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

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Given a contact sub-pseudo-Riemannian manifold (M,H,g) we study connections on the bundle OH,g(M) of orthonormal horizontal frames attached to (M,H,g). We prove that there exist connections on OH,g(M) with vanishing horizontal torsion. All such connections induce a unique covariant derivation ∇:Sec(H)×Sec(H)⟶Sec(H) satisfying ∇g=0 and ∇XY−∇YX=P([X,Y]) for all X,Y∈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 OH,g(M) with vanishing torsion. The induced covariant derivation ∇:Sec(TM)×Sec(H)⟶Sec(H) additionally satisfies the condition ∇RX=[R,X] for all X∈Sec(H). In the latter case, there exists a coframe on OH,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).