## Contact Projective Structures

Venue: | Indiana Univ. Math. J |

Citations: | 13 - 2 self |

### BibTeX

@ARTICLE{Fox_contactprojective,

author = {Daniel J. F. Fox},

title = {Contact Projective Structures},

journal = {Indiana Univ. Math. J},

year = {},

pages = {1547--1598}

}

### OpenURL

### Abstract

Abstract. A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a one-dimensional subspace of the contact distribution at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifold and using this the local equivalence problem for contact projective structures is solved by the construction of a canonical regular Cartan connection. This Cartan connection is normal if and only if an invariant contact torsion vanishes. Every contact projective structure determines canonical paths transverse to the contact structure which fill out the contact projective structure to give a full projective structure, and the vanishing of the contact torsion implies the contact

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Citation Context ...fference tensor of the contact projective structures. The facts about the irreducible representations of the symplectic group used in the sequel may be found in some form in Section 6.3 of [26] or in =-=[10]-=-. Let A, B, and C, respectively, be the bundles of tensors on H obtained by raising the third index of elements of, respectively, S 3 (H ∗ ); the subbundle of ⊗ 3 (H ∗ ) comprising tracefree tensors s... |

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Citation Context ...lts concerning projective structures are reviewed, following the approach of T. Y. Thomas, [23]. Some version of this material can be found in various modern sources, for instance [1], [11], [12], or =-=[14]-=-. 3.1.1. Projective Structures. The bundle of frames, F, in the canonical bundle, ∧n (T ∗M), of the smooth n-dimensional manifold, M, is the R × principal bundle of smooth, non-vanishing sections of t... |

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Citation Context ...nd hence Wijkl = 0, so that if the dimension is at least 5, the contact projective structure is flat. In three dimensions, R0ijk = 0, so Cijk = 1 n∇ (iRjk), which vanishes by (5.7). □ Example 5.1. In =-=[15]-=-, V. Matveev studies the space of Riemannian metrics projectively equivalent to a given Riemannian metric. An analogous problem is to describe the space of pseudo-hermitian structures determining the ... |

8 |
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(Show Context)
Citation Context ... the basic results concerning projective structures are reviewed, following the approach of T. Y. Thomas, [23]. Some version of this material can be found in various modern sources, for instance [1], =-=[11]-=-, [12], or [14]. 3.1.1. Projective Structures. The bundle of frames, F, in the canonical bundle, ∧n (T ∗M), of the smooth n-dimensional manifold, M, is the R × principal bundle of smooth, non-vanishin... |

6 |
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Citation Context ...termined by the geodesics of a Riemannian metric is flat if and only if the Riemannian metric has constant sectional curvatures. The proof, which is an exercise in using the Bianchi identities, is in =-=[9]-=-. Next there is proved an analogous theorem for pseudo-hermitian manifolds. A pseudo-hermitian structure is a contact manifold equipped with a distinguished contact one-form, θ, and an almost complex ... |

6 |
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Citation Context ...nnection, η, on G → M, determines a tractor connection as the induced covariant differentiation on the associated bundle T = G ×P V. Any η determines a development of paths in M onto G/P = P(V), (see =-=[17]-=-), and η induces on M the projective structure comprising those paths which develop onto straight lines in P(V). Evidently gauge equivalent Cartan connections induce the same projective structure. The... |

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Citation Context ...ional complex subbundle of the complexified tangent bundle, T1,0 ⊂ CTM, comprising vector fields of the form X+iJ(X) for X ∈ Γ(H). Define the Levi form, L(U, V ) = −iω(U, ¯ V ) for U, V ∈ Γ(T1,0). In =-=[19]-=- and [20], Tanaka constructed a canonical affine connection associated to a pseudo-hermitian structure. In the integrable case his construction specializes as in the following theorem, the statement o... |

3 |
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Citation Context ...o, [16], and A. Čap - H. Schichl, [4]. The projective and contact projective structures are exceptional for these theorems because in these cases a certain Lie algebra cohomology fails to vanish (see =-=[5]-=- and [27]). The local equivalence problem for projective structures was in any case solved by various methods by E. Cartan, T.Y. Thomas, and H. Weyl in the 1920’s. J. Harrison, [13], and Čap-Schichl s... |

3 |
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Citation Context ...asic results concerning projective structures are reviewed, following the approach of T. Y. Thomas, [23]. Some version of this material can be found in various modern sources, for instance [1], [11], =-=[12]-=-, or [14]. 3.1.1. Projective Structures. The bundle of frames, F, in the canonical bundle, ∧n (T ∗M), of the smooth n-dimensional manifold, M, is the R × principal bundle of smooth, non-vanishing sect... |

3 |
Parabolic geometries, IGA preprint 97/11
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(Show Context)
Citation Context ...J. F. FOX for conformal structures) or a foliation (as for generalized path geometries), of a P principal bundle supporting a (g, P) Cartan connection (for background on Cartan connections see [6] or =-=[18]-=-). For this there are various approaches, e.g. E. Cartan’s method of equivalence, T.Y. Thomas’s ambient constructions, or N. Tanaka’s Lie cohomological prolongations. Theorems associating canonical re... |

2 |
non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections
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(Show Context)
Citation Context ...tructed a canonical affine connection associated to a pseudo-hermitian structure. In the integrable case his construction specializes as in the following theorem, the statement of which is taken from =-=[21]-=-. Theorem 5.2 (N. Tanaka). On an integrable pseudo-hermitian manifold there exists a unique affine connection, ¯ ∇, the pseudo-hermitian connection, having torsion τ and satisfying: 1. ¯ ∇θ = 0. 2. ¯ ... |

1 |
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(Show Context)
Citation Context ...ails to vanish (see [5] and [27]). The local equivalence problem for projective structures was in any case solved by various methods by E. Cartan, T.Y. Thomas, and H. Weyl in the 1920’s. J. Harrison, =-=[13]-=-, and Čap-Schichl solved the local equivalence problem for structures corresponding to a subclass, characterized by the vanishing of an invariant contact torsion, of what are here called contact proje... |