Affine analogues of the Sasaki-Shchepetilov connection

Abstract

Abstract. For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle T M ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), [3893][3894][3895][3896][3897][3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on T M ⊕ (R × M ) and two on T M ⊕ (R 2 × M ) are constructed. It is shown that two of those connections -one from each pair -may be identified with the standard flat connection in R N , after suitable local affine embedding of (M, ∇) into R N .