## 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

### Citations

125 |
Representation Theory. Graduate Texts
- FULTON, HARRIS
- 1991
(Show Context)
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... |

84 |
Pseudo-hermitian structures on a real hypersurface
- Webster
- 1978
(Show Context)
Citation Context ...2.32 shows that (5.3) ¯Rijk l − Rijk l = ωijAk, l , ¯ R0ij k − R0ij k = − ¯ ∇iAj k = −∇iAj k . Define a quadratic form on T1,0 by Q(Z) = L( ¯ R(Z, ¯ Z)Z, Z) for Z ∈ Γ(T1,0). For Z ∈ Γ(T1,0), Webster, =-=[24]-=-, defined by Q(Z) = K(Z)L(Z, Z) 2 the holomorphic sectional curvature, K(Z), of the subspace spanned by Z and ¯ Z. The pseudohermitian structure has constant sectional curvature κ if there is a consta... |

83 |
Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Mathematics 22
- Yamaguchi
- 1993
(Show Context)
Citation Context ...the curvature function of the associated Cartan connection, and to prove Theorem C.CONTACT PROJECTIVE STRUCTURES 33 There is a co-boundary operator, ∂ : ∧ k ((g−) ∗ ) ⊗ g → ∧ k+1 ((g−) ∗ ) ⊗ g, (see =-=[27]-=-), defined when k = 2 by ∂φ(x1, x2, x3) = Cycle[x1 · φ(x2, x3) − φ([x1, x2], x3)] . Any invariant, symmetric bilinear form, B, on g is a constant multiple of the Killing form, and so B gives an identi... |

74 | Parabolic geometries and canonical Cartan connections - Čap, Schichl |

58 |
les variétés à connexion projective
- Cartan, Sur
- 1955
(Show Context)
Citation Context ...arameterized geodesics of any two representative connections are the same. Two projective structures are equivalent if they are equivalent as path geometries. Using his method of equivalence, Cartan, =-=[7]-=-, attached to each projective structure a unique regular, normal Cartan connection. T.Y. Thomas’s construction of an ‘ambient’ affine connection associated to a projective structure provides an altern... |

54 |
The Thomas structure bundle for conformal, projective and related structures
- Bailey, Eastwood, et al.
- 1994
(Show Context)
Citation Context ...ction 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-va... |

46 | Bernstein-Gelfand-Gelfand sequences
- Čap, Slovák, et al.
(Show Context)
Citation Context ...DANIEL 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 cano... |

41 |
The differential invariants of generalized spaces
- Thomas
- 1934
(Show Context)
Citation Context ...bient Connection 3.1. Thomas’s Ambient Construction for Projective Structures. In this section 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)... |

36 | Standard tractors and the conformal ambient metric construction - Čap, Gover |

35 |
Geometric structures on filtered manifolds
- Morimoto
- 1993
(Show Context)
Citation Context ...l prolongations. Theorems associating canonical regular, normal Cartan connections to broad classes of generalized G-structures on filtered manifolds have been proved by N. Tanaka, [22], T. Morimoto, =-=[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... |

30 | Weyl structures for parabolic geometries
- Cap, Szlovák
(Show Context)
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... |

22 |
Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung, Göttinger Nachr
- Weyl
- 1921
(Show Context)
Citation Context ...al R × -bundle ρ : L → M, a unique connection, φ, such that s is a parallel section and φ(X) = 1. The connection, φ, determines a horizontal lift, ˆ X, of each vector field, X, on M. Lemma 3.1 (Weyl, =-=[25]-=-). Two torsion-free affine connections have the same unparameterized geodesics if and only if there exists a one-form, γ, so that their difference tensor has the form γ (iδ j) k . Lemma 3.2 (Thomas, [... |

15 |
The geometry of the differential equation y ′′′ = f(x, y, y
- Chern
- 1940
(Show Context)
Citation Context ...hat contact path geometry determined by the solutions of a third order ODE considered modulo contact transformations, and the local equivalence problem for these structures was solved by S.-S. Chern, =-=[8]-=-. The study of the higher dimensional contact path geometries is an ongoing project of the author; in this paper attention is restricted to the contact projective structures. The study made here of co... |

14 |
On projective connections
- Kobayashi, Nagano
- 1964
(Show Context)
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... |

12 |
Hyperbolic manifolds are geodesically rigid
- Matveev
(Show Context)
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 ... |

9 |
Invariants and calculus for projective geometries
- Gover
- 1996
(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... |

7 |
Abel’s theorem and webs
- Chern, Griffiths
- 1978
(Show Context)
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 |
Differential Geometry”, Graduate Texts
- Sharpe
- 1997
(Show Context)
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... |

6 |
On the equivalence problems associated with a certain class of homogeneous spaces
- Tanaka
- 1965
(Show Context)
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 |
Weyl structures for parabolic geometries, ESI
- Čap, Slovák
- 1999
(Show Context)
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 |
Invariant theory of parabolic geometries, Complex geometry (Osaka
- Graham
- 1990
(Show Context)
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
- Slovák
- 1997
(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
- On
- 1976
(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 |
Some problems in the invariant theory of parabolic geometries
- Harrison
- 1995
(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... |