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 ⊕(R2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in RN, after suitable local affine embedding of (M, ∇) into RN. 1. Introduction In the article [9] R. Sasaki proposed to add the property of describing pseudospherical surfaces to other remarkable properties – such as applicability of the inverse scattering method, infinite number of conservation laws and Bäcklund transformations – which characterize soliton equations in 1 + 1 dimensions. He expressed the sl(2, R)-valued 1-form Ω, which arises in the corresponding linear scattering problem dv = Ωv, v = ( v1v2 ), by 1-forms ω1, ω2 and ω21 −12ω2 2(ω21 1 + ω1)! Ω= 12(−ω21 + ω1) 2ω21