Riemannian Manifold - Riemannian Metrics

Riemannian Metrics

Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of (positive definite) inner products

such that, for all differentiable vector fields X,Y on M,

defines a smooth function M → R.

More formally, a Riemannian metric g is a symmetric (0,2)-tensor that is positive definite (i.e. g(X, X) > 0 for all tangent vectors X ≠ 0).

In a system of local coordinates on the manifold M given by n real-valued functions x1,x2, …, xn, the vector fields

give a basis of tangent vectors at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p,

Equivalently, the metric tensor can be written in terms of the dual basis {dx1, …, dxn} of the cotangent bundle as

Endowed with this metric, the differentiable manifold (M, g) is a Riemannian manifold.