On some flat connection associated with locally symmetric surface

Abstract

Abstract. For every two-dimensional manifold M with locally symmetric linear connection ∇, endowed also with ∇-parallel volume element, we construct a flat connection on some principal fibre bundle P (M, G). Associated with -satisfying some particular conditions -local basis of T M local connection form of such a connection is an R(G)-valued 1-form Ω build from the dual basis ω 1 , ω 2 and from the local connection form ω of ∇. The structural equations of (M, ∇) are equivalent to the condition dΩ − Ω ∧ Ω = 0.This work was intended as an attempt to describe in a unified way the construction of similar 1-forms known for constant Gauss curvature surfaces, in particular of that given by R. Sasaki for pseudospherical surfaces.

Affine analogues of the Sasaki-Shchepetilov connection

“…Such immersions and transversal fields are described in [5]. If we use one of them, then we obtain D different from (6) and (8).…”

“…Those gl(N, R)-valued (N = 3 or N = 4) 1-forms were obtained in [8] as the local connection forms of connections on some principal GL(N, R)-bundle P and seem to be analogous to the Sasaki connection form. The bundle…”

“…In the construction of P and Ω in [8] and in the present paper we consider the left action of H on Q: a * q := qa −1 , where…”