The main result of this thesis is an asymptotic formula establishing a correspondence between two at first sight unrelated systems of local coordinates on a two-dimensional real analytic Riemannian manifold: the so-called isothermal local parameter and the Riemann normal coordinates. This formula was stated without proof in 1992 Fay’s memoir on analytic torsion, a weaker statement was proved in Walsh’s 2012 PhD thesis. A survey of basic facts from differential geometry used in our proof is also given.