our bilinear andHermitian forms are represented b = σ ǫ ij σ , h = σ ǫ ij σ . (2.2.2)Let B consist of all such adapted bases of V . With the standard basis of V serving as the identity element, B is identified with the Lie group G ( V , b ✁ H ) = O ( p + 2 , q + 2) whose Lie algebra is g = so ( p + 2 , q + 2). Inour representation, g is ( n + 4) × ( n + 4) matrices of the form ς κ i ǫ ij ζ j i ζ κ ς − i ǫ ij ζ j − i ζ η i η i γ ij i ζ i − i ζ i i η − ǫ ij η j − ς − κ − i η − ǫ ij η j − κ − ς , η , γ ij , ζ ∈ R ,ς, κ, η i , ζ i ∈ C ,ǫ ij γ ik + ǫ ik γ ij = 0 . (2.2.3)Here, we’ve used summation convention to write ǫ ij η j , which would otherwise be ǫ i η i for each fixed i .We adhere to the summation convention throughout the paper.The symbols ν, K ν, υ i , ν ′ , K ν ′ will continue to denote the vectors of a general basis in B , as well as thesmooth, V -valued functions on B which map each basis to the specified vector in it. These functions aredifferentiated using the g -valued Maurer-Cartan form on B , represented by (2.2.3) whose matrix entriesnow taken to be 1-forms on B ; e.g.,d ν = ς ⊗ ν + κ ⊗ K ν + η i ⊗ υ i + η ⊗ i ν ′ ∈ Ω ( B , V ) . (2.2.4)The Maurer-Cartan forms themselves are then differentiated according to the Maurer-Cartan equations,(2.2.5) d η = ( ς + ς ) ∧ η + i ǫ ij η i ∧ η j , d η i = ζ i ∧ η + ς ∧ η i − γ ij ∧ η j + κ ∧ η i , d κ = ( ς − ς ) ∧ κ + i ǫ ij ζ i ∧ η j , d γ ij = − γ ik ∧ γ kj , d ς = ζ ∧ η + i ǫ ij η i ∧ ζ j + κ ∧ κ, d ζ i = ζ ∧ η i − γ ij ∧ ζ j − ς ∧ ζ i + κ ∧ ζ i , d ζ = − ( ς + ς ) ∧ ζ − i ǫ ij ζ i ∧ ζ j . CURTIS PORTER
Name the two null cones N b = { v ∈ V | b ( v , v ) = 0 } ⇒ T v N b = { v ∈ V | b ( v , v ) = 0 } , N h = { v ∈ V | h ( v , v ) = 0 } ⇒ T v N h = { v ∈ V | ℜ h ( v , v ) = 0 } , and let N be their intersection, N = N b ∩ N h (dim R N = 2 n + 5) . Over each v ∈ N there are complex subbundles C { v } ⊂ ker h ( v , · ) ⊂ T v N , the latter having real corank one in T N . Let U be the image of N under complex projectivization P : V → CP n +3 , U = P ( N ) , (dim R U = 2 n + 3) , and name its corank-1, complex distribution D P ( v ) = P ∗ ker h ( v , · ) ⊂ T P ( v ) U . (2.2.6) B fibers over N and U via the projections(2.2.7) B → N → U ( ν, K ν, υ i , ν ′ , K ν ′ ) ν P ν, and we have T ν N = C { ν, K ν, υ i } | {z } ker h ( ν, · ) ⊕ R { i ν ′ } for each basis in the fiber of (2.2.7). In this sense B is an adapted (co)frame bundle of U : by composingwith a local section s : U → B , ν ∈ C ∞ ( B , V ) is a local section of the tautological line bundle on U ⊂ PV while K ν, υ i ∈ C ∞ ( B , V ) are vector fields whose P ∗ -images frame the distribution (2.2.6), and i ν ′ ∈ C ∞ ( B , V ) completes a local framing of T U . Dually, with a section s : U → B we can pull back theMaurer-Cartan forms on B to get a local coframing s ∗ η , s ∗ η i , s ∗ κ ∈ Ω ( U , C ) , s ∗ η ∈ Γ( D ⊥ ) , where the latter means that s ∗ η is a section of the annihilator of D in T ∗ U , as implied by (2.2.4). Thefirst of the Maurer-Cartan equations (2.2.5) then shows that D not integrable, as s ∗ d η ≡ i ǫ ij s ∗ η i ∧ s ∗ η j mod s ∗ η . (2.2.8)Indeed, (2.2.8) is a local representation of the Levi form of U , which measures the failure of the sheafΓ( D ) of local sections of D to be closed under the Lie bracket of vector fields.In this setting, the Levi form coincides with the Hermitian form h acting on the vector fields K ν, υ i which locally frame D (henceforth, we elide s in discussing local (co)framings of U ). In particular, Liebrackets of the υ i are no longer sections of D as h ( υ i , υ j ) = ǫ ij = 0. By contrast, the real and imaginaryparts of K ν span a rank-2 subbundle of D , the Levi kernel , which is integrable by virtue of h ( K ν, K ν ) = 0and the Newlander-Nirenberg Theorem. As such, the Levi kernel is tangent to the leaves of a foliationof U by complex curves, the Levi foliation . The foregoing argument is the vector-field equivalent of theobservation thatfor ( ν, K ν, υ i , ν ′ , K ν ′ ) ∈ B , { P ( ν + c K ν ) | c ∈ C } ⊂ U is a complex-projective line.(2.2.9)There is a natural equivalence relation on U : e u ∼ e u ⇐⇒ e u , e u lie in the same line (2.2.9); ( e u , e u ∈ U ) . The leaf space of the Levi foliation of U is the image of the canonical quotient projection Q : U → U / ∼ , U = Q ( U ) (dim R U = 2 n + 1) . NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 7
As we saw in (2.1.3), (2.1.2), the projective line (2.2.9) is the split-quaternionic line ν (cid:0) H ⊂ U , hence U = Gr V , the Grassmannian of totally b , h -null complex 2-planes in V . Complex-scalar multiples of K ν are the fibers of Q : U → U , but local trivializations U ≈ U × C of this fibration necessarily fail topreserve the geometry of U , which we discern from the Maurer-Cartan equations (2.2.5) as follows.It’s clear from (2.2.4) and (2.2.9) that the 1-form κ ∈ Ω ( U , C ) measures the infinitesimal variation of ν ∈ U in the direction of K ν , whose triviality under the Levi form is evidenced by the absence of κ (or κ )in (2.2.8). Subsequent Maurer-Cartan equations readd η i ≡ κ ∧ η i mod { η , η j } , (2.2.10)which shows that U is ; i.e., there is no local diffeomorphism U → U × C whose push-forward preserves the complex structure on D ⊂ T U . In other words, Lie derivatives of (sections of) D with respect to (sections of) the Levi kernel vary the fibers of D along those of Q : U → U in such a waythat Q ∗ D ⊂ T U has no canonical complex structure.We conclude this section by comparing U = P ( N ) and U = Gr V as homogeneous quotients of B ∼ = O ( p + 2 , q + 2). For ( ν, K ν, υ i , ν ′ , K ν ′ ) ∈ B we name the stabilizer subgroups of O ( p + 2 , q + 2):(2.2.11) R ⊂ O ( p + 2 , q + 2) stabilizing the complex line C { ν } ∈ PV , R ⊂ O ( p + 2 , q + 2) stabilizing the complex 2-plane C { ν, K ν } ∈ Gr V , so that U = O ( p + 2 , q + 2) / R and U = O ( p + 2 , q + 2) / R . Roughly illustrated, R = ∗ ∗ ∗ · · · ∗ ∗ ∗ · · · ∗ ∗ ∗ · · · ∗ ... ... ∗ · · · ∗ ∗ ∗ · · · ∗ , R = ∗ ∗ ∗ · · · ∗∗ ∗ ∗ · · · ∗ ∗ · · · ∗ ... ... ∗ · · · ∗ ∗ · · · ∗ ⊂ O ( p + 2 , q + 2) , (2.2.12)whence (2.2.3) shows that the Levi kernel 1-form κ is semibasic for B → U and vertical for
B → U .2.3. Quaternionic Lines as Leaves of a 2-Nondegenerate Levi Foliation.
The discussion in § mutatis mutandis , so we record what mutates. For p ≥
1, let n = 2 p and set V = R n +4 as before, with bilinear form and quaternionic structure b = σ p p σ , K = − i σ − p p − i σ , so that the Lie algebra g = so ∗ (2 p + 4) is represented ς − κ i ζ i i ζ p + i i ζ κ ς i ζ p + i − i ζ i − i ζ η i − η p + i ξ ij ξ ip + j i ζ i i ζ p + i η p + i η i ξ p + ij ξ p + ip + j i ζ p + i − i ζ i i η − η i η p + i − ς κ − i η − η p + i − η i − κ − ς ξ ij , ξ p + ij , ξ ip + j , ξ p + ip + j ∈ C , ≤ i ≤ p,ξ ij + ξ ji = 0 , ξ p + ip + j = ξ ij ,ξ p + ij + ξ p + ji = 0 , ξ ip + j = − ξ p + ij . (2.3.1)We highlight a few of the Maurer-Cartan equations,(2.3.2) d η = ( ς + ς ) ∧ η + i δ ij η i ∧ η j − i δ ij η p + i ∧ η p + j , d η i = ζ i ∧ η + ς ∧ η i − ξ ij ∧ η j − ξ ip + j ∧ η p + j − κ ∧ η p + i , d η p + i = ζ p + i ∧ η + ς ∧ η p + i − ξ p + ij ∧ η j − ξ p + ip + j ∧ η p + j + κ ∧ η i . CURTIS PORTER
In particular, the Levi form (2.2.8) of U matches (2.2.1) for q = p and ǫ i = 1 = − ǫ p + i (with all dueapology, we let i ≤ n in the quaternionic setting). Moreover, the 2-nondegeneracy of U which wasevinced by (2.2.10) for split-quaternionic lines is revealed here byd (cid:20) η i η p + i (cid:21) ≡ (cid:20) − κκ (cid:21) ∧ (cid:20) η i η p + i (cid:21) mod { η , η j , η p + j } . (2.3.3)In the felicitous language of the next section, the discrepancy between (2.2.10) and (2.3.3) is explainedby the fact that the leaf space U of U carries an almost-split-quaternionic structure in § § L-contact Manifolds
Almost (Hyper)CR Structures.
Let U be a smooth manifold. For any fiber bundle E → U , E o denotes the fiber of E over o ∈ U , and Γ( E ) is the sheaf of smooth local sections. If E is a vector bundle, E ∗ is its dual and C E its complexification, with fibers C E o = E o ⊗ R C = E o ⊕ i E o ( i = √− denotes the identity endomorphism field on E or C E .An almost-complex structure on an even-rank distribution D ⊂ T U is an endomorphism field J : D → D satisfying J = − . (3.1.1)An almost-complex structure determines a splitting C D = Λ ⊕ Λ(3.1.2)into ± i -eigenspaces of J ,Λ = { X − i JX | X ∈ D } , Λ = { X + i JX | X ∈ D } ⊂ C D. (3.1.3)For ε = ±
1, an almost- ε -hypercomplex structure is an almost-complex structure as in (3.1.1) along witha second endomorphism field K : D → D satisfying J ◦ K = − K ◦ J, K = ε . (3.1.4)In view of (3.1.3), it is apparent that the C -linear extension K : C D → C D echoes the conjugate -linear ε -quaternionic map K : V → V of § K : Λ → Λ , K : Λ → Λ , K ( Y ) = K ( Y ) ∀ Y ∈ Λ . Thus, almost-( − C Λ = rank D is even. Remark 3.1.
Typically, “almost-(hyper)complex structures” refer to those defined on D = T U for dim U even, and U is a (hyper)complex manifold if the endomorphisms J, K satisfy some additional integrabilityconditions. For proper subbundles D ( T U , the term “CR” takes the place of “complex,” subject to somesmoothness and integrability considerations for D . Regarding definition 3.1.4, we should also mention thatthe standard parlance treats (almost) hypercomplex structures as special cases of (almost) quaternionicstructures (see [6, § ε = − D ∗ ⊗ D which admits local framing by some { J, K, JK } , whereas “hypercomplex” is reservedfor distinguished, global J, K .Let U have odd dimension ≥ D ⊂ T U . D is integrable in theFrobenius sense if its sections are closed under the Lie bracket of vector fields, [Γ( D ) , Γ( D )] ⊂ Γ( D ). The complexified Levi bracket (cf. [6, Def. 3.1.7]) L : C D × C D → C ( T U /D )(3.1.5)measures the failure of D to be integrable,(3.1.6) L ( y , y ) = i [ Y , Y ]( o ) mod C D o y , y ∈ C D o ,Y , Y ∈ Γ( C D ) ,Y ( o ) = y , Y ( o ) = y . NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 9
Because it takes values in the quotient C ( T U /D ), L is tensorial and, up to a choice of local trivialization C ( T U /D ) ≈ C , may be considered a skew-symmetric, C -bilinear form on C D .An almost-CR structure J : D → D is an almost-complex structure (3.1.1) which satisfies the partial-integrability condition, L ( JX, JY ) = L ( X, Y ) ⇐⇒ L ( JX, Y ) + L ( X, JY ) = 0 ∀ X, Y ∈ Γ( D ) . (3.1.7)In terms of (3.1.3), (3.1.7) can be rephrased[Γ(Λ) , Γ(Λ)] ⊂ Γ( C D ) ⇐⇒ L| Λ × Λ = 0 = L| Λ × Λ , (3.1.8)so that (3.1.2) is an L -null splitting. Definition 3.2. A CR structure on an odd-dimensional manifold U with a corank-1 distribution D ⊂ T U is a splitting C D = H ⊕ H satisfying the CR integrability condition:(3.1.9) [Γ( H ) , Γ( H )] ⊂ Γ( H ) . Remark 3.3.
A CR structure determines an almost-CR structure as follows: for X ∈ D , X = ℜ Z forsome Z ∈ H , and we can define JX = −ℑ Z (where ℜ Z = ( Z + Z ), ℑ Z = − i ( Z − Z )). Thus, thedistinction between an almost-CR structure and a CR structure as we’ve defined them – aside from thechoice of labels Λ and H for the i -eigenspaces – is exactly the distinction between partial integrability(3.1.8) and CR integrability (3.1.9), stated in their complex-conjugate forms as[Γ(Λ) , Γ(Λ)] ⊂ Γ(Λ ⊕ Λ) vs [Γ( H ) , Γ( H )] ⊂ Γ( H ) . This difference is quantified by the
Nijenhuis tensor N Λ : Λ ∧ Λ → Λ of Λ: N Λ ( Y , Y ) = [ Y , Y ] mod Λ , Y , Y ∈ Γ(Λ) , which obviously vanishes for H . As discussed in [6, § § U that give rise toa complex-parameter-family of (partially integrable) almost-CR structures on the leaf space of U , so wehope that investing in the distinction now pays off in conceptual clarity later.An almost- ε -hyper-CR structure on a manifold equipped with an almost-CR structure (3.1.7) is an-other endomorphism field K : D → D satisfying K = ε , KJ = − JK, L ( KX, Y ) + L ( X, KY ) = 0 ∀ X, Y ∈ Γ( D ) . (3.1.10)As in § K will be called split-quaternionic when ε = 1 and quaternionic when ε = −
1. To register afinal analogy between K and K , note that (2.1.4) implies the Hermitian identity h ( K v , K v ) = ε h ( v , v ) v , v ∈ V , whereas (3.1.6) and (3.1.10) show L ( KY , KY ) = ε L ( Y , Y ) , Y , Y ∈ C D. (3.1.11) Hyper-CR structures are defined differently in [11], extending the terminology of [13], where “hyper-Hermitian” refersto hypercomplex structures (as in Remark 3.1) that are orthogonal for some metric. Our usage does not contradict that of[11]; in fact, the hyper-CR structure of a unit tangent bundle ( § Let U be a smooth manifold ofdimension 2 m +1 equipped with a CR structure as in Definition 3.2; i.e., a corank-1 subbundle D m ⊂ T U whose complexification splits into the CR subbundle and its complex conjugate (anti-CR) bundle, C D = H ⊕ H (rank C H = rank C H = m ) , satisfying the CR integrability condition (3.1.9). In particular, H and H Levi-null, L| H × H = 0 = L| H × H , where L is the Levi bracket (3.1.5), (3.1.6). The Levi kernel
K ⊂ H is K = { X ∈ H | L ( X, Y ) = 0 ∀ Y ∈ C D } , and we assume rank C K = k > U is foliated bycomplex submanifolds of complex dimension k which are tangent to K ⊕ K , the
Levi-foliation . TheLevi-foliation introduces an equivalence relation on U , e u ∼ e u ⇐⇒ e u , e u lie in the same leaf of the Levi-foliation; ( e u , e u ∈ U ) . The leaf space of U is the image of the canonical quotient map Q : U → U = U / ∼ . U is a smoothmanifold of dimension 2 n + 1 where n = m − k . Whether a bundle or tensor on U descends along Q to bewell-defined on U depends on its behavior subject to the Lie derivative with respect to the Levi kernel;e.g., [Γ( K ) , Γ( C D )] ⊂ Γ( C D ) , hence ∆ = Q ∗ D ⊂ T U is a well-defined contact distribution on the leaf space. Similarly, the Levi form L of U descends to a symplectic form L on C ∆ ∼ = C D/ ( K ⊕ K ).However, the leaf space does not necessarily inherit a CR structure from U , as [Γ( K ) , Γ( H )] Γ( K⊕ H )in general. More precisely, for each e u ∈ U in a given leaf Q ( e u ) = u ∈ U , we have complementary L -Lagrangian subspacesΛ e u = Q ∗ ( H e u ) ∼ = H e u / K e u , Λ e u = Q ∗ ( H e u ) ∼ = H e u / K e u ⊂ C ∆ u , (3.2.1)and integrability of the Levi kernel ensures the Lie derivatives [ X, Y ], [
X, Y ] ( X ∈ Γ( K ) , Y ∈ Γ( H ))descend to well-defined, tensorial operatorsad X e u : ( Λ e u −→ Λ e u Λ e u def −→ , ad X e u : ( Λ e u −→ Λ e u Λ e u def −→ , X e u ∈ K e u , which we have extended trivially to the rest of C ∆ as (3.1.9) obviates the need to consider [ K , H ] or[ K , H ]. U is called straightenable if ad X = 0 ∀ X ∈ K , meaning the subspaces { Λ e u ⊂ C ∆ u | e u ∈ u } are allcanonically identified and thus determine a well-defined CR structure on the leaf space U . The oppositeextreme is ad X = 0 for every nonzero X ∈ K , in which case U has no canonical CR structure and U iscalled .In any case, the maps ad K = { ad X | X ∈ K} and ad K serve to quantify the obstruction to straight-enability of U in a manner that we will exploit geometrically. Varying e u smoothly within a fixed leaf Q ( e u ) = u ∈ U will vary the Lagrangian subspaces (3.2.1) of C ∆ u , sweeping out two submanifolds of theLagrangian-Grassmannian, L u = { Λ e u ⊂ C ∆ u | e u ∈ u } , L u = { Λ e u ⊂ C ∆ u | e u ∈ u } ⊂ LaGr ( C ∆ u ) . (3.2.2)If U is straightenable then L u and L u are singletons, but in the 2-nondegenerate setting, they encode theLevi foliation of U within the bundle LaGr ( C ∆) → U over the leaf space, as we now demonstrate.The Jacobi identity for the Lie bracket of vector fields implies L (ad X ( Y ) , Y ) + L ( Y , ad X ( Y )) = 0 , ∀ X ∈ K e u , Y , Y ∈ Λ e u ⊕ Λ e u . (3.2.3) NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 11 As L -Lagrangian complements, Λ e u and Λ e u and may be regarded as each other’s dual spaces, whencead K e u ⊂ S (Λ e u ) ∗ ∼ = T Λ e u LaGr ( C ∆ u ) ⊂ Hom (Λ e u , Λ e u ) , ad K e u ⊂ S (Λ e u ) ∗ ∼ = T Λ e u LaGr ( C ∆ u ) ⊂ Hom (Λ e u , Λ e u ) . When U is 2-nondegenerate, ad : K → ad K is injective and serves to identify K and K with the tangentbundles of their respective factors in the product space L u × L u = { (Λ e u , Λ e u ) | e u , e u ∈ u } ⊂ LaGr ( C ∆ u ) × LaGr ( C ∆ u ) , which features an involution combining transposition and conjugation, τ : L u × L u → L u × L u (Λ e u , Λ e u ) (Λ e u , Λ e u )whose fixed-point set L u = { (Λ e u , Λ e u ) ∈ L u × L u | e u ∈ u } is a submanifold of LaGr ( C ∆ u ) sitting diagonallyin LaGr ( C ∆ u ) × LaGr ( C ∆ u ). Now τ ∗ : K e u ⊕ K e u → K e u ⊕ K e u is the identity map on T L u , which istherefore ℜ ( K ⊕ K ).The total space L = { L u | u ∈ U } defines a bundle over the leaf space from which U is recoverable ,in the language of [25] (at least under some additional hypotheses on ad K which will always be satisfiedfor our purposes). Generalizing this construction enables us to circumvent U while we investigate thegeometry of U . Definition 3.4. (cf. [25, Def. 2.3]) Let U be a smooth manifold of dimension 2 n + 1. A dynamicalLegendrian-contact structure on U consists of: a contact distribution ∆ ⊂ T U with “Levi form” L : C ∆ × C ∆ → C ( T U/ ∆) given by L ( X, Y ) = i [ X, Y ] mod C ∆ , X, Y ∈ Γ( C ∆); LaGr ( C ∆) → U whose fiber over u ∈ U is LaGr ( C ∆ u ) = { Λ ⊂ C ∆ u | dim C Λ = n, L ( v , v ) = 0 ∀ v , v ∈ Λ } ; the product bundle LaGr ( C ∆) × LaGr ( C ∆) → U with two involutions τ , τ : LaGr ( C ∆ u ) × LaGr ( C ∆ u ) → LaGr ( C ∆ u ) × LaGr ( C ∆ u )given by transposition τ (Λ , Λ ) = (Λ , Λ ) and its composition with complex conjugation withrespect to the real subspace ∆ u ⊂ C ∆ u , τ (Λ , Λ ) = (Λ , Λ ) for Λ , Λ ∈ LaGr ( C ∆ u ); a smooth submanifold L ⊂ LaGr ( C ∆) × LaGr ( C ∆) such that, ∀ u ∈ U , a. L u = L ∩ LaGr ( C ∆ u ) × LaGr ( C ∆ u ) is a complex manifold of complex dimension k ; b. τ ( L u ) ∩ L u = ∅ ; c. τ | L u is the identity map on L u . Remark 3.5.
It is implicit in the definition that we identify C T (Λ , Λ) L u with the subspace of T (Λ , Λ) ( LaGr ( C ∆ u ) × LaGr ( C ∆ u )) ⊂ Hom (Λ , Λ) ⊕ Hom (Λ , Λ)that contains T (Λ , Λ) L as its real subspace with respect to the conjugation operator τ ∗ . Thus, we extendthe rationale for leaf spaces above to obtain a CR splitting of C T (Λ , Λ) L u by taking K and K to be(the complex spans of) the images of the projections T (Λ , Λ) L → Hom (Λ , Λ) and T (Λ , Λ) L → Hom (Λ , Λ),respectively.
Remark 3.6.
The fibers
LaGr ( C ∆ u ) of the Lagrangian-Grassmannian bundle are homogeneous for theaction of the symplectic group Sp ( C ∆ u , L u ) ∼ = Sp (2 n ) ([14]), and dynamical Legendrian contact struc-tures are especially symmetric when the fibers L u are foliated by distinguished curves generated by theaction of Sp (2 n ) ([6, § L u are complex curves ( k = 1 in Def.3.4 ), L is completelydetermined by a local section U → L (i.e., a choice of splitting C ∆ = Λ ⊕ Λ for (Λ , Λ) ∈ L ) along withnonvanishing endomorphism fields a : Λ → Λ, a : Λ → Λ that span K and K as in Remark 3.5. In par-ticular, there is a dynamical Legendrian contact structure naturally associated to any contact manifold ( U, ∆) carrying an almost- ε -quaternionic structure K as in § K , K share a commonfiber coordinate a ∈ C ∞ ( U, C ) with respect to which K = { aK : Λ → Λ } , K = { aK : Λ → Λ } . To compare to [25], note that formulating [25, Def. 2.3] for complex C U (the analytic continuationof real-analytic U ) facilitates the “recovery” process ([25, Rem. 2.5, Prop. 2.6]) of reconstructing 2-nondegenerate U from the dynamical Legendrian contact structure of its leaf space. Since we are notconcerned with recovering 2-nondegenerate CR manifolds, we formulate Definition 3.4 in the smooth(real) category. Still, it is instructive to analyze the definition in light of 2-nondegenerate CR geometry.Condition is automatic if U is the leaf space of a 2-nondegenerate CR manifold; b. and c. ensure thatthe two complex submanifolds (3.2.2) are appropriately non-intersecting and conjugate to one another.Item a. applies to leaf spaces of 2-nondegenerate CR manifolds whose Levi kernel has complex rank k so that the Levi foliation consists of complex k -submanifolds. When k = 1 so that U is foliated bycomplex curves, 2-nondegeneracy is equivalent to non-straightenability (higher nondegeneracy conditionscome into play if k ≥
2; see [1, Ch. 11]), and this case will be our focus.
Definition 3.7.
A dynamical Legendrian contact manifold U as in Definition 3.4 is called L-contact when L u as in Def.3.4 has complex dimension k = 1; for each (Λ , Λ) ∈ L u and ( a , a ) ∈ T (Λ , Λ) L u ⊂ ℜ ( Hom (Λ , Λ) ⊕ Hom (Λ , Λ)), L ( a ( v ) , a ( v )) = εr L ( v , v ) ∀ v , v ∈ Λ , for ε = ± r ∈ R which is zero if and only if a = 0.Item of Definition 3.7, called the “conformal unitary” condition in [19] or “strongly regular” in[20], together with situates L-contact structures within the category modeled on the homogeneousspaces presented in § § ε -quaternionic structure;as Def. 3.7 and (3.1.11) suggest, an almost ε -quaternionic structure can be similarly instrumental.Indeed, [20, Cor. 4.6] assures us that L-contact manifolds are among the favorable dynamical Legendriancontact structures described in Remark 3.6, which validates our continued application of the term split-quaternionic to describe the case ε = 1 of Definition 3.7 and quaternionic for ε = − Adapted Coframings.
We now construct a bundle whose sections are local coframings adaptedto an L-contact structure L → U as in Definition 3.7. At first we consider a local L -Lagrangian splitting C ∆ = Λ ⊕ Λ with (Λ u , Λ u ) ∈ L u for each u ∈ U – i.e., a local section U → L – though the ambiguityassociated with such a selection will eventually be incorporated into the construction. With a Lagrangiansplitting we get an almost-complex structure on ∆, J Λ : ∆ → ∆ given by J Λ | Λ = i , J Λ | Λ = − i , (3.3.1)and C ∆ = Λ ⊕ Λ is a partially-integrable CR structure on U .Let β : B Λ → U be the bundle whose fiber over u ∈ U consists of R -linear isomorphisms adapted tothe almost-complex structure (3.3.1), B Λ u = β − ( u ) = n ϕ : T u U ∼ = −→ R ⊕ C n | ϕ (∆ u ) = C n , ϕ ◦ J Λ = i ϕ o . Writing R ⊕ C n as column vectors, a section θ : U → B Λ is a local coframing (cid:20) θ θ i (cid:21) ∈ Ω (cid:16) U, R ⊕ C n (cid:17) , θ ∈ Γ(∆ ⊥ ) , θ i ∈ Γ(Λ ⊥ ) 1 ≤ i ≤ n, where ∆ ⊥ ⊂ T ∗ U and Λ ⊥ ⊂ C T ∗ U denote the annihilators of ∆ , Λ. In particular, θ is a contact formwhich determines a local matrix representation of the (symplectic) Levi form,d θ ≡ i ℓ ij θ i ∧ θ j mod θ [ L ] = (cid:20) ℓ ij − ℓ ij (cid:21) , ℓ ij = ℓ ji ∈ C ∞ ( B Λ , C ) . (3.3.2) NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 13
We reduce to the subbundle B ⊂ B Λ of coframes that “block-diagonalize” L ,(3.3.3) B u = { ϕ ∈ B Λ u | ℓ ij ( ϕ ) = ǫ ij as in (2.2.1) } . This is a principal bundle β : B → U with structure group G = g = (cid:20) | c | c i c c ij (cid:21) ∈ GL (cid:16) R ⊕ C n (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c , c i , c ij ∈ C c = 0 ǫ ij c ik c jl = ǫ kl , (3.3.4)whose Lie algebra g is matrices of the form g = (cid:20) ξ + ξ ξ i ξ + ξ ij (cid:21) , ǫ ik ξ kj + ǫ kj ξ ki = 0 . (3.3.5)The tautological 1-form λ ∈ Ω ( B , R ⊕ C n ) is λ | ϕ = ϕ ◦ β ∗ . (3.3.6)A section U → B given by θ ∈ Ω ( U, R ⊕ C n ) determines a local trivialization B ≈ G × U with respectto which λ can be written λ = g − β ∗ θ, (3.3.7)left-matrix-multiplication by the inverse of g ∈ C ∞ ( U, G ) in (3.3.4) giving B a right -principal- G action.The C -valued parameters (3.3.4) now serve as fiber coordinates on B , and (3.3.7) reads (cid:20) λ λ i (cid:21) = (cid:20) | c | c i c c ij (cid:21) − β ∗ (cid:20) θ θ j (cid:21) . (3.3.8)Differentiating the local expression (3.3.7) yieldsd λ = − g − d g ∧ λ + g − β ∗ d θ, (3.3.9)with − g − d g ∈ Ω ( B , g ) completing λ to a local coframing of B in decidedly non -canonical fashion.To wit, with (3.3.5) and (3.3.8) we rewrite (3.3.9),d (cid:20) λ λ i (cid:21) = − (cid:20) ξ + ξ ξ i ξ + ξ ij (cid:21) ∧ (cid:20) λ λ j (cid:21) + " i ǫ ij λ i ∧ λ j M ijk λ j ∧ λ k + N ijk λ j ∧ λ k , (3.3.10)and having absorbed what torsion we can into ξ , ξ i , ξ ij ∈ Ω ( B , C ), these 1-forms are still only deter-mined by the structure equations (3.3.10) up to a transformation of the form (cid:20) ξ ξ i (cid:21) (cid:20) ξ ξ i (cid:21) + (cid:20) s s i s (cid:21) (cid:20) λ λ i (cid:21) , s , s i ∈ C ∞ ( B , C ) . (3.3.11)The G -structure equations (3.3.10) are exactly those of the partially integrable CR structure C ∆ =Λ ⊕ Λ, with the (harmonic) torsion coefficients N ijk representing the Nijenhuis tensor of Λ (see Remark3.3). By pulling back an adapted coframing of U along the bundle projection L → U , we generalizethe structure equations (3.3.10) to those of a general Lagrangian splitting of C ∆ in L ; i.e., an arbitrarysection U → L . Here the equivalence problem branches based on the value of ε = ± , and is continued in § § LaGr ( C ∆) → U to an arbitrary Lagrangian splittingsatisfying condition of Definition 3.7, and then reduce to L . Split-Quaternionic Structure Equations.
Over the bundle β : B → U of 1-adapted coframingsof U , we construct ˆ β : ˆ B → B with fibersˆ β − ( ϕ ) = ˆ ϕ : T ϕ B → R ⊕ C n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ ϕ = (cid:2) a (cid:3) ϕ ◦ β ∗ − [ A ] ϕ ◦ β ∗ A ∈ Hom ( C n , C n ) ,A t = A,AA = r ,a = 1 + i r, r ∈ R . (3.4.1)By definition, β ∗ ker ˆ ϕ | C T ϕ B ⊂ C ∆ is a Lagrangian subspace framed by an L -“orthonormal” basis (thatis, a basis so that L is represented as in (3.3.2), (3.3.3)), with trivial fiber coordinates A = 0, r = 0corresponding to ˆ ϕ = ϕ ◦ β ∗ annihilating the original Λ ⊂ C ∆.The tautological 1-form η ∈ Ω ( ˆ B , R ⊕ C n ) is η | ˆ ϕ = ˆ ϕ ◦ ( ˆ β ) ∗ , or in terms of the tautological forms on B , η = (cid:20) η η i (cid:21) , η = ( ˆ β ) ∗ λ , η i = (1 − i r )( ˆ β ) ∗ λ i − a ij ( ˆ β ) ∗ λ j , (3.4.2)for fiber coordinates r ∈ C ∞ ( ˆ B ) , a ij = a ji ∈ C ∞ ( ˆ B , C ) , ǫ ij a ik a jl = r ǫ kl . In particular, ( ˆ β ) ∗ λ i = (1 + i r ) η i + a ij η j , which we plug into (3.3.10) to obtaind η = − ( ξ + ξ ) ∧ η + i ǫ ij η i ∧ η j , ( ˆ β ) ∗ d λ i = − ξ i ∧ η − a ( δ ik ξ + ξ ik ) ∧ η k − ( a ik ξ + a lk ξ il ) ∧ η k + ( a M ikn a nl + N imn a mk a nl ) η k ∧ η l + ( a M iml a mk + a N ikl ) η k ∧ η l + ( | a | M ikl − M imn a ml a nk + 2 a N iml a mk ) η k ∧ η l , where we have recycled the names of the pseudoconnection forms ξ = ( ˆ β ) ∗ ξ , ξ i = ( ˆ β ) ∗ ξ i , ξ ij = ( ˆ β ) ∗ ξ ij .These give way to(3.4.3) d η = ( ς + ς ) ∧ η + i ǫ ij η i ∧ η j , d η i = ζ i ∧ η + ς ∧ η i + ( δ ij r d r − a kj d a ik ) ∧ η j − ζ ij ∧ η j + κ ij ∧ η j + O ikl η k ∧ η l + P ikl η k ∧ η l + Q ikl η k ∧ η l , for 1-forms(3.4.4) ς = − ( ξ + i d r + r ( ξ − ξ )) , ζ ij = (1 + r ) ξ ij − a ik a lj ξ kl ,ζ i = − ((1 − i r ) ξ i − a ij ξ j ) , κ ij = − i a ij d r − (1 − i r )(d a ij + a ij ( ξ − ξ ) + ( a kj ξ ik − a ik ξ kj )) , and torsion coefficients(3.4.5) O ikl = | a | M ikn a nl + a N imn a mk a nl − a a ij ( M jml a mk + a N jkl ) ,P ikl = a ( | a | M ikl − M imn a ml a nk + 2 a N iml a mk ) + a ij ( | a | M jlk − M jmn a mk a nl + 2 a N jmk a ml ) ,Q ikl = a ( M iml a mk + a N ikl ) − a ij a nl ( a M jkn + N jmn a mk ) . To reduce from general coframes of
LaGr ( C ∆) to those of L , we construct a bundle B ⊂ ˆ B whosecoframes (3.4.1) annihilate Λ A = β ∗ ker ˆ ϕ ⊂ C ∆ such that (Λ A , Λ A ) ∈ L . Condition of Definition 3.7states that the fiber coordinates A = a ij of B will be functions of a single complex variable a ∈ C ∞ ( B , C ).In fact, we can bring the coordinates into normal form ([20, Rem. 1.3]) and reduce the structure group NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 15 of B even further: B is not a bundle over B , but over B ⊂ B whose coframes are adapted such thatˆ β : B → B has fibers (3.4.1) with A diagonalized ([20, Cor. 4.6(1)]), a ij = aδ ij , a ∈ C ∞ ( B , C ) , r = | a | . (3.4.6)Revisiting the language of § A = a ij encode the maps ad K : Λ → Λ, which are represented in thestructure equations by the matrix κ ij (3.4.4) (cf. [19, § § L is obtained by complex-conjugating the second argument, so that the pair ( L , A ) can be brought intonormal form by choosing appropriate bases of (Λ , Λ) ∈ L ([20, § ensures that the pair ( L , A ) is strongly non-nilpotent regular so that its normal form specializes from thedescription in [20, Thm. 4.4] to that of [20, Cor. 4.6(1)]. With A diagonalized, κ ij = δ ij κ ∈ Ω ( B , C ) (compare to (2.2.10)) . In particular, after reducing B ⊂ B , the forms ξ ij ∈ Ω ( B , su ( p, q )) as in (3.3.5) are no longer inde-pendent, but satisfy a kj ξ ik − a ik ξ kj = a ( ξ ij − ξ ij ) ≡ { λ , λ i , λ j } , (3.4.7)and we are left with γ ij = ℜ ξ ij ∈ Ω ( B , so ( p, q )) (compare to (2.2.3)) , along with the expansion of (3.4.7), ξ ij − ξ ij = i ( R ij λ + R ijk λ k + R ijk λ k ) for some R ij ∈ C ∞ ( B ) , R ijk ∈ C ∞ ( B , C ) . (3.4.8)As before, we keep the same names of the pseudoconnection forms ξ = ( ˆ β ) ∗ ξ , ξ i = ( ˆ β ) ∗ ξ i , γ ij = ( ˆ β ) ∗ γ ij and torsion coefficients M, N when we pull back along ˆ β : B → B to get(3.4.9) d η = ( ς + ς ) ∧ η + i ǫ ij η i ∧ η j , d η i = ζ i ∧ η + ς ∧ η i − γ ij ∧ η j + κ ∧ η i + O ikl η k ∧ η l + P ikl η k ∧ η l + Q ikl η k ∧ η l , for 1-forms ς = − ( ξ + i d r + r ( ξ − ξ ) + ( a d a − a d a )) , ζ i = − ((1 − i r ) ξ i − aξ i + i R ij ( (1 + 2 r ) η j + aη j )) , (3.4.10)and torsion coefficients(3.4.11) O ikl = − i (1 + 2 r )( a R ikl + aR ikl ) + a | a | M ikl + a a N ikl − a r M ikl − aa N ikl ,P ikl = − i (1 + 2 r )( a R ikl + aR ikl ) + a | a | M ikl − a r M ilk + 2 aa N ikl + a | a | M ilk − ar M ikl + 2 a a N ilk − i a ( a R ikl + aR ikl ) ,Q ikl = aa M ikl + a N ikl − a a M ikl − a N ikl − i a ( a R ikl + aR ikl ) . As in § ς, ζ i , κ ∈ Ω ( B , C ) are not uniquely determinedby the structure equations (3.4.9), but only up to a substitution ςζ i κ ςζ i κ + s s i s s s η η i η i , s , s i , s ∈ C ∞ ( B , C ) . (3.4.12)To complete the construction of an absolute parallelism over U via Cartan’s method of equivalence, onewould pull back the structure equations (3.4.9) to the bundle of all such pseudoconnection forms (3.4.12)over B and differentiate them with the help of the identity d = 0 to determine the exterior derivativesof the tautological forms on the prolonged bundle ([19, § §§ R -valued function s whose corresponding 1-form in the structure equations (2.2.5) of the homogeneousmodel was denoted ζ . Remark 3.8.
We have no need to continue with the prolongation procedure, which was shown toterminate as expected in [20] in the 2-nondegenerate CR case (and this carries over to the general L-contact case by [25]). Indeed, the structure equations (3.4.9) are already in their final form, and aresufficient to compare to the homogeneous model (2.2.5) for our purposes. Namely, in order that (3.4.9)describes an L-contact structure which is locally equivalent to the homogeneous model, it is necessarythat the torsion coefficients (3.4.11) are zero.3.5.
Quaternionic Structure Equations.
We carry out the same sequence of constructions as in § B is normalized tod λ = − ( ξ + ξ ) ∧ λ + i δ ij λ i ∧ λ j − i δ ij λ p + i ∧ λ p + j (cf. § , (3.5.1)and in this section indices range from 1 to p = n . The bundle ˆ β : ˆ B → B has fibersˆ β − ( ϕ ) = ˆ ϕ : T ϕ B → R ⊕ C n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 + r ) ˆ ϕ = ϕ ◦ β ∗ − h − A A i ϕ ◦ β ∗ A ∈ Hom ( C p , C p ) ,A t = A,AA = r , r ∈ R . , and tautological forms η = r ˆ β ∗ λ , η i = r ( ˆ β ∗ λ i + a ij ˆ β ∗ λ p + j ) , η p + i = r ( ˆ β ∗ λ p + i − a ij ˆ β ∗ λ j ) , (3.5.2)for fiber coordinates r ∈ C ∞ ( ˆ B ) , a ij = a ji ∈ C ∞ ( ˆ B , C ) , δ ij a ik a jl = r δ jl . In particular,ˆ β ∗ λ = (1 + r ) η , ˆ β ∗ λ i = η i − a ij η p + j , ˆ β ∗ λ p + i = η p + i + a ij η j , and we pull back (3.3.10) to getd η = ( ς + ς ) ∧ η + i δ ij η i ∧ η j − i δ ij η p + i ∧ η p + j , d η i ≡ ζ i ∧ η + ς ∧ η i + r ) ( a kj d a ik − a ik d a kj ) ∧ η j − ζ ij ∧ η j − ζ ip + j ∧ η p + j + κ ij ∧ η j + κ ip + j ∧ η p + j mod { η ∧ η, η ∧ η, η ∧ η } , d η p + i ≡ ζ p + i ∧ η + ς ∧ η p + i + r ) ( a kj d a ik − a ik d a kj ) ∧ η p + j − ζ p + ij ∧ η j − ζ p + ip + j ∧ η p + j + κ p + ij ∧ η j + κ p + ip + j ∧ η p + j mod { η ∧ η, η ∧ η, η ∧ η } , (having reduced modulo torsion terms), where ς = − (cid:16) r d r + r ( ξ + r ξ ) (cid:17) , ζ i = − ( ξ i + a ij ξ p + j ) , ζ p + i = − ( ξ p + i − a ij ξ j ) ,ξ ij ∈ Ω ( B su ( p, p )) have pulled back to (cid:20) ζ ij ζ ip + j ζ p + ij ζ p + ip j (cid:21) = 11 + r " ξ ij + a ik a lj ξ p + kp + l ξ ip + j − a ik a lj ξ p + kl ξ p + ij − a ik a lj ξ kp + l ξ p + ip + k + a ij a lk ξ jl , and the “ad K ” maps are represented (cid:20) κ ij κ ip + j κ p + ij κ p + ip + j (cid:21) = 11 + r " − a lk ξ ip + l − a ij ξ p + jk d a ij + a ik ( ξ − ξ ) + a lk ξ il − a ij ξ p + jp + k − d a ij − a ik ( ξ − ξ ) − a lk ξ p + ip + l + a ij ξ jk a lk ξ p + il + a ij ξ jp + k . (3.5.3)As in § B ⊂ ˆ B over B ⊂ B consisting of coframes of L such that the fibers of B → B have A diagonalized (3.4.6). With (3.5.3) normalized as in (2.3.3) NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 17 (modulo torsion terms), the pseudoconnection forms ξ ∈ Ω ( B , su ( p, p )) are no longer independent over B , but satisfy (2.3.1) (modulo tautological forms). Modulo torsion terms, the final structure equationson B are equivalent to (2.3.2), with ς = − (cid:16) r d r + r ( ξ + r ξ ) (cid:17) + r ) ( a d a − a d a ) . Here again, the pseudoconnection forms are not uniquely determined by the structure equations, butrequire prolongation before they may be canonically defined by higher order structure equations (see thediscussion at the end of § L-contact Structure on Unit Tangent Bundles
The Unit Tangent Bundle of a Semi-Riemannian Manifold.
Let M be a smooth manifoldand µ : T M → M its tangent bundle; µ ( y ) = x ∀ y ∈ T x M . The pushforward µ ∗ : T T M → T M annihilates the vertical bundle T ↑ M = ker µ ∗ , which is naturally identified with T M as follows. For f ∈ C ∞ ( M ), d f ∈ Ω ( M ) ⊂ C ∞ ( T M ) is regarded as a function on
T M , linear on the fibers, and each Y ∈ Γ( T M ) determines Y ↑ ∈ Γ( T ↑ M ) by their actions as derivations, Y ( f ) = Y ↑ (d f ) . Now suppose M is equipped with an affine connection ∇ : Γ( T M ) → Γ( T ∗ M ⊗ T M ). Each y ∈ T x M can be locally extended to Y ∈ Γ( T M ) so that ∇ X Y = 0 ∀ X ∈ Γ( T M ), which uniquely determines the1-jet of Y : M → T M at x ∈ M . The pushforward Y ∗ : T M → T T M , satisfies µ ∗ ◦ Y ∗ = , hence Y ∗ ( T x M ) ⊂ T y T x M defines the horizontal subbundle −→ T M ⊂ T T M which is complementary to T ↑ M andcanonically isomorphic to T M via µ ∗ ; i.e., each Y ∈ Γ( T M ) determines −→ Y ∈ Γ( −→ T M ) ([6, § J : −→ T M ⊕ T ↑ M → −→ T M ⊕ T ↑ M on T M will satisfy J = − if we set J ( −→ Y ) = Y ↑ , J ( Y ↑ ) = −−→ Y , ∀ Y ∈ T M. (4.1.1)Similarly, K : T T M → T T M with K = and JK = − KJ is given by K ( −→ Y ) = Y ↑ , K ( Y ↑ ) = −→ Y , ∀ Y ∈ T M. (4.1.2)Now suppose (
M, g ) is semi-Riemannian and ∇ is the Levi-Civita connection of g ∈ S T ∗ M . TheSasaki metric ˆ g ∈ S T ∗ T M is defined byˆ g ( −→ X , −→ Y ) = ˆ g ( X ↑ , Y ↑ ) = g ( X, Y ) , ˆ g ( −→ X , Y ↑ ) = 0 , ∀ X, Y ∈ T M, from which it follows immediatelyˆ g ( J ˆ X, J ˆ Y ) = ˆ g ( ˆ X, ˆ Y ) ⇒ ˆ g ( J ˆ X, ˆ Y ) + ˆ g ( ˆ X, J ˆ Y ) = 0 ∀ ˆ X, ˆ Y ∈ T T M. (4.1.3)On the other hand,ˆ g ( K ˆ X, ˆ Y ) = ˆ g ( ˆ X, K ˆ Y ) ⇒ ˆ g ( K ˆ X, J ˆ Y ) + ˆ g ( ˆ X, JK ˆ Y ) = 0 ∀ ˆ X, ˆ Y ∈ T T M. (4.1.4)We depict the triple ˆ g, J, K schematically asˆ g = (cid:20) g g (cid:21) , J = (cid:20) − (cid:21) , K = (cid:20) (cid:21) on T T M = −→ T M ⊕ T ↑ M . (4.1.5)If we suppose further that g has at least one positive eigenvalue, we can consider the unit tangentbundle (1.0.2) of M . For each u ∈ U x M , u ⊥ = ker g ( u, · ) ⊂ T x M is a hyperplane implicitly identifiedwith T u ( U x M ) (as one would for a sphere in Euclidean space). Explicitly, T u U x M = −−−→ T x M ⊕ ( u ⊥ ) ↑ ⊂−→ T M ⊕ T ↑ M and we name the corank-1 distribution of T U M ,(4.1.6) ∆ u = −−→ ( u ⊥ ) ⊕ ( u ⊥ ) ↑ ⊂ T u U M.
Equivalently, ∆ = ker θ for the contact form θ ∈ Ω ( U M ) given by θ | u = ˆ g ( −→ u , · ) . (4.1.7)The endomorphism field J on T M restricts to an almost-complex structure on ∆, whose complexificationsplits(4.1.8) C ∆ = Λ ⊕ Λinto i and − i eigenspaces of J , respectively. Remark 4.1.
The splitting (4.1.8) is the standard CR structure of U M ([26]), in spite of the fact that it’srarely CR-integrable (see Remark 3.3). Indeed, the almost-complex structure (4.1.1) of
T M is integrableif and only if M is flat as a semi-Riemannian manifold ([15] , [10]), so for dim M > M , e.g. Riemannian space forms of sectional curvature 1 ([2]).By definition (4.1.1),Λ u = C n −→ Y − i Y ↑ (cid:12)(cid:12)(cid:12) Y ∈ u ⊥ o , Λ u = C n −→ Y + i Y ↑ (cid:12)(cid:12)(cid:12) Y ∈ u ⊥ o , (4.1.9)and (4.1.5) restricts toˆ g = (cid:20) g | u ⊥ g | u ⊥ (cid:21) , J = (cid:20) i − i (cid:21) , K = (cid:20) i − i (cid:21) on C ∆ = Λ u ⊕ Λ u . (4.1.10)The Levi form of U M is L ( ˆ Y , ˆ Y ) = − i d θ ( ˆ Y , ˆ Y ) ˆ Y , ˆ Y ∈ Γ( C ∆) . Lemma 4.2. L ( ˆ Y , ˆ Y ) = ˆ g ( ˆ Y , J ˆ Y ) ∀ ˆ Y , ˆ Y ∈ Γ( C ∆) .Proof. This follows from (4.1.3), and will be shown in § U M is homothetic to the Sasaki metric; see [3], keeping in mind that (4.1.7) is equivalentto ˆ g ( u ↑ , J · ), and the Levi form is only defined up to a choice of local trivialization C ( T U M/ ∆) ≈ C ,provided here by θ . (cid:3) In light of Lemma 4.2 and (4.1.3), (4.1.4), we offer the following
Definition 4.3. On U M with its contact distribution (4.1.6), the endomorphisms
J, K : ∆ → ∆ definedby restriction of (4.1.1) and (4.1.2) define the standard split-quaternionic structure of the unit tangentbundle of M . The corresponding L-contact structure (see Remark 3.6) is also called standard (cf. Remark4.1).The homogeneous leaf space U presented as split-quaternionic lines in § U M for M = R p +1 ,q (see Appendix A) with its standard L-contact structure.Here, R p +1 ,q is semi-Euclidean space with its diagonalized (flat) metric of signature ( p + 1 , q ). An L-contact manifold is “non-flat” if it is not equivalent to the homogeneous model (see Remark 3.8), whichis measured at lowest order by torsion terms in the structure equations (3.4.9). Similarly, M is non-flatif it is not locally equivalent to R p +1 ,q , which is measured by the Riemann curvature tensor of M . Inorder to relate these two notions of curvature (“non-flatness”) in § g is R ∈ Ω ( M, so ( T M )) , so for X, Y ∈ T x M , R ( X, Y ) : T x M → T x M satisfies g ( R ( X, Y ) y , y ) + g ( y , R ( X, Y ) y ) = 0 , ∀ y , y ∈ T x M. (4.1.11) [15] attributes this to [18], but I was unable to locate a copy of Nagano’s article. Note that these sources only explicitlyrefer to the Riemannian (definite-signature) case. NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 19
In particular, if we set y = y = u , we conclude that R ( · , · ) u : ∧ ( u ⊥ ) → u ⊥ is well-defined, hence wecan extend by conjugate-bilinearity and lift to R ( · , · ) −→ u : Λ u ∧ Λ u → Λ u . (4.1.12)Compare (4.1.12) to the Nijenhuis tensor discussed in Remark 3.3.Contracting R with g yields the Ricci tensor Ric ∈ S T ∗ M , which we use to construct the Ricci-shiftoperator , −→ X
7→ −→ X + Ric( u, X ) X ↑ , X ↑ X ↑ + Ric( u, X ) −→ X ∀ u, X ∈ T x M −→ X , X ↑ ∈ T u U M. (4.1.13)
Definition 4.4. On U M with its contact distribution (4.1.6), the
Ricci-shifted L-contact structure isdefined by application of the Ricci shift operator (4.1.13) to each Lagrangian subspace in the standardL-contact structure (Definition 4.3),Note that if M is Ricci-flat, the Ricci-shifted L-contact structure coincides with the standard one.4.2. The Orthonormal Frame Bundle.
Let W = R n +1 with its standard basis of column vectors e , . . . , e n ∈ W . Indices written in typewriter font range from 0 to n while those in standard font startcounting at 1. The metric h· , ·i ∈ S W ∗ of signature ( p + 1 , q ) is h e i , e j i = ǫ ij as in (2.2.1) , adding ǫ = 1 . ( M, g ) is a semi-Riemannian manifold of signature ( p +1 , q ), p + q = n . Over each x ∈ M , the orthonormalframe bundle π : F → M has fiber π − ( x ) = { φ : W → T x M | h w , w i = g ( φ ( w ) , φ ( w )) ∀ w , w ∈ W } , which carries a right principal action of the orthogonal group O ( W, h· , ·i ) = O ( p +1 , q ) given by composing φ with orthogonal transformations of W . The tautological 1-form ω ∈ Ω ( F , W ) is(4.2.1) ω | φ = φ − ◦ π ∗ , and the Levi-Civita connection form γ ∈ Ω ( F , so ( W, h· , ·i )) is determined by the torsion-free structureequation(4.2.2) d ω = − γ ∧ ω. With respect to the standard basis of W , ω = ω i ⊗ e i for ω i ∈ Ω ( F ) so that(4.2.3) π ∗ g = ǫ ij ω i ⊗ ω j , and (4.2.2) becomes d ω i = − γ ij ∧ ω j , ǫ i γ ij + ǫ j γ ji = 0 , where the latter holds for each fixed i , j (not summed over), reflecting the fact that γ ij is so ( p + 1 , q )-valued. It will be convenient to introduce ˜ γ ij taking values in so ( p + q + 1) viad ω i = − ǫ jk ˜ γ ij ∧ ω k , ǫ jk = ǫ jk , ˜ γ ij + ˜ γ ji = 0 . (4.2.4)The rest of the semi-Riemannian structure equations readd γ ij = − γ ik ∧ γ kj + R ijkl ω k ∧ ω l . (4.2.5)For each 1-form ω i , γ ij ∈ Ω ( F ) in the coframing of F , ∂ ω i , ∂ γ ij ∈ Γ( T F ) will denote their dual vectorfields. In particular, by definition (4.2.1),(4.2.6) ω ( ∂ ω i ) = e i . F fibers over the unit tangent bundle (1.0.2) via the projection π : F →
U M mapping each frame to itsfirst basis vector, π ( φ ) = φ ( e ) ∈ U x M. In light of (4.2.6), π ( φ ) = π ∗ ∂ ω , and since π = µ ◦ π for the basepoint projection µ : T M → M , wehave ( π ) ∗ ∂ ω = −→ u ∈ −→ T M for u = π ( φ ) ∈ U M . Along with (4.2.3), the fact that µ ∗ g | −→ T M = ˆ g | −→ T M thenimplies that the contact form (4.1.7) pulls back to(4.2.7) ( π ) ∗ θ = ω . Moreover, d ω vanishes separately on the following subbundles of ker ω , which map isomorphically tothe summands of (4.1.6),( π ) ∗ ( R { ∂ ω i } ni =1 ) = −−−−→ π ( φ ) ⊥ , ( π ) ∗ (cid:16) R n ∂ ˜ γ i o ni =1 (cid:17) = ( π ( φ ) ⊥ ) ↑ ;name their direct sum ∆ ⊂ T F so that ( π ) ∗ ∆ = ∆. We lift the almost-complex structure J on∆ ⊂ T U M to ∆ to get ( π ) ∗ L -Lagrangian subbundles Λ , Λ ⊂ C ∆ mapping isomorphically under ( π ) ∗ to (4.1.9), Λ = C n ∂ ω i − i ∂ ˜ γ i o , Λ = C n ∂ ω i + i ∂ ˜ γ i o . The lifted almost-complex structure is encoded in 1-forms λ i ∈ Ω ( F , C ) given by λ = ω , λ i = ( ω i − i ˜ γ i ) . (4.2.8)With this coframing, (4.2.4) ( i = 0) becomes(4.2.9) d λ = i ǫ ij λ i ∧ λ j , and for i ≥ λ i = (cid:16) − γ i ∧ ω − γ ij ∧ ω j + i ( γ ij ∧ ˜ γ j − R i k ω k ∧ ω − R i kl ω k ∧ ω l ) (cid:17) = − ( γ i + i R i k ω k ) ∧ λ − γ ij ∧ λ j − i R i kl ( λ k + λ k ) ∧ ( λ l + λ l ) . These are the structure equations (3.3.10) for ξ i = γ i + i R i k ω k , ξ ≡ ξ ij ≡ γ ij (cid:27) mod { i ℜ λ k } N ijk = − i R i jk . (4.2.11) Remark 4.5.
Continuing the discussion of Remarks 3.3 and 4.1, we now observe that the Nijenhuistensor of the standard CR structure on
U M is given, up to scale, by (4.1.12). Note that this is trivialif dim M = 2 – the one component of curvature having been absorbed into the definition (4.2.11) of ξ – reflecting the fact that 3-dimensional CR manifolds are automatically CR-integrable. To reproducethe theorem of [2] that Riemannian space forms can give rise to integrable CR structures on U M , it isconvenient to replace the Riemannian structure equations (4.2.5) with the “model-mutated” version (see[24, Ch.5 § γ ij = − γ ik ∧ γ kj ± ω i ∧ ω j + R ijkl ω k ∧ ω l , whose curvature tensor measures deviation from the structure equations of space forms with nonzerosectional curvature. The resulting equations (4.2.10), (4.2.11) on U M would see the 1-forms ξ i replacedby γ i + i ( R i k ω k ∓ ω i ), and vanishing curvature would ensure CR integrability.The 1-forms (4.2.8) may be considered a pull-back of the tautological form (3.3.6) along a section U M → B of the bundle of coframings which are 1-adapted to (4.1.8). We realize the tautological forms(3.4.2), (3.4.6) of B by setting η = λ , η i = (1 − i r ) λ i − aλ i , which brings the structure equations (4.2.9), (4.2.10) into agreement with (3.4.9) for 1-forms(4.2.12) ς = − i d r + ( a d a − a d a ) + i | z − | η ,κ = − i a d r − (1 − i r )d a − i ( z − ) η − i ( z + ) R k η k ,ζ i = i | z − | η i − i ( z − ) η i − ( z − ) γ i − i ( z + ) R i k ω k , NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 21 and torsion coefficients O ikl = − i z + ( z + ) R i kl , P ikl = − i ( z + ) z + R i kl , Q ikl = − i ( z + ) C i kl , (4.2.13)where we have introduced z ± = (1 − i r ± a ) , R jk = n +1 ǫ mn R njmk , C ijkl = R ijkl − ǫ ik R jl . (4.2.14) Remark 4.6.
The last term in the expression (4.2.12) of κ is exactly the Ricci-shift (4.1.13), anddistinguishes the torsion coefficients (4.2.13) from those of the standard L-contact structure in that Q is given by components the Weyl curvature tensor (4.2.14) rather than the full Riemannian curvaturetensor (like O and P ). One could also absorb the Ricci components of P into κ , but the analogous effortto use ς to absorb the Ricci components of O and/or P fails to be maintain the identity for d η . Theorem 4.7.
In order for the Ricci-shifted L-contact structure of
U M to be the leaf space of a 2-nondegenerate CR manifold, it is sufficient that M is conformally flat and real-analytic.Proof. If U M is the leaf space of a 2-nondegenerate CR structure U , then the Nijenhuis tensor of U isrepresented by the torsion coefficients Q in the structure equations (3.4.9). For the Ricci-shifted L-contactstructure equations (4.2.13), these torsion coefficients are components of the Weyl curvature tensor of M . Because the L-contact structure of U M (standard or Ricci-shifted) arises from a split-quaternionicstructure, it meets the criteria [25, Cor. 2.8] for recoverability, provided M is real-analytic. (cid:3) Quaternionic L-contact in Semi-Riemannian Signature ( p +1 , p ) . ( M, g ) is semi-Riemannian,sig( g ) = ( p + 1 , p ). With the same notation as § ∆ = R n ∂ ω i , ∂ ω p + i , ∂ γ i , ∂ γ p + i o (4.3.1)has the L -Lagrangian splitting C ∆ = Λ ⊕ Λ for Λ = C n ∂ ω i − i ∂ γ i , ∂ ω p + i − i ∂ γ p + i o . The quaternionic structure K : ∆ → ∆ maps K : ∂ ω i ∂ ω p + i ,∂ γ i
7→ − ∂ γ p + i . (4.3.2)The contact form still pulls back to F according to (4.2.7), and the first semi-Riemannian structureequation (4.2.4) is d ω = − γ j ∧ ω j + γ p + j ∧ ω p + j , (4.3.3)so we define the 1-adapted coframing as before, λ = ω , λ i = ( ω i − i γ i ) , λ p + i = ( ω p + i − i γ p + i ) , whereby (4.3.3) becomes (3.5.1) with ξ = 0. The rest of the construction proceeds by analogy to theend of § § § B encodes the induced action of (4.3.2) on Λ , Λ . Details will be included in anupdated pre-print. Appendix A. The Homogeneous Model of § In this appendix, we exhibit a local hypersurface realization of the homogeneous models presented in § ǫ ij = δ ij (2.2.1),the mixed signature case being a straightforward modification of this. Second, we use representations ofthe bilinear and Hermitian forms that differ from (2.2.2) in order to bring the local defining equations in § A.2 into more familiar form. Finally, the local appearance of this model depends on our non-canonicalchoices of these representations. To better explain this last caveat, we take a detour in § A.1 to compareL-contact geometry to a closely related field.A.1.
Lie Contact Geometry.
Strongly regular 2-nondegenerate CR manifolds ([20, Thms 3.2, 5.1])and L-contact manifolds are generalizations – in Cartan’s sense of espaces g´en´eraliz´es [24, Preface] – ofthe homogeneous models presented in §§ O ( n + 2 ,
2) (in the definite signature case ǫ ij = δ ij ) for the stabilizer subgroups (2.2.12) preserving complex lines and 2-planes in C n +4 . In thisrespect these two geometries are closely related to conformal and Lie contact geometry, respectively.Lie contact geometry ([6, § n -spheres of arbitrary radius in R n +1 –including limiting cases of points (zero radius) and oriented hyperplanes (infinite radius) – which can bein “signed” contact (oriented or not) with each other at some common point of tangency. Symmetries ofinterest map these objects to each other while preserving their contact relationships, so we are interestedin rigid motions O ( n + 1) of R n +1 , Lorentz boosts and time-translations of R n +1 , , conformal symmetries O ( n + 2 , / {± } of the conformal compactification S n +1 of R n +1 , and everything else generated by these([7, Thm 3.16], cf. [17, § n + 2)-dimensional Lie quadric Q ⊂ RP n +3 whose symmetries O ( n + 2 , / {± } preserve the real projectivization of the null cone in R n +2 , . Liecontact geometry generalizes the space of null projective lines on Q , the Grassmannian Gr R n +2 , oftotally null, real 2-planes.The representation of O ( n + 2 ,
2) pertinent to conformal and Lie contact geometry differs from thatof § K = (nevertheless, the significance of split-quaternionic structures to Liecontact geometry has been noticed – see [27, § real lines and 2-planes are parabolic in this real representation, hence the Cartan geometries modeled onhomogeneous coset spaces of O ( n +2 ,
2) with respect to these stabilizer subgroups are parabolic geometries– Lorentzian-conformal geometry generalizing Q and Lie contact generalizing Gr R n +2 , , respectively –unlike 2-nondegenerate CR and L-contact geometries.One motivation for the present work was the prominent role of unit tangent bundles as examples of Liecontact manifolds. For a Riemannian manifold ( M, g ), U M (1.0.2) has a natural Lie contact structureincorporating many of the same constructions as in § Gr R n +2 , or the null lines on the Lie quadric Q – is the unit tangent bundle of theRiemannian sphere M = S n +1 . However, it should noted that the Lie contact structure of U M is onlyassociated to the conformal class of the metric g rather than g itself. In other words, flatness of U M as a Lie contact manifold is equivalent to M being conformally flat ([16]), and any such U M is locallyequivalent to the homogeneous model. Similar considerations for L-contact manifolds should be kept inmind while reading § A.2, which presents a local realization of a homogeneous L-contact manifold as aunit tangent bundle.A.2.
The Future Tube as a Unit Tangent Bundle.
The simplest examples of Levi-degenerate CRhypersurfaces which are not straightenable are tube conical surfaces U = C × i R m +1 ⊂ C m +1 where C ⊂ R m +1 is a cone rC ⊂ C ∀ r > § NIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 23
CR geometry (where the nondegenerate part of the Levi form has definite signature) is the tube over thefuture light cone U = ρ − (0) for ρ : C m +1 → R given by2 ρ = ( z + z ) + · · · + ( z m + z m ) − ( z m +1 + z m +1 ) . U is the tube over the future light cone because we impose z m +1 + z m +1 > . (A.2.1)Note that if we take coordinates ( z , . . . , z m +1 , z m +2 ) ∈ C m +2 to lie in an affine subset of CP m +2 ,[1 : z : · · · : z m +1 : z m +2 ] = [ Z : Z : · · · : Z m +1 : Z m +2 ] , then { ρ = 0 } = P ( N b ∩ N h ) where N b , N h ⊂ C m +3 are the null cones of the bilinear and Hermitian forms b ( Z, Z ) = − Z Z m +2 + Z + · · · + Z m − Z m +1 , h ( Z, Z ) = Z Z m +2 + Z Z m +2 + | Z | + · · · + | Z m | − | Z m +1 | . The Levi form of U is L = ∂∂ρ = d z ∧ d z + · · · + d z m ∧ d z m − d z m +1 ∧ d z m +1 . Evidently, ∂ρ ( X ) = 0 = L ( X, X ) for X = ( z + z ) ∂∂z + · · · + ( z m + z m ) ∂∂z m + ( z m +1 + z m +1 ) ∂∂z m +1 , which is the infinitesimal generator of the scaling operator z δ R ( z, z ) z, δ R ( z, z ) = x + · · · + x m +1 , (A.2.2)where we have broken the complex coordinates into their real an imaginary parts z i = x i + i y i . Observe that if ( z , . . . , z m +1 ) ∈ U and c = r + i s ∈ C ( r, s ∈ R , r > cx + i y , . . . , cx m +1 + i y m +1 ) = ( rx + i ( sx + y ) , . . . , rx m +1 + i ( sx m +1 + y m +1 )) ∈ U . (A.2.3)On the real subspace R m +1 ⊂ C m +1 away from the origin, (A.2.3) is just a constant (rescaled) versionof (A.2.2). Thus, the “complex half rays” on U given by all such multiples (A.2.3) are the leaves of theLevi foliation of U . We claim that the leaf space U of U is the unit tangent bundle U M for (
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