Riemannian Manifolds and Two of Hopf’s Theorems

Abstract

A Riemannian metric g on a manifold M is a family of inner products {g x}x∈M such that the quantities

$${g_{ij}}(x): = g\left( {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^i}}}} \right)$$

are smooth in local coordinates. The Finsler function F(x, y) of a Riemannian manifold has the characteristic structure

$$F(x,y) = \sqrt {{g_{ij(x)}}{y^i}{y^j}} $$

.