Affine analogues of the Sasaki-Shchepetilov connection

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 $TM ⊕ E$, 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), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on $TM ⊕ (R × M)$ and two on $TM ⊕ (R^2 × M)$ 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 $R^N$.