There is one more crucial ingredient which we need to introduce for dealing with manifolds: lengths and angles. Given a smooth manifold, since we know what it means for a curve in the manifold to be smooth, and we have a well-defined notion of the tangent vector to a curve, all we need in order to have a notion of distance on the manifold is a way of defining the speed of a curve — that is, the length of its tangent vector.
Metric Tensors
Metric Tensor & Semi-Riemannian Manifold
A (Semi-Riemannian) metric tensor
on is a -tensor, which at each , takes a pair of tangent vectors and returns a real number such that
is symmetric, that is . is nondegenerate, that is for every , there is a such that . A semi-Riemannian manifold is a smooth manifold furnished with a metric tensor.
Riemannian Metric & Riemannian Manifold
A Riemannian metric
on a smooth manifold is a nonnegative metric tensor, that is for all and if and only if . A pair is called a Riemannian manifold.
e.g.
- The standard inner product on Euclidean space is a trivial example of a Riemannian metric.
- Consider
with standard spherical coordinates with and , the metric tensor from is given by
Metric Signature
The signature of a metric tensor
is the pair of integers (or sometimes the integer difference ) that records, respectively, the number of positive and negative eigenvalues of the symmetric matrix representation of the metric at a point. Sometimes it may also be denoted as an explicit list such as or .
e.g. A Riemannian metric has signature
Killing Vector Field
A killing vector field on a semi-Riemannian manifold is a vector field
for which the Lie derivative of the metric tensor vanishes: .
Length and Distance
Def Length of a Curve
Let
Def Distance
Given two points
Prop If