On some flat connection associated with locally symmetric surface

Abstract

For every two-dimensional manifold M with locally symmetric linear connection nabla, endowed also with nabla-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 Omega build from the dual basis omega1, omega 2 and from local connection form omega of nabla. The structural equations of (M, nabla) are equivalent to the condition dOmega-Omega Lambda Omega=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.