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 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.