On some flat connection associated with locally symmetric surface

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 TM local connection form of such connection is a R(G)-valued 1-form Ω build from the dual basis ω¹, ω² and from local connection form ω of ∇. The structural equations of (M, ∇) are equivalent to the condition dΩ-ΩΛΩ=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 surface.