g-Derivatives and Gauss Structures on Differentiable Manifolds

Abstract

<p>Some results are given connecting the concepts of g-derivatives and Jacobians on differentiable manifolds. Also some general properties of Gauss structures on manifolds important for our problems are discussed here. The connection between g-derivatives and Jacobians is given by studying the following problem: Given two differentiable manifolds Mn and M'n and a differentiable map Φ: Mn → M'n with ⎮JU, U'Φ(x)⎮ > O* for each x ε Mn, find a g-function f and families of coverings (V,V') such that f: (Mn, V) → (M'n, V') generates Φ, and for suitable Gauss structures F, F' the g-derivatives Df generates a continuous function ψ, such that for all x ε Mn: ψ(x) = ⎮JU, U'Φ(x)⎮ for convenient local charts U,U.</p>