Connections on bundles of horizontal frames associated with contact sub-pseudo-Riemannian manifolds

[摘要] Given a contact sub-pseudo-Riemannian manifold (M, H, g) we study connections on the bundle O-H.g (M) of orthonormal horizontal frames attached to (M, H, g). We prove that there exist connections on O-H.g (M) with vanishing horizontal torsion. All such connections induce a unique covariant derivation del:Sec(H) x Sec(H) -> Sec(H) satisfying del g = 0 and del Y-X - del X-Y = P([X, Y]) for all X, Y is an element of Sec(H), where P: TM -> H is the canonically defined projection. If we additionally assume that the Reeb vector field R is an infinitesimal isometry, then one can prove that there exists a unique connection on O-H.g(M) with vanishing torsion. The induced covariant derivation del: Sec(TM) x Sec(H) -> Sec(H) additionally satisfies the condition del X-R = [R, X] for all X is an element of Sec(H). In the latter case, there exists a coframe on O-H.g(M) which is invariant with respect to the isometries of (M, H, g), and which enables one to obtain the complete system of differential invariants of the metric (H, g). (C) 2019 Elsevier B.V. All rights reserved.

[发布日期] 2019-12-01

[关键词] Contact distributions;Pseudo-Riemannian metrics;Connections;Sub-Riemannian geometry;Sub-Lorentzian geometry