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 con struct a flat connection on some principal fibre bundle P (M, G). Associated with – satisfying some particular conditions – local basis of T M local con nection 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.