Unit Tangent Bundles, Cr Leaf Spaces, and Hypercomplex Structures 3

arXiv:2102.11909v1 [math.DG] 23 Feb 2021 ..TeFtr uea ntTnetBundle Tangent Unit a as Tube Future References The Geometry Contact A.2. Lie A.1. pedxA h ooeeu oe of Model Homogeneous The A. Appendix ..TeUi agn udeo eiReana aiod17 ( Signature Semi-Riemannian in L-contact Quaternionic Manifold Bundle Semi-Riemannian Frame a Orthonormal 4.3. of The Bundle Tangent Bundles Unit 4.2. Tangent The Unit on Structure 4.1. L-contact Equations Structure 4. Quaternionic Equations Structure 3.5. Split-Quaternionic Spaces Leaf Coframings 3.4. Their Adapted and Manifolds CR 3.3. 2-Nondegenerate Structures (Hyper)CR Foliat3.2. Almost Levi 2-Nondegenerate a Manifolds 3.1. of L-contact Levi Leaves 2-Nondegenerate as a Lines 3. of Quaternionic Leaves as Structures Lines 2.3. Quaternionic Split-Quaternionic (Split) of Algebra 2.2. Linear Models 2.1. Homogeneous 2. Introduction 1. 2010 NTTNETBNLS RLA PCS N HYPERCOMPLEX AND SPACES, LEAF CR BUNDLES, TANGENT UNIT ahmtc ujc Classification. Subject Mathematics h efsaeo -odgnrt Rmnfl hc is which manifold CR 2-nondegenerate a of space leaf the aihswhen vanishes uetno of bund tangent tensor unit ture more structures, and hypercomplex respectively, with manifolds line, quaternionic or quaternionic feape f2nneeeaeC structures. CR 2-nondegenerate of examples of whose 2-planes complex of of Grassmannians isotropic by given structures -odgnrt Rsrcue hc eecalled Le were dynamical which treats structures paper CR present The 2-nondegenerate space. 2-nondegenerat leaf regular, their of from In [20] manifolds. Porter-Zelenko CR in 2-nondegenerate define fication of were spaces which leaf structures, generalize contact Legendrian dynamical Abstract. so ( p 2 + lsl eae oLecnatsrcue,Lcnatman L-contact structures, Lie-contact to related Closely . q , nttnetbundles tangent Unit )or 2) + M M sue odfiethe define to used is scnomlyflt ntelnug fSksZlno(for Sykes-Zelenko of language the In flat. conformally is so ∗ (2 p )for 4) + UM 20,5C0 31,53C15. 53C12, 53C10, 32V05, p Ricci-shifted ≥ fsm-imninmanifolds semi-Riemannian of STRUCTURES 1, UTSPORTER CURTIS q Contents § . saShr ude22 Bundle Sphere a as 2.2 ≥ .Ec -ln ntehmgnosmdli split- a is model homogeneous the in 2-plane Each 0. togyregular strongly 1 -otc tutr on structure L-contact Rsrcue otoeta a erecovered be can that those to structures CR e recoverable eea -otc tutrsaieo contact on arise structures L-contact general e en rm xml.TeRcicurva- Ricci The example. prime a being les on o ye-eek xeddteclassi- the extended Sykes-Zelenko so, doing edincnatsrcue soitdwith associated structures contact gendrian nrcn ok[5 fSksZlnoto Sykes-Zelenko of [25] work recent in d ler fifiieia ymtisi one is symmetries infinitesimal of algebra p nPre-eek,named Porter-Zelenko, in 1 + from flshv ooeeu models homogeneous have ifolds M p , oito 5 Foliation o 7 ion r hw ob xmlsof examples be to shown are UM UM 21 ) M hs iehi tensor Nijenhuis whose , rvdn e source new a providing , nltc,such analytic), L-contact UM is 24 22 22 19 17 16 14 12 10 8 8 4 4 2 2 CURTIS PORTER

1. Introduction A CR manifold whose Levi form has k-dimensional kernel is foliated by complex submanifolds of complex dimensionU k, hence has a localL product structure U Ck, where U is the leaf space of the Levi foliation. For concreteness, we can take to be a regularU ≈ level× set of a smooth, non-constant function ρ : Cn+k+1 R so that T = ker dρ Ucontains a corank-1 distribution given by the kernel D = ker ∂ρ T of the→ holomorphicU differential. The CR structure of is the splitting CD = H H of the complexification⊂ U of D, where H, H are the intersections with CTU of the holomorphic and anti-⊕ holomorphic bundles of the ambient space Cn+k+1. The Levi kernel UH consists of null directions for = ∂∂ρ, so that is the complexified tangent bundle of the leavesK⊂ of the Levi foliation (we always Lassume has constantK⊕ K rank). The LeviK foliation of introduces an equivalence relation on – two points being equivalent if they lie in the same leaf – and theU quotient map Q : U sends each pointU to its equivalence class in the (2n+1)- dimensional leaf space (since our considerationsU → are local, we can assume without further comment that the leaf space is a manifold and the quotient map is smooth). A local trivialization U Ck adapted to the Levi foliation is a straightening ([8, §5.2], [12]) if it preserves the CR structureU ≈ of ;× equivalently, U is straightenable if Q∗H CTU is a well-defined CR structure on the leaf space. In this article we Uconsider the opposite extreme,⊂ where is 2-nondegenerate (§3.2), so that U does not inherit a canonical CR structure from , but rather a k-(complex)-dimensionalU family U (1.0.1) Q∗Hue CTuU u Q(u)= u { ⊂ | ∈ } of almost-CR structures at each u U. In this sense, a 2-nondegenerate CR manifold determines a highly nontrivial fibration over its leaf space.∈ e e Classification of 2-nondegenerate CR manifolds has been an active research program for the past decade, 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 realizing them as curved versions of a homogeneous model in the spirit of E. Cartan’s espaces généralizés [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. Regularity is a stringent requirement, as it is generically absent in the moduli space of all possibleU 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-regular symbols do not admit homogeneous models. A key strategy of Sykes-Zelenko is to shift focus from the 2-nondegenerate CR manifold to its leaf U space U, as follows. U has a contact distribution ∆ = Q∗D, and descends to a symplectic form L L : C∆ C∆ C(TU/∆) so that (1.0.1) and its complex conjugate Q∗H define two k-dimensional complex× 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 contact manifold (U, ∆) is exactly that: a pair of submanifolds of LaGr(C∆) that are appropriately related by complex conjugation. Working in the analytic category1, Sykes-Zelenko pass to the complexification of U in order to show that a 2-nondegenerate CR structure is recoverable from the dynamical Legendrian contact structure on its leaf space, at least under certain hypotheses ([25, Prop. 2.6]) on Lie derivatives along which will always be satisfied for our purposes. Though we work in the smooth (real) category, and weK are only interested in regular 2-nondegenerate CR structures, this article aims to illustrate the scope of dynamical Legendrian contact geometry by presenting a large family of examples: unit tangent bundles of semi-Riemannian manifolds. Tangent sphere bundles of Riemannian manifolds are rich with structure. They have been studied extensively as examples of Riemannian manifolds in their own right [4] (or, extrinsically, as Riemannian hypersurfaces [9]) and in their capacity as contact-metric [5], Lie contact [23], and CR manifolds [26, 2].

1The geometric PDE underlying dynamical Legendrian contact structures is determined at ﬁnite jet order, so analyticity is only employed in [25] to analytically continue local coordinate charts on U, because the complex-conjugation map on CU simpliﬁes several constructions on bundles over U. UNIT 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 tangent bundle of M is (1.0.2) UM = u TM g(u,u)=1 . { ∈ | } The Levi-Civita connection of g determines a splitting TTM = −−→TM T ↑M into horizontal and vertical bundles canonically isomorphic to TM, and g is lifted to the Sasaki⊕ metric ([22])g ˆ on TM evaluating on −−→TM and its orthogonal complement T ↑M exactly as g evaluates on TM. Just as one identifies n ⊥ n+1 ⊥ TuS ∼= u R , the vertical part of Tu(UxM) is the vertical lift of u = ker g(u, ) TxM. On all of ⊂ 1 · ⊂ Tu(UM), θ u =g ˆ(−→u , ) Ω (UM) is a contact 1-form, where −→u −−→TM is the horizontal lift of u UM. Thus, we have| a contact· ∈ distribution ∆ TUM and symplectic form∈ L : C∆ C∆ C, ∈ ⊂ × → ∆ = ker θ, L = idθ. − It remains to appoint a submanifold of LaGr(C∆u) for each u UM, and to this end we return to 2-nondegenerate CR structures for inspiration. Among regular structures∈ classified in [20], strongly regular 2-nondegenerate manifolds (with k = rankC = 1) are modeled on homogeneous spaces whose Lie algebra g of infinitesimal symmetries is a real formK of so(n +4, C). Specifically, if has signature (p, q) for p + q = n, g is one of so(p +2, q +2) or so∗(2p + 4) (the latter is only possible ifLq = p). An essential observation for the present work is that the dynamical Legendrian contact structure on the leaf space U of a strongly regular 2-nondegenerate CR structure is generated by a hyper-CR structure (§3.1) on U;

U 2 2 ½ i.e., a pair of endomorphisms J, K : ∆ ∆ satisfying J = ½, J K = K J, and K = ε (where → − ◦ − ◦

½ is the identity and ε = 1). This is the “CR version” of a hypercomplex structure, and the case ε =1 ± 1 is called split-quaternionic while ε = 1 (which requires 2 rank∆ to be even) is quaternionic. Our primary object of study is therefore− a contact manifold (U, ∆) with a dynamical Legendrian contact structure given by a complex curve in the bundle LaGr(C∆) which is generated by a hyper-CR structure on J, K : ∆ ∆; in short, we call U an L-contact2 manifold. For U = UM, where the signature of the semi-Riemannian→ metric g is (p +1, q), the standard L-contact structure is generated by the split-quaternionic structure

−−→TM ½ 0 ½ 0

J, K : TTM TTM, J = − ,K = on TTM = . ½ → ½ 0 0 T ↑⊕M An alternative L-contact structure is given by “shifting” the standard one by the Ricci curvature tensor of M (Definition 4.4 in §4.1), which leads to the article’s main result, Theorem. (4.7, §4.2) In order for the Ricci-shifted L-contact structure of UM to be the leaf space of a 2-nondegenerate CR manifold, it is sufficient that M is conformally flat and real-analytic. L-contact structures of the quaternionic variety are available on UM when the signature of g is (p +1,p). These are discussed in §4.3. Homogeneous models of strongly regular 2-nondegenerate CR manifolds and their leaf spaces are presented in §2, emphasizing the algebraic consequences of a (split) quaternionic structure. This offers a casual encounter with 2-nondegenerate CR geometry before the formal definitions of §3. Leaf spaces of 2-nondegenerate CR manifolds motivate the definition of L-contact manifolds in §3.2. Once defined, the initial steps of Cartan’s method of equivalence indicate how L-contact structures may be classified in §§3.3-3.5. The constructions are similar in flavor to the author’s thesis work ([21], see also [19] and its references), which may be more accessible than [20], but the conclusions of §3.4 and §3.5 fundamentally rely on those of [20] and [25]. The L-contact structure of the unit tangent bundle is the subject of §4. We show in §4.2 that the orthonormal frame bundle of (M,g) fibers over UM, realizing the structure equations of §§3.3-3.5 for U = UM. In particular, this is sufficient to compare the L-contact structure of UM to the homogeneous models in §2. Finally, Appendix A exhibits local hypersurface realizations of the 2-nondegenerate CR and L-contact models of §2.2 in order to explicitly realize the leaf space as

2Here “L” could plausibly refer to Lagrange, Legendre, Levi, Lie, or just leaf, so the reader can choose their favorite. 4 CURTIS PORTER a unit tangent bundle in §A.2. §A.1 is a brief account of some similarities between L-contact and Lie contact geometry, which was a principal motivation for this work.

2. Homogeneous Models 2.1. Linear Algebra of (Split) Quaternionic Structures. Denote i = √ 1, and recall the Pauli matrices, − 1 0 0 1 0 i 1 0 (2.1.1) σ = , σ = , σ = , σ = . 0 0 1 1 1 0 2 i −0 3 0 1 − The R-algebra A will refer to one of w z Quaternions : A = H = R σ0, iσ1, iσ2, iσ3 = − w,z C ; { − } z w ∈ (2.1.2) w z Split-Quaternions : A = H = R σ , σ , σ , iσ = w,z C . { 0 1 2 3} z w ∈

Let V be a real vector space and V = V C its complexiﬁcation, so that an R-basis of V is a C-basis

⊗ 2 ½ of V. An ε-quaternionic structure on V is given by a linear map K : V V with K = ε½ (where is the identity and ε = 1) which is extended to V by conjugate-linearity: K(cv→)= cKv for c C and v V . The case ε = 1 is called± split-quaternionic and the case ε = 1 is called quaternionic. Quaternionic∈ structures∈ − may exist only when dimR V = dimC V is even. Write the external direct sum V V as length-2 row vectors 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 Kv v V = V. V | ∈ ∼ The latter is an isomorphism of real vector spaces, by which we can say that the A-span of v V is ∈ w εz (2.1.3) vA = v Kv w,z C . z w ∈ If v Kv = 0, vA is the complex 2-plane C v, Kv Gr2(V ) in the Grassmannian of complex 2-planes in

V. Equivalently,∧ 6 v Kv =0 vA is the complex-projective{ } ∈ line P(v + cKv) PV (c C) through P(v) in the complex-projectivization∧ 6 ⇒ of V. We also call (2.1.3) a split-quaternionic⊂ line (ε =∈ 1) or quaternionic line (ε = 1). The action− of A GL(V) on V V is given by the standard action on the first summand and the ∈ ⊕ conjugate A on the second, where conjugation on GL(V) is determined by conjugation on V with respect to the real subspace V V. Evidently, A( ) when AK = KA. G(V, K) GL(V) denotes the corresponding subgroup of⊂ symmetries of . V ⊂ V ⊂ Let V be equipped with a symmetric, nondegenerateV bilinear form b S2V ∗ with respect to which K is symmetric, ∈ b(Kv , v )= b(v , Kv ), v , v V. 1 2 1 2 1 2 ∈ Note that K is b-orthogonal when ε = 1, and when ε = 1 it must be that b has split signature. G(V,b) GL(V ) is the real orthogonal group specified by the− signature of b. The C-bilinear extension of 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 (2.1.4) b(Kv , v )= b(v , Kv ), v , v V, 1 2 1 2 1 2 ∈ which defines a Hermitian form h(v1, v2) on V. Moreover, (2.1.4) shows that b extends naturally to an A-valued form on , V b(v1, v2) b(v1, Kv2) √εb(v1, v2) √εh(v1, v2) bA( v1 Kv1 , v2 Kv2 )= √ε = . b(Kv1, v2) b(Kv1, Kv2) √εh(v , v ) ε√εb(v , v ) 1 2 1 2 UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 5

Thus, the symmetries G( ,bA) of the pair (b, K) on V may be thought of as the intersection of complex- orthogonal G(V, b) and unitaryV symmetries G(V, h) of V. Writing g for the Lie algebra of the symmetry ∗ group G( ,bA), we have in the quaternionic case g = so (dim V ) and in the split-quaternionic case g = so(sig(Vb)) 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 = Rn+4 with b S2V ∗ and split-quaternionic K : V V represented in the standard basis of≥ column vectors by ∈ →

0 0 σ1 σ1 0 0

b = 0 ǫij 0 , K = 0 ½n 0 ,     σ1 0 0 0 0 σ1 where    

ǫi = 1 for 1 i = j n p of the ǫi’s are 1 (2.2.1) ǫij = ± ≤ ≤ , . 0 for i = j q of the ǫi’s are 1 6 − To assemble a basis of V = Cn+4 adapted to b and h = Ktb, begin with ν, Kν V spanning a totally null ∈ 2-plane, add to these mutually orthogonal unit vectors υi V – i.e., b(υi,υj ) = h(υi,υj) = ǫij – and ′ ′ ∈ ′ ′ name ν , Kν the h-duals of ν, Kν. With respect to the ordered basis ν, Kν,υi,ν , Kν V, our bilinear and Hermitian forms are represented ∈

0 0 σ1 0 0 σ0 (2.2.2) b = 0 ǫij 0 , h = 0 ǫij 0 .     σ1 0 0 σ0 0 0 Let consist of all such adapted bases of V. With the standard basis of V serving as the identity element, isB identified with the Lie group G( ,b )= O(p +2, q + 2) whose Lie algebra is g = so(p +2, q + 2). In B V H✁ our representation, g is (n + 4) (n +4) matrices of the form × j 0 0 i 0 ς κ iǫij ζ iζ 0 η ,γj , ζ R, j 0 ∈ κ ς iǫijζ 0 iζ  − −  (2.2.3) i i i i i , ς,κ,ηi, ζi C, η η γj iζ iζ  0 j −  ∈ iη 0 ǫij η ς κ   0 − j − −  ǫ γi + ǫ γi =0.  0 iη ǫij η κ ς  ij k ik j  − − − −    j i Here, we’ve used summation convention to write ǫij η , which would otherwise be ǫiη for each fixed i. We adhere to the summation convention throughout the paper. ′ ′ The symbols ν, Kν,υi,ν , Kν will continue to denote the vectors of a general basis in , as well as the smooth, V-valued functions on which map each basis to the specified vector in it. TheseB functions are differentiated using the g-valuedB Maurer-Cartan form on , represented by (2.2.3) whose matrix entries now taken to be 1-forms on ; e.g., B B i 0 ′ 1 (2.2.4) dν = ς ν + κ Kν + η υi + η iν Ω ( , V). ⊗ ⊗ ⊗ ⊗ ∈ B The Maurer-Cartan forms themselves are then differentiated according to the Maurer-Cartan equations, 0 0 i j dη = (ς + ς) η + iǫij η η , ∧ ∧ dηi = ζi η0 + ς ηi γi ηj + κ ηi, ∧ ∧ − j ∧ ∧ i j dκ = (ς ς) κ + iǫij ζ η , − ∧ ∧ dγi = γi γk, (2.2.5) j − k ∧ j 0 0 i j dς = ζ η + iǫij η ζ + κ κ, ∧ ∧ ∧ i dζi = ζ0 ηi γi ζj ς ζi + κ ζ , ∧ − j ∧ − ∧ ∧ 0 0 i j dζ = (ς + ς) ζ iǫijζ ζ . − ∧ − ∧ 6 CURTIS PORTER

Name the two null cones

b = v V b(v , v )=0 Tv b = v V b(v , v)=0 , N { 0 ∈ | 0 0 } ⇒ 0 N { ∈ | 0 } h = v V h(v , v )=0 Tv h = v V h(v , v)=0 , N { 0 ∈ | 0 0 } ⇒ 0 N { ∈ | ℜ 0 } and let be their intersection, N = b h (dimR =2n + 5). N N ∩ N N Over each v there are complex subbundles 0 ∈ N C v ker h(v , ) Tv , { 0}⊂ 0 · ⊂ 0 N the latter having real corank one in T . Let be the image of under complex projectivization P : V CPn+3, N U N → = P( ), (dimR =2n + 3), U N U and name its corank-1, complex distribution

(2.2.6) DP = P∗ ker h(v , ) TP . (v0) 0 · ⊂ (v0)U fibers over and via the projections B N U B→N→U (2.2.7) ′ ′ (ν, Kν,υi,ν , Kν ) ν Pν, 7→ 7→ and we have ′ Tν = C ν, Kν,υi R iν N { } ⊕ { } ker h(ν,·) for each basis in the fiber of (2.2.7). In this sense| {zis an} adapted (co)frame bundle of : by composing with a local section s : , ν C∞( , V)B is a local section of the tautologicalU line bundle on U∞ →B ∈ B PV while Kν,υi C ( , V) are vector fields whose P∗-images frame the distribution (2.2.6), and Uiν′ ⊂ C∞( , V) completes∈ aB local framing of T . Dually, with a section s : we can pull back the Maurer-Cartan∈ B forms on to get a local coframingU U→B B s∗η0,s∗ηi,s∗κ Ω1( , C), s∗η0 Γ(D⊥), ∈ U ∈ where the latter means that s∗η0 is a section of the annihilator of D in T ∗ , as implied by (2.2.4). The first of the Maurer-Cartan equations (2.2.5) then shows that D not integrable,U as ∗ 0 ∗ i ∗ j ∗ 0 (2.2.8) s dη iǫijs η s η mod s η . ≡ ∧ Indeed, (2.2.8) is a local representation of the Levi form of , which measures the failure of the sheaf Γ(D) of local sections of D to be closed under the Lie bracketU of vector fields. In this setting, the Levi form coincides with the Hermitian form h acting on the vector fields Kν,υi which locally frame D (henceforth, we elide s in discussing local (co)framings of ). In particular, Lie U brackets of the υi are no longer sections of D as h(υi,υj)= ǫij = 0. By contrast, the real and imaginary parts of Kν span a rank-2 subbundle of D, the Levi kernel, which6 is integrable by virtue of h(Kν, Kν)=0 and the Newlander-Nirenberg Theorem. As such, the Levi kernel is tangent to the leaves of a foliation of by complex curves, the Levi foliation. The foregoing argument is the vector-field equivalent of the observationU that ′ ′ (2.2.9) for (ν, Kν,υi,ν , Kν ) , P(ν + cKν) c C is a complex-projective line. ∈B { | ∈ }⊂U There is a natural equivalence relation on : U u u u , u lie in the same line (2.2.9); (u , u ). 1 ∼ 2 ⇐⇒ 1 2 1 2 ∈U The leaf space of the Levi foliation of is the image of the canonical quotient projection e e Ue e e e Q : / ,U = Q( ) (dimR U =2n + 1). U→U ∼ U UNIT 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 H , hence Gr0 b h ⊂ U U = 2V, the Grassmannian of totally , -null complex 2-planes in V. Complex-scalar multiples of Kν are the fibers of Q : U, but local trivializations U C of this fibration necessarily fail to preserve the geometry ofU →, which we discern from the Maurer-CartanU ≈ × equations (2.2.5) as follows. It’s clear from (2.2.4) andU (2.2.9) that the 1-form κ Ω1( , C) measures the infinitesimal variation of ν in the direction of Kν, whose triviality under the∈ LeviU form is evidenced by the absence of κ (or κ) in∈U (2.2.8). Subsequent Maurer-Cartan equations read (2.2.10) dηi κ ηi mod η0, ηj , ≡ ∧ { } which shows that is 2-nondegenerate; i.e., there is no local diffeomorphism U C whose pushforward preserves theU complex structure on D T . In other words, Lie derivativesU → of× (sections of) D with respect to (sections of) the Levi kernel vary⊂ theU fibers of D along those of Q : U in such a way U → that Q∗D TU has no canonical complex structure. ⊂ Gr0 We conclude this section by comparing = P( ) and U = 2V as homogeneous quotients of ′ ′ U N = O(p +2, q + 2). For (ν, Kν,υi,ν , Kν ) we name the stabilizer subgroups of O(p +2, q + 2): B ∼ ∈B 1 O(p +2, q + 2) stabilizing the complex line C ν PV, (2.2.11) R ⊂ { } ∈ O(p +2, q + 2) stabilizing the complex 2-plane C ν, Kν Gr0V, R2 ⊂ { } ∈ 2 so that = O(p +2, q + 2)/ and U = O(p +2, q + 2)/ . Roughly illustrated, U R1 R2

0∗ ∗ ∗···∗ ∗ ∗ ∗···∗ 0 ∗ ∗···∗ 0∗ 0∗ ∗···∗ (2.2.12) 1 = ∗ ∗···∗ , 2 = ∗···∗ O(p +2, q + 2), R . .  R . .  ⊂ . .  . .   ∗···∗  ∗···∗ 0  0 0   ∗ ∗···∗  ∗···∗     whence (2.2.3) shows that the Levi kernel 1-form κ is semibasic for and vertical for U. B→U B →

2.3. Quaternionic Lines as Leaves of a 2-Nondegenerate Levi Foliation. The discussion in §2.2 carries over to the quaternionic case mutatis mutandis, so we record what mutates. For p 1, let n =2p and set V = Rn+4 as before, with bilinear form and quaternionic structure ≥ 0 0 0 σ iσ 0 0 0

1 − 2 ½ 0 0 ½p 0 0 0 p 0

b =   , K =  −  , ½ 0 ½p 0 0 0 p 0 0 σ 0 0 0   0 0 0 iσ   1   − 2     so that the Lie algebra g = so∗(2p + 4) is represented

i p+i 0 ς κ iζ iζ iζ 0 ξi , ξp+i, ξi , ξp+i C, 1 i p, κ −ς iζp+i iζi 0 iζ0 j j p+j p+j ∈ ≤ ≤  − −p+i i p+i i i i j i η η ξj ξp+j iζ iζ i p+i (2.3.1) − ξj + ξi =0, ξp j = ξj ,  p+i i p+i p+i p+i i  + η η ξj ξp+j iζ iζ   0 i p+i −   iη 0 η η ς κ  p+i p+j i p+i  − −  ξj + ξi =0, ξp+j = ξj .  0 iη0 ηp+i ηi κ ς  −    − − − − −  We highlight a few of the Maurer-Cartan equations,

0 0 i j p+i p+j dη = (ς + ς) η + iδij η η iδij η η , ∧ ∧ − ∧ (2.3.2) dηi = ζi η0 + ς ηi ξi ηj ξi ηp+j κ ηp+i, ∧ ∧ − j ∧ − p+j ∧ − ∧ dηp+i = ζp+i η0 + ς ηp+i ξp+i ηj ξp+i ηp+j + κ ηi. ∧ ∧ − j ∧ − p+j ∧ ∧ 8 CURTIS PORTER

In particular, the Levi form (2.2.8) of matches (2.2.1) for q = p and ǫi =1= ǫp+i (with all due 1 U − apology, we let i 2 n in the quaternionic setting). Moreover, the 2-nondegeneracy of which was evinced by (2.2.10)≤ for split-quaternionic lines is revealed here by U ηi 0 κ ηi (2.3.3) d − mod η0, ηj , ηp+j . ηp+i ≡ κ 0 ∧ ηp+i { } In the felicitous language of the next section, the discrepancy between (2.2.10) and (2.3.3) is explained by the fact that the leaf space U of carries an almost-split-quaternionic structure in §2.2 and an almost-quaternionic structure in §2.3. U

3. L-contact Manifolds

3.1. Almost (Hyper)CR Structures. Let be a smooth manifold. For any fiber bundle E , Eo denotes the fiber of E over o , and Γ(E) isU the sheaf of smooth local sections. If E is a vector→U bundle, ∗ ∈U E is its dual and CE its complexification, with fibers CEo = Eo R C = Eo iEo (i = √ 1), and ½ denotes the identity endomorphism field on E or CE. ⊗ ⊕ − An almost-complex structure on an even-rank distribution D T is an endomorphism field ⊂ U 2

(3.1.1) J : D D satisfying J = ½. → − An almost-complex structure determines a splitting (3.1.2) CD =Λ Λ ⊕ into i-eigenspaces of J, ± (3.1.3) Λ= X iJX X D , Λ= X + iJX X D CD. { − | ∈ } { | ∈ } ⊂ For ε = 1, an almost-ε-hypercomplex structure is an almost-complex structure as in (3.1.1) along with a second± endomorphism ﬁeld 2

(3.1.4) K : D D satisfying J K = K J, K = ε½. → ◦ − ◦ In view of (3.1.3), it is apparent that the C-linear extension K : CD CD echoes the conjugate-linear ε-quaternionic map K : V V of §2.1 in that → → K : Λ Λ,K :Λ Λ,K(Y )= K(Y ) Y Λ. → → ∀ ∈ 1 Thus, almost-( 1)-hypercomplex structures require that rankCΛ= rankD is even. − 2 Remark 3.1. Typically, “almost-(hyper)complex structures” refer to those defined on D = T for dim even, and is a (hyper)complex manifold if the endomorphisms J, K satisfy some additional integrabilityU U conditions.U For proper subbundles D ( T , the term “CR” takes the place of “complex,” subject to some smoothness and integrability considerationsU for D. Regarding definition 3.1.4, we should also mention that the standard parlance treats (almost) hypercomplex structures as special cases of (almost) quaternionic structures (see [6, §4.1.8], which only pertains to ε = 1), the latter characterized by a 3-dimensional subbundle of D∗ D which admits local framing by some− J,K,JK , whereas “hypercomplex” is reserved for distinguished,⊗ global J, K. { } Let have odd dimension 3 and a smooth, corank-1 distribution D T . D is integrable in the FrobeniusU sense if its sections are≥ closed under the Lie bracket of vector fields,⊂ [Γ(U D), Γ(D)] Γ(D). The complexified Levi bracket (cf. [6, Def. 3.1.7]) ⊂ (3.1.5) : CD CD C(T /D) L × → U measures the failure of D to be integrable,

y ,y CDo, 1 2 ∈ (3.1.6) (y ,y )= i[Y , Y ](o) mod CDo Y1, Y2 Γ(CD), L 1 2 1 2  ∈  Y1(o)= y1, Y2(o)= y2.  UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 9

Because it takes values in the quotient C(T /D), is tensorial and, up to a choice of local trivialization C(T /D) C, may be considered a skew-symmetric,U L C-bilinear form on CD. AnU almost-CR≈ structure J : D D is an almost-complex structure (3.1.1) which satisﬁes the partial- integrability condition, →

(3.1.7) (JX,JY )= (X, Y ) (JX,Y )+ (X,JY )=0 X, Y Γ(D). L L ⇐⇒ L L ∀ ∈ In terms of (3.1.3), (3.1.7) can be rephrased

(3.1.8) [Γ(Λ), Γ(Λ)] Γ(CD) × =0= , ⊂ ⇐⇒ L|Λ Λ L|Λ×Λ so that (3.1.2) is an -null splitting. L Definition 3.2. A CR structure on an odd-dimensional manifold with a corank-1 distribution D T is a splitting CD = H H satisfying the CR integrability condition:U ⊂ U ⊕ (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 = Z for 1 i ∈ ℜ some Z H, and we can define JX = Z (where Z = 2 (Z + Z), Z = 2 (Z Z)). Thus, the distinction∈ between an almost-CR structure−ℑ and a CRℜ structure as we’veℑ defined− them− – aside from the choice 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 of Λ: Λ ∧ → N (Y , Y ) = [Y , Y ] mod Λ, Y , Y Γ(Λ), Λ 1 2 1 2 1 2 ∈ which obviously vanishes for H. As discussed in [6, §4.2.4] (in the Levi-nondegenerate case), there is no need to exclude partially-integrable structures from the CR category, as they can simply be considered “CR structures with torsion” – the harmonic component of which is the Nijenhuis tensor – and the usual classification scheme of Cartan’s method of equivalence still applies. We choose to name and label them differently because in §3.2 we consider Levi-degenerate (CR integrable) structures on that give rise to a complex-parameter-family of (partially integrable) almost-CR structures on the leafU space of , so we hope that investing in the distinction now pays off in conceptual clarity later. U

An almost-ε-hyper-CR structure3 on a manifold equipped with an almost-CR structure (3.1.7) is another endomorphism ﬁeld K : D D satisfying → 2

(3.1.10) K = ε½, KJ = JK, (KX,Y )+ (X,KY )=0 X, Y Γ(D). − L L ∀ ∈ As in §2.1, K will be called split-quaternionic when ε = 1 and quaternionic when ε = 1. To register a ﬁnal analogy between K and K, note that (2.1.4) implies the Hermitian identity −

h(Kv , Kv )= εh(v , v ) v , v V, 1 2 1 2 1 2 ∈ whereas (3.1.6) and (3.1.10) show

(3.1.11) (KY ,KY )= ε (Y , Y ), Y , Y CD. L 1 2 L 1 2 1 2 ∈

3Hyper-CR structures are deﬁned diﬀerently in [11], extending the terminology of [13], where “hyper-Hermitian” refers to 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 (§4.1) is orthogonal for the Sasaki metric. 10 CURTIS PORTER

3.2. 2-Nondegenerate CR Manifolds and Their Leaf Spaces. Let be a smooth manifold of dimension 2m+1 equipped with a CR structure as in Deﬁnition 3.2; i.e., a corank-1U subbundle D2m T whose complexiﬁcation splits into the CR subbundle and its complex conjugate (anti-CR) bundle,⊂ U

CD = H H (rankCH = rankCH = m), ⊕ satisfying the CR integrability condition (3.1.9). In particular, H and H Levi-null,

H×H =0= , L| L|H×H where is the Levi bracket (3.1.5), (3.1.6). The Levi kernel H is L K⊂ = X H (X, Y )=0 Y CD , K { ∈ | L ∀ ∈ } and we assume rankC = k > 0 is constant. By the Newlander-Nirenberg Theorem, is foliated by complex submanifoldsK of complex dimension k which are tangent to , the Levi-foliationU . The Levi-foliation introduces an equivalence relation on , K ⊕ K U u u u , u lie in the same leaf of the Levi-foliation; (u , u ). 1 ∼ 2 ⇐⇒ 1 2 1 2 ∈U The leaf space of is the image of the canonical quotient map Q : U = / . U is a smooth manifolde of dimensione U 2n +1 wheree en = m k. Whether a bundle or tensorU → on descendsU ∼e e along Q to be well-defined on U depends on its behavior− subject to the Lie derivative with respectU to the Levi kernel; e.g., [Γ( ), Γ(CD)] Γ(CD), K ⊂ hence ∆ = Q∗D TU is a well-defined contact distribution on the leaf space. Similarly, the Levi form ⊂ of descends to a symplectic form L on C∆ = CD/( ). L U ∼ K⊕ K However, the leaf space does not necessarily inherit a CR structure from , as [Γ( ), Γ(H)] Γ( H) in general. More precisely, for each u in a given leaf Q(u) = u UU, we haveK complementary6⊂ K⊕ L- Lagrangian subspaces ∈ U ∈ e e (3.2.1) Λue = Q∗(Hue) = Hue/ ue, Λue = Q∗(H ue) = Hue/ ue C∆u, ∼ K ∼ K ⊂ and integrability of the Levi kernel ensures the Lie derivatives [X, Y ], [X, Y ] (X Γ( ), Y Γ(H)) descend to well-defined, tensorial operators ∈ K ∈

Λue Λue Λue Λue ad e : −→ , ad : −→ , Xe e, Xu def Xue def u u Λue 0 Λue 0 ∈ K ( −→ ( −→ which we have extended trivially to the rest of C∆ as (3.1.9) obviates the need to consider [ ,H] or K [ , H]. is called straightenable if adX =0 X , meaning the subspaces Λue C∆u u u are all canonicallyK U identified and thus determine a well-defined∀ ∈ K CR structure on the{ leaf⊂ space U|. The∈ opposite} extreme is adX = 0 for every nonzero X , in which case U has no canonical CR structuree and is called 2-nondegenerate6 . ∈ K U In any case, the maps adK = adX X and adK serve to quantify the obstruction to straightenability of in a manner that{ we will| exploit∈ K} geometrically. Varying u smoothly within a fixed leaf U Q(u)= u U will vary the Lagrangian subspaces (3.2.1) of C∆u, sweeping out two submanifolds of the ∈ Lagrangian-Grassmannian, e e (3.2.2) Lu = Λue C∆u u u , Lu = Λue C∆u u u LaGr(C∆u). { ⊂ | ∈ } { ⊂ | ∈ } ⊂ If is straightenable then L and L are singletons, but in the 2-nondegenerate setting, they encode the u e u e LeviU foliation of within the bundle LaGr(C∆) U over the leaf space, as we now demonstrate. The Jacobi identityU for the Lie bracket of vector→ fields implies

(3.2.3) L(ad (Y ), Y )+ L(Y , ad (Y ))=0, X ue, Y , Y Λue Λue. X 1 2 1 X 2 ∀ ∈ K 1 2 ∈ ⊕ UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 11

As L-Lagrangian complements, Λue and Λue and may be regarded as each other’s dual spaces, whence 2 ∗ LaGr ad S (Λue) = TΛ e (C∆u) Hom(Λue, Λue), Kue ⊂ ∼ u ⊂ 2 ∗ LaGr adK e S (Λue) = T (C∆u) Hom(Λue, Λue). u ⊂ ∼ Λue ⊂ When is 2-nondegenerate, ad : adK is injective and serves to identify and with the tangent bundlesU of their respective factorsK in → the product space K K

Lu Lu = (Λue , Λue ) u , u u LaGr(C∆u) LaGr(C∆u), × { 1 2 | 1 2 ∈ }⊂ × which features an involution combining transposition and conjugation, e e τ : Lu Lu Lu Lu × → × (Λue , Λue ) (Λue , Λue ) 1 2 7→ 2 1 whose fixed-point set Lu = (Λue, Λue) Lu Lu u u is a submanifold of LaGr(C∆u) sitting diagonally { ∈ × | ∈ } in LaGr(C∆u) LaGr(C∆u). Now τ ∗ : ue ue ue ue is the identity map on T Lu, which is × K 1 ⊕ K 2 → K 2 ⊕ K 1 therefore ( ). ℜ K⊕ K e The total space L = Lu u U defines a bundle over the leaf space from which is recoverable, { | ∈ } U in the language of [25] (at least under some additional hypotheses on adK which will always be satisfied for our purposes). Generalizing this construction enables us to circumvent while we investigate the geometry of U. U Definition 3.4. (cf. [25, Def. 2.3]) Let U be a smooth manifold of dimension 2n +1. A dynamical Legendrian-contact structure on U consists of: 1. a contact distribution ∆ TU with “Levi form” L : C∆ C∆ C(TU/∆) given by ⊂ × → L(X, Y )= i[X, Y ] mod C∆, X,Y Γ(C∆); ∈ 2. LaGr(C∆) U whose fiber over u U is → ∈ LaGr(C∆u)= Λ C∆u dimC Λ= n, L(v , v )=0 v , v Λ ; { ⊂ | 1 2 ∀ 1 2 ∈ } 3. 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(Λ1, Λ2) =(Λ2, Λ1) and its composition with complex conjugation with respect to the real subspace ∆u C∆u, τ(Λ1, Λ2) = (Λ2, Λ1) for Λ1, Λ2 LaGr(C∆u); 4. a smooth submanifold L LaGr(⊂C∆) LaGr(C∆) such that, u U, ∈ ⊂ × ∀ ∈ a. Lu = L LaGr(C∆u) LaGr(C∆u) is a complex manifold of complex dimension k; ∩ × b. τ(Lu) Lu = ∅; ∩ c. τ L is the identity map on Lu. | u Remark 3.5. It is implicit in the definition that we identify CT(Λ,Λ)Lu with the subspace of

T (LaGr(C∆u) LaGr(C∆u)) Hom(Λ, Λ) Hom(Λ, Λ) (Λ,Λ) × ⊂ ⊕ that contains T(Λ,Λ)L as its real subspace with respect to the conjugation operator τ ∗. Thus, we extend the rationale for leaf spaces above to obtain a CR splitting of CT Lu by taking and to be (Λ,Λ) K K (the complex spans of) the images of the projections T L Hom(Λ, Λ) and T L Hom(Λ, Λ), (Λ,Λ) → (Λ,Λ) → respectively.

Remark 3.6. The fibers LaGr(C∆u) of the Lagrangian-Grassmannian bundle are homogeneous for the action of the symplectic group Sp(C∆u, Lu) ∼= Sp(2n) ([14]), and dynamical Legendrian contact structures are especially symmetric when the fibers Lu are foliated by distinguished curves generated by the action of Sp(2n) ([6, §1.4.11]). When such Lu are complex curves (k = 1 in Def.3.4 4.a.), L is completely determined by a local section U L (i.e., a choice of splitting C∆=Λ Λ for (Λ, Λ) L) along with → ⊕ ∈ nonvanishing endomorphism fields a : Λ Λ, a :Λ Λ that span and as in Remark 3.5. In particular, there is a dynamical Legendrian→ contact structure→ naturallyK associatedK to any contact manifold 12 CURTIS PORTER

(U, ∆) carrying an almost-ε-quaternionic structure K as in §3.1, whose bundles , share a common fiber coordinate a C∞(U, C) with respect to which K K ∈ = aK : Λ Λ , = aK :Λ Λ . K { → } K { → } To compare to [25], note that formulating [25, Def. 2.3] for complex CU (the analytic continuation of real-analytic U) facilitates the “recovery” process ([25, Rem. 2.5, Prop. 2.6]) of reconstructing 2- nondegenerate from the dynamical Legendrian contact structure of its leaf space. Since we are not concerned withU 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 4. is automatic if U is the leaf space of a 2-nondegenerate CR manifold; b. and c. ensure that the 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 is foliated by complex curves, 2-nondegeneracy is equivalent to non-straightenability (higher nondegeneracyU conditions come 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

1. Lu as in Def.3.4 4. has complex dimension k = 1; 2. for each (Λ, Λ) Lu and (a, a) T Lu (Hom(Λ, Λ) Hom(Λ, Λ)), ∈ ∈ (Λ,Λ) ⊂ℜ ⊕ L(a(v ), a(v )) = εr2L(v , v ) v , v Λ, 1 2 1 2 ∀ 1 2 ∈ for ε = 1 and some r R which is zero if and only if a = 0. ± ∈ Item 2. of Deﬁnition 3.7, called the “conformal unitary” condition in [19] or “strongly regular” in [20], together with 1. situates L-contact structures within the category modeled on the homogeneous spaces presented in §2.2 and §2.3. There, the L-contact structure arose from an ε-quaternionic structure; as Def. 3.7 2. 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 Legendrian contact 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 2. and quaternionic for ε = 1. −

3.3. Adapted Coframings. We now construct a bundle whose sections are local coframings adapted to an L-contact structure L U as in Deﬁnition 3.7. At ﬁrst we consider a local L-Lagrangian splitting → C∆=Λ Λ with (Λu, Λu) Lu for each u U – i.e., a local section U L – though the ambiguity associated⊕ with such a selection∈ will eventually∈ be incorporated into the construction.→ With a Lagrangian

splitting we get an almost-complex structure on ∆, ½ (3.3.1) J : ∆ ∆ given by J = i½, J = i , Λ → Λ|Λ Λ|Λ − and C∆=Λ Λ is a partially-integrable CR structure on U. ⊕ Let β : Λ U be the bundle whose ﬁber over u U consists of R-linear isomorphisms adapted to the almost-complexB → structure (3.3.1), ∈ ∼ −1 = n n = β (u)= ϕ : TuU R C ϕ(∆u)= C , ϕ J = iϕ . BΛu −→ ⊕ | ◦ Λ Writing R Cn as column vectors,n a section θ : U is a local coframing o ⊕ →BΛ θ0 R ⊥ Ω1 U, ⊕ , θ0 Γ(∆⊥), θi Γ(Λ ) 1 i n, θi ∈ Cn ∈ ∈ ≤ ≤ ⊥ where ∆⊥ T ∗U and Λ CT ∗U denote the annihilators of ∆, Λ. In particular, θ0 is a contact form which determines⊂ a local matrix⊂ representation of the (symplectic) Levi form,

0 i j 0 0 ℓij ∞ (3.3.2) dθ iℓijθ θ mod θ [L]= , ℓij = ℓji C ( Λ, C). ≡ ∧ ℓij 0 ∈ B − UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 13

We reduce to the subbundle 1 of 1-adapted coframes that “block-diagonalize” L, BΛ ⊂BΛ 1 (3.3.3) = ϕ ℓij (ϕ)= ǫij as in (2.2.1) . BΛu { ∈BΛu | } This is a principal bundle β : 1 U with structure group BΛ → c ,ci,ci C 2 R 0 j c0 0 ∈ ⊕ c =0 (3.3.4) G1 = g1 = | i| i GL n 0 ,  c c0c ∈ C j6  j ǫ ci c = ǫ  ij k l kl 

whose Lie algebra g1 is matrices of the form  ξ + ξ 0 k (3.3.5) g = 0 0 , ǫ ξk + ǫ ξ =0. 1 ξi ξ + ξi ik j kj i 0 j The tautological 1-form λ Ω1( 1 , R Cn) is ∈ BΛ ⊕ (3.3.6) λ ϕ = ϕ β∗. | ◦ 1 1 n 1 A section U Λ given by θ Ω (U, R C ) determines a local trivialization Λ G1 U with respect to which λ can→B be written ∈ ⊕ B ≈ ×

−1 ∗ (3.3.7) λ = g1 β θ,

∞ 1 left-matrix-multiplication by the inverse of g C (U, G1) in (3.3.4) giving Λ a right-principal-G1 action. The C-valued parameters (3.3.4) now serve as∈ ﬁber coordinates on 1 , andB (3.3.7) reads BΛ

0 2 −1 0 λ c0 0 ∗ θ (3.3.8) i = | i| i β j . λ c c0c θ j Diﬀerentiating the local expression (3.3.7) yields

(3.3.9) dλ = g−1dg λ + g−1β∗dθ, − 1 1 ∧ 1 −1 1 1 1 with g1 dg1 Ω ( Λ, g1) completing λ to a local coframing of Λ in decidedly non-canonical fashion. To wit,− with (3.3.5)∈ andB (3.3.8) we rewrite (3.3.9), B

0 0 i j λ ξ + ξ 0 λ iǫijλ λ (3.3.10) d = 0 0 + , i i i j i j k ∧ i j k λ − ξ ξ0 + ξj ∧ λ "M λ λ + N λ λ # jk ∧ jk ∧ and having absorbed what torsion we can into ξ , ξi, ξi Ω1( 1 , C), these 1-forms are still only deter- 0 j ∈ BΛ mined by the structure equations (3.3.10) up to a transformation of the form

0 ξ0 ξ0 s0 0 λ i ∞ 1 (3.3.11) i i + i i , s0,s C ( Λ, C). ξ 7→ ξ s s0 λ ∈ B

The G1-structure equations (3.3.10) are exactly those of the partially integrable CR structure C∆= Λ Λ, with the (harmonic) torsion coefficients N i representing the Nijenhuis tensor of Λ (see Remark ⊕ jk 3.3). By pulling back an adapted coframing of U along the bundle projection L U, we generalize the structure equations (3.3.10) to those of a general Lagrangian splitting of C∆ in→L; i.e., an arbitrary section U L. Here the equivalence problem branches based on the value of ε = 1 in Definition 3.7 2., and is→ continued in §3.4 for the split-quaternionic case and §3.5 for the quaternionic± case. In both cases, we first pull back adapted coframings along LaGr(C∆) U to an arbitrary Lagrangian splitting satisfying condition 2. of Definition 3.7, and then reduce to L.→ 14 CURTIS PORTER

3.4. Split-Quaternionic Structure Equations. Over the bundle β : 1 U of 1-adapted coframings BΛ → of U, we construct βˆ : ˆ1 1 with ﬁbers B →BΛ A Hom(Cn, Cn), ∈ t ˆ−1 1 n 1 0 0 0 A = A,

(3.4.1) β (ϕ)= ϕˆ : Tϕ Λ R C ϕˆ = 0 a0 ½ ϕ β∗ [ 0 A ] ϕ β∗  2  B → ⊕ ◦ − ◦ AA = r ½,      a0 =1+ ir, r R.  ∈    By definition, β∗ kerϕ ˆ CT B1 C∆ is a Lagrangian subspace framed by an L-“orthonormal” basis (that  | ϕ Λ ⊂   is, a basis so that L is represented as in (3.3.2), (3.3.3)), with trivial fiber coordinates A = 0, r = 0 corresponding toϕ ˆ = ϕ β∗ annihilating the original Λ C∆. ◦ ⊂ The tautological 1-form η Ω1( ˆ1 , R Cn) is ∈ BΛ ⊕ η ϕ =ϕ ˆ (βˆ)∗, | ˆ ◦ or in terms of the tautological forms on 1 , BΛ η0 j (3.4.2) η = , η0 = (βˆ)∗λ0, ηi = (1 ir)(βˆ)∗λi ai (βˆ)∗λ , ηi − − j for fiber coordinates ∞ 1 i j ∞ 1 i j 2 r C ( ˆ ), a = a C ( ˆ , C), ǫij a a = r ǫkl. ∈ B j i ∈ B k l In particular, ˆ ∗ i i i j (β) λ = (1+ ir)η + aj η , which we plug into (3.3.10) to obtain 0 0 i j dη = (ξ + ξ ) η + iǫijη η , − 0 0 ∧ ∧ (βˆ)∗dλi = ξi η0 a (δi ξ + ξi ) ηk (ai ξ + al ξi) ηk − ∧ − 0 k 0 k ∧ − k 0 k l ∧ + (a M i an + N i aman)ηk ηl + (a M i am + a2N i )ηk ηl 0 kn l mn k l ∧ 0 ml k 0 kl ∧ + ( a 2M i M i aman +2a N i am)ηk ηl, | 0| kl − mn l k 0 ml k ∧ ˆ ∗ i ˆ ∗ i i ˆ ∗ i where we have recycled the names of the pseudoconnection forms ξ0 = (β) ξ0, ξ = (β) ξ , ξj = (β) ξj . These give way to 0 0 i j dη = (ς + ς) η + iǫijη η , ∧ ∧ (3.4.3) dηi = ζi η0 + ς ηi + (δi rdr akdai ) ηj ζi ηj + κi ηj ∧ ∧ j − j k ∧ − j ∧ j ∧ + Oi ηk ηl + P i ηk ηl + Qi ηk ηl, kl ∧ kl ∧ kl ∧ for 1-forms k ς = (ξ + idr + r2(ξ ξ )), ζi =(1+ r2)ξi ai al ξ , − 0 0 − 0 j j − k j l (3.4.4) j k ζi = ((1 ir)ξi ai ξ ), κi = iai dr (1 ir)(dai + ai (ξ ξ ) + (akξi ai ξ )), − − − j j − j − − j j 0 − 0 j k − k j and torsion coefficients j j Oi = a 2M i an + a N i aman a ai (M am + a N ), kl | 0| kn l 0 mn k l − 0 j ml k 0 kl j j j (3.4.5) P i = a ( a 2M i M i aman +2a N i am)+ ai ( a 2M M aman +2a N am), kl 0 | 0| kl − mn l k 0 ml k j | 0| lk − mn k l 0 mk l j j Qi = a2(M i am + a N i ) ai an(a M + N am). kl 0 ml k 0 kl − j l 0 kn mn k To reduce from general coframes of LaGr(C∆) to those of L, we construct a bundle 2 ˆ1 whose B ⊂ B coframes (3.4.1) annihilate ΛA = β∗ kerϕ ˆ C∆ such that (ΛA, ΛA) L. Condition 1. of Definition 3.7 states that the fiber coordinates A = ai of ⊂2 will be functions of a single∈ complex variable a C∞( 2, C). j B ∈ B In fact, we can bring the coordinates into normal form ([20, Rem. 1.3]) and reduce the structure group UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 15

of 2 even further: 2 is not a bundle over 1 , but over 2 1 whose coframes are adapted such that B B BΛ BΛ ⊂BΛ βˆ : 2 2 has fibers (3.4.1) with A diagonalized ([20, Cor. 4.6(1)]), B →BΛ (3.4.6) ai = aδi , a C∞( 2, C), r = a . j j ∈ B | | Revisiting the language of §3.2, for L-contact structures that arise as leaf spaces of 2-nondegenerate CR i manifolds, the fiber coordinates A = aj encode the maps adK : Λ Λ, which are represented in the i → structure equations by the matrix κj (3.4.4) (cf. [19, §2.4, §3.2]). The Hermitian version of our Levi form L is obtained by complex-conjugating the second argument, so that the pair (L, A) can be brought into normal form by choosing appropriate bases of (Λ, Λ) L ([20, §4 esp. Def. 4.3]). Even in the generalized setting of Definition 3.7 (i.e., an L-contact structure∈ not necessarily given by a leaf space), condition 2. ensures that the pair (L, A) is strongly non-nilpotent regular so that its normal form specializes from the description in [20, Thm. 4.4] to that of [20, Cor. 4.6(1)]. With A diagonalized, κi = δi κ Ω1( 2, C) (compare to (2.2.10)). j j ∈ B 2 1 i 1 1 In particular, after reducing Λ Λ, the forms ξj Ω ( Λ, su(p, q)) as in (3.3.5) are no longer independent, but satisfy B ⊂ B ∈ B k i j (3.4.7) akξi ai ξ = a(ξi ξ ) 0 mod λ0, λi, λ , j k − k j j − j ≡ { } and we are left with γi = ξi Ω1( 2 , so(p, q)) (compare to (2.2.3)), j ℜ j ∈ BΛ along with the expansion of (3.4.7), i i k (3.4.8) ξi ξ = i(Ri λ0 + Ri λk + R λ ) forsome Ri C∞( 2 ), Ri C∞( 2 , C). j − j j0 jk jk j0 ∈ BΛ jk ∈ BΛ ˆ ∗ i ˆ ∗ i i ˆ ∗ i As before, we keep the same names of the pseudoconnection forms ξ0 = (β) ξ0, ξ = (β) ξ ,γj = (β) γj and torsion coefficients M,N when we pull back along βˆ : 2 2 to get B →BΛ 0 0 i j dη = (ς + ς) η + iǫij η η , (3.4.9) ∧ ∧ dηi = ζi η0 + ς ηi γi ηj + κ ηi + Oi ηk ηl + P i ηk ηl + Qi ηk ηl, ∧ ∧ − j ∧ ∧ kl ∧ kl ∧ kl ∧ for 1-forms (3.4.10) i ς = (ξ + idr + r2(ξ ξ )+ 1 (ada ada)), ζi = ((1 ir)ξi aξ + iRi ( 1 (1+2r2)ηj + aηj )), − 0 0 − 0 2 − − − − j0 2 and torsion coefficients i i i Oi = i (1+2r2)(a Ri + aR )+ a a 2M i + a a2N i a r2M aa2N , kl − 2 0 kl kl | 0| kl 0 kl − 0 kl − 0 kl i P i = i (1+2r2)(a R + aRi )+ a a 2M i a r2M i +2aa2N i kl − 2 0 kl kl 0| 0| kl − 0 lk 0 kl (3.4.11) i i i i + a a 2M ar2M +2a a2N ia(a Ri + aR ), | 0| lk − kl 0 lk − 0 kl kl i i i Qi = aa2M i + a3N i a a2M a3N ia(a R + aRi ). kl 0 kl 0 kl − 0 kl − kl − 0 kl kl As in §3.3 (see (3.3.11)), the pseudoconnection 1-forms ς, ζi,κ Ω1( 2, C) are not uniquely determined by the structure equations (3.4.9), but only up to a substitution∈ B 0 ς ς s0 0 0 η i i i i i ∞ 2 (3.4.12) ζ ζ + s s0 s1 η , s0,s ,s1 C ( , C).   7→      i  ∈ B κ κ s1 0 0 η To complete the construction   of an absolute parallelism   over U via Cartan’s method of equivalence, one would pull back the structure equations (3.4.9) to the bundle of all such pseudoconnection forms (3.4.12) over 2 and differentiate them with the help of the identity d2 = 0 to determine the exterior derivatives of theB tautological forms on the prolonged bundle ([19, §3.5]). Then, normalizing torsion in the new structure equations ([19, §§3.4-3.5]) would eventually reduce the fiber coordinates (3.4.12) to a single, 16 CURTIS PORTER

R-valued function s0 whose corresponding 1-form in the structure equations (2.2.5) of the homogeneous model was denoted ζ0. Remark 3.8. We have no need to continue with the prolongation procedure, which was shown to terminate 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 are sufficient 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 necessary that the torsion coefficients (3.4.11) are zero.

3.5. Quaternionic Structure Equations. We carry out the same sequence of constructions as in §3.4, 1 adjusting the structure equations as needed. To begin, we note that the Levi form (3.3.2), (3.3.3) on Λ is normalized to B 0 0 i j p+i p+j (3.5.1) dλ = (ξ + ξ ) λ + iδij λ λ iδij λ λ (cf. §2.3), − 0 0 ∧ ∧ − ∧ and in this section indices range from 1 to p = 1 n. The bundle βˆ : ˆ1 1 has ﬁbers 2 B →BΛ A Hom(Cp, Cp), 0 0 0 ˆ−1 1 n 2 − ∈ t β (ϕ)= ϕˆ : Tϕ Λ R C (1 + r )ˆϕ = ϕ β∗ 0 0 A ϕ β∗ A = A, ,  B → ⊕ ◦ − 0 A 0 ◦  2  AA = r ½, r R.  h i  ∈  and tautological forms    0 1 ˆ∗ 0 i 1 ˆ∗ i i ˆ∗ p+j p+i 1 ˆ∗ p+i i ˆ∗ j (3.5.2) η = 2 β λ , η = 2 (β λ + a β λ ), η = 2 (β λ a β λ ), 1+r 1+r j 1+r − j for ﬁber coordinates ∞ 1 i j ∞ 1 i j 2 r C ( ˆ ), a = a C ( ˆ , C), δij a a = r δjl. ∈ B j i ∈ B k l In particular, βˆ∗λ0 = (1+ r2)η0, βˆ∗λi = ηi ai ηp+j , βˆ∗λp+i = ηp+i + ai ηj , − j j and we pull back (3.3.10) to get 0 0 i j p+i p+j dη = (ς + ς) η + iδij η η iδijη η , ∧ ∧ − ∧ i i 0 i 1 k i i k j dη ζ η + ς η + 2 (a da a da ) η ≡ ∧ ∧ 2(1+r ) j k − k j ∧ ζi ηj ζi ηp+j + κi ηj + κi ηp+j mod η η, η η, η η , − j ∧ − p+j ∧ j ∧ p+j ∧ { ∧ ∧ ∧ } p+i p+i 0 p+i 1 k i i k p+j dη ζ η + ς η + 2 (a da a da ) η ≡ ∧ ∧ 2(1+r ) j k − k j ∧ ζp+i ηj ζp+i ηp+j + κp+i ηj + κp+i ηp+j mod η η, η η, η η , − j ∧ − p+j ∧ j ∧ p+j ∧ { ∧ ∧ ∧ } (having reduced modulo torsion terms), where

1 2 i i i p+j p+i p+i i j ς = rdr + 2 (ξ + r ξ ) , ζ = (ξ + a ξ ), ζ = (ξ a ξ ), − 1+r 0 0 − j − − j ξi Ω1( 1 su(p,p)) have pulled back to j ∈ BΛ i i i i l p+k i i l p+k ζj ζp+j 1 ξj + akaj ξp+l ξp+j akaj ξl p i = − , ζ + ζp+i 2 p+i i l k p+i i l j j pj 1+ r "ξ a a ξ ξ + a a ξ # j − k j p+l p+k j k l and the “adK ” maps are represented (3.5.3) p+j p+j κi κi 1 al ξi ai ξ dai + ai (ξ ξ )+ al ξi ai ξ j p+j = − k p+l − j k j k 0 − 0 k l − j p+k . κp+i κp+i 2 i i l p+i i j l p+i i j j p+j 1+ r " da a (ξ0 ξ ) a ξ + a ξk a ξ + a ξp k # − j − k − 0 − k p+l j k l j + 2 ˆ1 2 1 As in §3.4, we invoke [20, Cor. 4.6(2)] to reduce to over Λ Λ consisting of coframes of L such that the ﬁbers of 2 2 have A diagonalizedB (3.4.6).⊂ B WithB (3.5.3)⊂ B normalized as in (2.3.3) B → BΛ UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 17

1 1 (modulo torsion terms), the pseudoconnection forms ξ Ω ( Λ, su(p,p)) are no longer independent over 2 ∈ B Λ, but satisfy (2.3.1) (modulo tautological forms). Modulo torsion terms, the ﬁnal structure equations Bon 2 are equivalent to (2.3.2), with B 1 2 1 ς = rdr + 2 (ξ + r ξ ) + 2 (ada ada). − 1+r 0 0 2(1+r ) − Here again, the pseudoconnection forms are not uniquely determined by the structure equations, but require prolongation before they may be canonically deﬁned by higher order structure equations (see the discussion at the end of §3.4).

4. L-contact Structure on Unit Tangent Bundles 4.1. The Unit Tangent Bundle of a Semi-Riemannian Manifold. Let M be a smooth manifold and µ : TM M its tangent bundle; µ(y) = x y TxM. The pushforward µ∗ : TTM TM → ↑ ∀ ∈ → annihilates the vertical bundle T M = ker µ∗, which is naturally identified with TM as follows. For f C∞(M), df Ω1(M) C∞(TM) is regarded as a function on TM, linear on the fibers, and each Y ∈ Γ(TM) determines∈ Y ↑⊂ Γ(T ↑M) by their actions as derivations, ∈ ∈ Y (f)= Y ↑(df). ∗ Now suppose M is equipped with an affine connection : Γ(TM) Γ(T M TM). Each y TxM ∇ → ⊗ ∈ can be locally extended to Y Γ(TM) so that X Y =0 X Γ(TM), which uniquely determines the ∈ ∇ ∀ ∈ 1-jet of Y : M TM at x M. The pushforward Y∗ : TM TTM, satisfies µ∗ Y∗ = ½, hence → ∈ → ◦ ↑ Y∗(TxM) TyTxM defines the horizontal subbundle −→TM TTM which is complementary to T M and ⊂ ⊂ canonically isomorphic to TM via µ∗; i.e., each Y Γ(TM) determines −→Y Γ(−→TM) ([6, §1.3.1]). The ↑ ↑ ∈ 2 ∈ endomorphism field J : −→TM T M −→TM T M on TM will satisfy J = ½ if we set ⊕ → ⊕ − (4.1.1) J(−→Y )= Y ↑, J(Y ↑)= −→Y, Y TM. − ∀ ∈ 2 Similarly, K : TTM TTM with K = ½ and JK = KJ is given by → − (4.1.2) K(−→Y )= Y ↑,K(Y ↑)= −→Y, Y TM. ∀ ∈ Now suppose (M,g) is semi-Riemannian and is the Levi-Civita connection of g S2T ∗M. The Sasaki metricg ˆ S2T ∗TM is defined by ∇ ∈ ∈ gˆ(−→X, −→Y )=ˆg(X↑, Y ↑)= g(X, Y ), gˆ(−→X, Y ↑)=0, X, Y TM, ∀ ∈ from which it follows immediately (4.1.3) gˆ(JX,Jˆ Yˆ )=ˆg(X,ˆ Yˆ ) gˆ(JX,ˆ Yˆ )+ˆg(X,Jˆ Yˆ )=0 X,ˆ Yˆ TTM. ⇒ ∀ ∈ On the other hand, (4.1.4) gˆ(KX,ˆ Yˆ )=ˆg(X,Kˆ Yˆ ) gˆ(KX,Jˆ Yˆ )+ˆg(X,JKˆ Yˆ )=0 X,ˆ Yˆ TTM. ⇒ ∀ ∈ We depict the tripleg,J,K ˆ schematically as

−→TM ½ g 0 0 ½ 0

(4.1.5) gˆ = , J = − ,K = on TTM = . ½ 0 g ½ 0 0 T ↑⊕M If we suppose further that g has at least one positive eigenvalue, we can consider the unit tangent ⊥ bundle (1.0.2) of M. For each u UxM, u = ker g(u, ) TxM is a hyperplane implicitly identiﬁed ∈ · ⊂ ⊥ ↑ with Tu(UxM) (as one would for a sphere in Euclidean space). Explicitly, TuUxM = −−−→TxM (u ) ⊕ ⊂ −→TM T ↑M and we name the corank-1 distribution of TUM, ⊕ ⊥ ⊥ ↑ (4.1.6) ∆u = (−−→u ) (u ) TuUM. ⊕ ⊂ 18 CURTIS PORTER

Equivalently, ∆ = ker θ for the contact form θ Ω1(UM) given by ∈ (4.1.7) θ u =ˆg( u , ). | −→ · The endomorphism field J on TM restricts to an almost-complex structure on ∆, whose complexification splits (4.1.8) C∆=Λ Λ ⊕ into i and i eigenspaces of J, respectively. − Remark 4.1. The splitting (4.1.8) is the standard CR structure of UM ([26]), in spite of the fact that it’s rarely CR-integrable (see Remark 3.3). Indeed, the almost-complex structure (4.1.1) of TM is integrable if and only if M is flat as a semi-Riemannian manifold ([15]4, [10]), so for dim M > 2 the Nijenhuis tensor of Λ only vanishes for highly symmetrical M, e.g. Riemannian space forms of sectional curvature 1 ([2]). By definition (4.1.1),

↑ ⊥ ↑ ⊥ (4.1.9) Λu = C −→Y iY Y u , Λu = C −→Y + iY Y u , − ∈ ∈ n o n o and (4.1.5) restricts to

Λu

⊥ ½ 0 2g i½ 0 0 i

(4.1.10) gˆ = |u , J = ,K = on C∆= . ½ 2g u⊥ 0 0 i½ i 0 ⊕ | − − Λu The Levi form of UM is L(Yˆ , Yˆ )= idθ(Yˆ , Yˆ ) Yˆ , Yˆ Γ(C∆). 1 2 − 1 2 1 2 ∈ Lemma 4.2. L(Yˆ , Yˆ )= 1 gˆ(Yˆ ,JYˆ ) Yˆ , Yˆ Γ(C∆). 1 2 2 1 2 ∀ 1 2 ∈ Proof. This follows from (4.1.3), and will be shown in §4.2. Here we wish to acknowledge the fact that the contact metric on UM is homothetic to the Sasaki metric; see [3], keeping in mind that (4.1.7) is equivalent tog ˆ(u↑,J ), and the Levi form is only defined up to a choice of local trivialization C(TUM/∆) C, provided here· by θ. ≈ In light of Lemma 4.2 and (4.1.3), (4.1.4), we offer the following Definition 4.3. On UM with its contact distribution (4.1.6), the endomorphisms J, K : ∆ ∆ defined by restriction of (4.1.1) and (4.1.2) define the standard split-quaternionic structure of the→ unit tangent bundle of M. The corresponding L-contact structure (see Remark 3.6) is also called standard (cf. Remark 4.1). The homogeneous leaf space U presented as split-quaternionic lines in §2.2 can be locally realized as the unit tangent bundle UM for M = Rp+1,q (see Appendix A) with its standard L-contact structure. Here, Rp+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), which is measured at lowest order by torsion terms in the structure equations (3.4.9). Similarly, M is non-flat if it is not locally equivalent to Rp+1,q, which is measured by the Riemann curvature tensor of M. In order to relate these two notions of curvature (“non-flatness”) in §4.2, let us establish some notation. The Riemann curvature tensor of g is R Ω2(M, so(TM)), ∈ so for X, Y TxM, ∈ (4.1.11) R(X, Y ): TxM TxM satisfies g(R(X, Y )y ,y )+ g(y , R(X, Y )y )=0, y ,y TxM. → 1 2 1 2 ∀ 1 2 ∈

4[15] attributes this to [18], but I was unable to locate a copy of Nagano’s article. Note that these sources only explicitly refer to the Riemannian (deﬁnite-signature) case. UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 19

2 ⊥ ⊥ In particular, if we set y1 = y2 = u, we conclude that R( , )u : (u ) u is well-deﬁned, hence we can extend by conjugate-bilinearity and lift to · · ∧ →

(4.1.12) R( , ) u :Λu Λu Λu. · · −→ ∧ → Compare (4.1.12) to the Nijenhuis tensor discussed in Remark 3.3. Contracting R with g yields the Ricci tensor Ric S2T ∗M, which we use to construct the Ricci-shift operator, ∈ ↑ ↑ ↑ ↑ (4.1.13) −→X −→X + Ric(u,X)X , X X + Ric(u,X)−→X u,X TxM −→X,X TuUM. 7→ 7→ ∀ ∈ ∈ Definition 4.4. On UM with its contact distribution (4.1.6), the Ricci-shifted L-contact structure is defined by application of the Ricci shift operator (4.1.13) to each Lagrangian subspace in the standard L-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 = Rn+1 with its standard basis of column vectors e0,...,en W . Indices written in typewriter font range from 0 to n while those in standard font start counting at∈ 1. The metric , S2W ∗ of signature (p +1, q) is h· ·i ∈ ei,ej = ǫij as in (2.2.1), adding ǫ =1. h i 0 (M,g) is a semi-Riemannian manifold of signature (p+1, q), p+q = n. Over each x M, the orthonormal frame bundle π : M has fiber ∈ F → −1 π (x)= φ : W TxM w , w = g(φ(w ), φ(w )) w , w W , { → | h 1 2i 1 2 ∀ 1 2 ∈ } which carries a right principal action of the orthogonal group O(W, , )= O(p+1, q) given by composing φ with orthogonal transformations of W . The tautological 1-form h·ω ·i Ω1( , W ) is ∈ F −1 (4.2.1) ω φ = φ π∗, | ◦ and the Levi-Civita connection form γ Ω1( , so(W, , )) is determined by the torsion-free structure equation ∈ F h· ·i (4.2.2) dω = γ ω. − ∧ i i 1 With respect to the standard basis of W , ω = ω ei for ω Ω ( ) so that ⊗ ∈ F ∗ i j (4.2.3) π g = ǫijω ω , ⊗ and (4.2.2) becomes i i j i j dω = γ ω , ǫiγ + ǫjγ =0, − j ∧ j i i where the latter holds for each fixed i, j (not summed over), reflecting the fact that γj is so(p +1, q)- i valued. It will be convenient to introduceγ ˜j taking values in so(p + q + 1) via i j i k j i j (4.2.4) dω = ǫ γ˜ ω , ǫ = ǫjk, γ˜ +˜γ =0. − k j ∧ k j i The rest of the semi-Riemannian structure equations read (4.2.5) dγi = γi γk + 1 Ri ωk ωl. j − k ∧ j 2 jkl ∧ i i 1 For each 1-form ω ,γ Ω ( ) in the coframing of , ∂ωi , ∂γi Γ(T ) will denote their dual vector j ∈ F F j ∈ F fields. In particular, by definition (4.2.1),

(4.2.6) ω(∂ωi )= ei.

ﬁbers over the unit tangent bundle (1.0.2) via the projection π0 : UM mapping each frame to its Fﬁrst basis vector, F →

π (φ)= φ(e ) UxM. 0 0 ∈ 20 CURTIS PORTER

In light of (4.2.6), π0(φ) = π∗∂ω0 , and since π = µ π0 for the basepoint projection µ : TM M, we ◦ ∗ → have (π )∗∂ 0 = u −→TM for u = π (φ) UM. Along with (4.2.3), the fact that µ g −→ =g ˆ −→ then 0 ω −→ ∈ 0 ∈ | T M | T M implies that the contact form (4.1.7) pulls back to ∗ 0 (4.2.7) (π0) θ = ω . Moreover, dω0 vanishes separately on the following subbundles of ker ω0, which map isomorphically to the summands of (4.1.6), n n −−−−→⊥ ⊥ ↑ (π0)∗ (R ∂ωi i=1)= π0(φ) , (π0)∗ R ∂γ˜0 = (π0(φ) ) ; { } i i=1 n o name their direct sum ∆ T so that (π0)∗∆ = ∆. We lift the almost-complex structure J on ⊂∗ F ∆ TUM to ∆ to get (π0) L-Lagrangian subbundles Λ, Λ C∆ mapping isomorphically under (π0)∗ to⊂ (4.1.9), ⊂

Λ = C ∂ωi i∂γ˜0 , Λ = C ∂ωi + i∂γ˜0 . − i i The lifted almost-complexn structureo is encoded in 1-forms λi Ω1(n , C) giveno by ∈ F (4.2.8) λ0 = 1 ω0, λi = 1 (ωi iγ˜i ). 2 2 − 0 With this coframing, (4.2.4) (i = 0) becomes

0 i j (4.2.9) dλ = iǫij λ λ , ∧ and for i 1, ≥ i 1 i 0 i j i j i k 0 1 i k l dλ = 2 γ0 ω γj ω + i(γj γ˜0 R0k0ω ω 2 R0klω ω ) (4.2.10) − ∧ − ∧ ∧ − ∧ − ∧ i k l = (γi + iRi ωk) λ0 γi λj Ri (λk + λ ) (λl + λ ). − 0 0k0 ∧ − j ∧ − 4 0kl ∧ These are the structure equations (3.3.10) for

ξ 0 i (4.2.11) ξi = γi + iRi ωk, 0 ≡ mod i λk N i = Ri . 0 0k0 ξi γi { ℜ } jk − 4 0jk j ≡ j Remark 4.5. Continuing the discussion of Remarks 3.3 and 4.1, we now observe that the Nijenhuis tensor of the standard CR structure on UM is given, up to scale, by (4.1.12). Note that this is trivial if dim M = 2 – the one component of curvature having been absorbed into the deﬁnition (4.2.11) of ξ1 – reﬂecting the fact that 3-dimensional CR manifolds are automatically CR-integrable. To reproduce the theorem of [2] that Riemannian space forms can give rise to integrable CR structures on UM, it is convenient to replace the Riemannian structure equations (4.2.5) with the “model-mutated” version (see [24, Ch.5 §6]), dγi = γi γk ωi ωj + 1 Ri ωk ωl, j − k ∧ j ± ∧ 2 jkl ∧ whose curvature tensor measures deviation from the structure equations of space forms with nonzero sectional curvature. The resulting equations (4.2.10), (4.2.11) on UM would see the 1-forms ξi replaced by γi + i(Ri ωk ωi), and vanishing curvature would ensure CR integrability. 0 0k0 ∓ The 1-forms (4.2.8) may be considered a pull-back of the tautological form (3.3.6) along a section 1 UM Λ of the bundle of coframings which are 1-adapted to (4.1.8). We realize the tautological forms (3.4.2),→B (3.4.6) of 2 by setting B i η0 = λ0, ηi = (1 ir)λi aλ , − − which brings the structure equations (4.2.9), (4.2.10) into agreement with (3.4.9) for 1-forms 1 i 2 0 ς = idr + (ada ada)+ z− η , − 2 − 2 | | i 2 0 i 3 k (4.2.12) κ = iadr (1 ir)da (z−) η (z ) R kη , − − − − 2 − 4 + 0 i i 2 i i 2 i i i k ζ = z− η (z−) η (z−)γ i(z )R ω , 2 | | − 2 − 0 − + 0k0 UNIT TANGENT BUNDLES, CR LEAF SPACES, AND HYPERCOMPLEX STRUCTURES 21

and torsion coeﬃcients

(4.2.13) Oi = i z (z )2Ri , P i = i (z )2z Ri ,Qi = i (z )3Ci , kl − 4 + + 0kl kl − 2 + + 0kl kl − 4 + 0kl where we have introduced

1 m n i i i (4.2.14) z± = (1 ir a), Rjk = ǫ R , C = R ǫ Rjl. − ± n+1 n jmk jkl jkl − k Remark 4.6. The last term in the expression (4.2.12) of κ is exactly the Ricci-shift (4.1.13), and distinguishes 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 curvature tensor (like O and P ). One could also absorb the Ricci components of P into κ, but the analogous eﬀort to use ς to absorb the Ricci components of O and/or P fails to be maintain the identity for dη0.

Theorem 4.7. In order for the Ricci-shifted L-contact structure of UM to be the leaf space of a 2- nondegenerate CR manifold, it is suﬃcient that M is conformally ﬂat and real-analytic.

Proof. If UM is the leaf space of a 2-nondegenerate CR structure , then the Nijenhuis tensor of is represented by the torsion coeﬃcients Q in the structure equations (3.4.9).U For the Ricci-shifted L-contacU t structure equations (4.2.13), these torsion coeﬃcients are components of the Weyl curvature tensor of M. Because the L-contact structure of UM (standard or Ricci-shifted) arises from a split-quaternionic structure, it meets the criteria [25, Cor. 2.8] for recoverability, provided M is real-analytic.

4.3. 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 §4.2,

(4.3.1) ∆ = R ∂ i , ∂ p+i , ∂ 0 , ∂ 0 ω ω γi γp+i n o has the L-Lagrangian splitting C∆ = Λ Λ for ⊕

Λ = C ∂ωi i∂γ0 , ∂ωp+i i∂γ0 . − i − p+i n o The quaternionic structure K : ∆ ∆ maps →

∂ i ∂ p+i , ω 7→ ω (4.3.2) K :  ∂γ0 ∂γ0 .  i 7→ − p+i The contact form still pulls back to according to (4.2.7), and the ﬁrst semi-Riemannian structure equation (4.2.4) is F

(4.3.3) dω0 = γ0 ωj + γ0 ωp+j , − j ∧ p+j ∧ so we deﬁne the 1-adapted coframing as before,

λ0 = 1 ω0, λi = 1 (ωi iγi ), λp+i = 1 (ωp+i iγp+i), 2 2 − 0 2 − 0 whereby (4.3.3) becomes (3.5.1) with ξ0 = 0. The rest of the construction proceeds by analogy to the end of §4.2, following §3.5 rather than §3.4. In particular, the deﬁnition (3.5.2) (with (3.4.6)) of the tautological forms on 2 encodes the induced action of (4.3.2) on Λ, Λ. Details will be included in an updated pre-print. B 22 CURTIS PORTER

Appendix A. The Homogeneous Model of §2.2 as a Sphere Bundle In this appendix, we exhibit a local hypersurface realization of the homogeneous models presented in §2.2 in an effort to provide a more analytical perspective on that discussion. Specifically, we offer an explicit local identification of that L-contact structure with the unit tangent bundle of a Riemannian manifold. Three caveats are in order. First, we only present the definite-signature case ǫij = δij (2.2.1), the mixed signature case being a straightforward modification of this. Second, we use representations of the 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-canonical choices of these representations. To better explain this last caveat, we take a detour in §A.1 to compare L-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 généralizés [24, Preface] – of the homogeneous models presented in §§2.2-2.3: coset spaces of O(n +2, 2) (in the definite signature case n+4 ǫij = δij ) for the stabilizer subgroups (2.2.12) preserving complex lines and 2-planes in C . In this respect these two geometries are closely related to conformal and Lie contact geometry, respectively. Lie contact geometry ([6, §4.2.5]) is the generalization (in Cartan’s sense, but also generalization to arbitrary signature) of Lie Sphere Geometry ([7]), introduced by Sophus himself in his 1872 dissertation. The objects of interest in Lie sphere geometry are oriented n-spheres of arbitrary radius in Rn+1 – including limiting cases of points (zero radius) and oriented hyperplanes (infinite radius) – which can be in “signed” contact (oriented or not) with each other at some common point of tangency. Symmetries of interest map these objects to each other while preserving their contact relationships, so we are interested in rigid motions O(n +1) of Rn+1, Lorentz boosts and time-translations of Rn+1,1, conformal symmetries n+1 n+1 O(n+2, 1)/ ½ of the conformal compactification S of R , and everything else generated by these ([7, Thm 3.16],{± cf.} [17, §2.2]). All of this is encoded in the (n + 2)-dimensional Lie quadric RPn+3 Qn+2 ⊂ ,2 whose symmetries O(n +2, 2)/ ½ preserve the real projectivization of the null cone in R . Lie {± } Gr0 n+2,2 contact geometry generalizes the space of null projective lines on , the Grassmannian 2R of totally null, real 2-planes. Q The representation of O(n +2, 2) pertinent to conformal and Lie contact geometry differs from that of §2.2 in that it is totally real; in other words, it is the representation corresponding to the trivial

split-quaternionic structure K = ½ (nevertheless, the significance of split-quaternionic structures to Lie contact geometry has been noticed – see [27, §2.5]). Moreover, the stabilizer subgroups (2.2.12) of real lines and 2-planes are parabolic in this real representation, hence the Cartan geometries modeled on homogeneous coset spaces of O(n+2, 2) with respect to these stabilizer subgroups are parabolic geometries Gr0 n+2,2 – Lorentzian-conformal geometry generalizing and Lie contact generalizing 2R , respectively – unlike 2-nondegenerate CR and L-contact geometries.Q One motivation for the present work was the prominent role of unit tangent bundles as examples of Lie contact manifolds. For a Riemannian manifold (M,g), UM (1.0.2) has a natural Lie contact structure incorporating many of the same constructions as in §4.1. The “flat” (homogeneous) model of Lie contact Gr0 n+2,2 geometry – i.e., 2R or the null lines on the Lie quadric – is the unit tangent bundle of the Riemannian sphere M = Sn+1. However, it should noted that theQ Lie contact structure of UM is only associated to the conformal class of the metric g rather than g itself. In other words, flatness of UM as a Lie contact manifold is equivalent to M being conformally flat ([16]), and any such UM is locally equivalent to the homogeneous model. Similar considerations for L-contact manifolds should be kept in mind while reading §A.2, which presents a local realization of a homogeneous L-contact manifold as a unit tangent bundle.

A.2. The Future Tube as a Unit Tangent Bundle. The simplest examples of Levi-degenerate CR hypersurfaces which are not straightenable are tube conical surfaces = C iRm+1 Cm+1 where C Rm+1 is a cone rC C r > 0 ([8, §5.2]). The ﬂat model of stronglyU × regular, 2-nondegenerate⊂ ⊂ ⊂ ∀ UNIT 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 the future light cone = ρ−1(0) for ρ : Cm+1 R given by U → 2 2 2 2ρ = (z + z ) + + (zm + zm) (zm + zm ) . 1 1 ··· − +1 +1 is the tube over the future light cone because we impose U (A.2.1) zm+1 + zm+1 > 0.

m+2 m+2 Note that if we take coordinates (z ,...,zm ,zm ) C to lie in an aﬃne subset of CP , 1 +1 +2 ∈ [1 : z : : zm : zm ] = [Z : Z : : Zm : Zm ], 1 ··· +1 +2 0 1 ··· +1 +2 m then ρ =0 = P( b h) where b, h C +3 are the null cones of the bilinear and Hermitian forms { } N ∩N N N ⊂ 2 2 2 b(Z,Z)= 2Z Zm + Z + + Z Z , − 0 +2 1 ··· m − m+1 2 2 2 h(Z,Z)= Z Zm + Z Zm + Z + + Zm Zm . 0 +2 0 +2 | 1| ··· | | − | +1| The Levi form of is U = ∂∂ρ = dz dz + + dzm dzm dzm dzm . L 1 ∧ 1 ··· ∧ − +1 ∧ +1 Evidently, ∂ρ(X)=0= (X, X) for L ∂ ∂ ∂ X = (z1 + z1) + + (zm + zm) + (zm+1 + zm+1) , ∂z1 ··· ∂zm ∂zm+1 which is the inﬁnitesimal generator of the scaling operator

2 2 (A.2.2) z δR(z,z)z, δR(z,z)= x + + x , 7→ 1 ··· m+1 where we have broken the complex coordinates into their real an imaginary parts

zi = xi + iyi.

Observe that if (z ,...,zm ) and c = r + is C (r, s R, r> 0), then we also have 1 +1 ∈U ∈ ∈ (A.2.3) (cx + iy ,...,cxm + iym ) = (rx + i(sx + y ),...,rxm + i(sxm + ym )) . 1 1 +1 +1 1 1 1 +1 +1 +1 ∈U On the real subspace Rm+1 Cm+1 away from the origin, (A.2.3) is just a constant (rescaled) version of (A.2.2). Thus, the “complex⊂ half rays” on given by all such multiples (A.2.3) are the leaves of the Levi foliation of . We claim that the leaf spaceU U of is the unit tangent bundle UM for (M,g)= Rm with its EuclideanU metric; i.e., U = Rn+1 Sn for n =Um 1. n+1× − n To see this, let t = (t1,...,tm) R be coordinates for M and u = (u1,...,um) S coordinates constrained by ∈ ∈ g(u,u)= u2 + + u2 =1. 1 ··· m Deﬁne the inclusion U by →U (A.2.4) (t,u) (u + it ,...,um + itm, 1). 7→ 1 1 Note that we implicitly rely on the futuristic condition (A.2.1). From here, it is straightforward to verify: distinct points of UM map into distinct leaves in , • U every (z1,...,zm+1) lies in the image (A.2.4) composed with (A.2.3) for some c = r + is with • r> 0. ∈U is therefore realized as a complex “half-ray” bundle over U = URn+1 whose ﬁber over u U is Ucu c C, c> 0 . ∈ { | ∈ ℜ } 24 CURTIS PORTER

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Department of Mathematics, Duke University, 120 Science Dr. Durham, NC 27710 Email address: [email protected]