A Riemannian metric on a smooth manifold is defined as a covariant symmetric 2-tensor – a section of the symmetrized second tensor power of the tangent bundle – such that for all . For convenience, we will write for . In other words, a Riemannian metric is a collection of (positive) inner products on each of the tangent spaces such that if are (smooth) vector fields, the function defined by taking the inner product at each point, is smooth. A manifold together with a Riemannian metric is called a Riemannian manifold.
There are several ways to get Riemannian metrics:
On , there is a standard Riemannian metric coming from the usual inner product. More generally, if are smooth functions such that the matrix is symmetric and positive definite for all , we get a Riemannian metric on , where the sum is to be interpreted as a covariant tensor.
Given an immersion , a Riemannian metric on induces one on in the natural way, simply by pulling back. For instance, any surface in has a Riemannian structure based upon the standard Riemannian structure on —based simply on the usual inner product—and induced on the surface.
Given an open covering on , Riemannian metrics on , and a partition of unity subordinate to the covering , we get a Riemannian metric on by
Thus, using 1) above, any smooth manifold—which necessarily admits partitions of unity—can be given a Riemannian metric.
A Riemannian metric allows us to take the length of a curve in a manner resembling the standard case. Given , use the notation . If is a smooth curve for an interval in , we define
this is easily checked to be independent of parametrization, just as in the usual case. Using this, we can make a Riemannian manifold into a metric space: for , let
The metric on induces the standard topology on . To see this, first note that it is a local question, so we can reduce to the case of an open ball in euclidean space . Each tangent vector can be viewed as an element of in a natural way. Now let be the standard norm on . By continuity, we can find by shrinking if necessary such that for all ,
in particular, the lengths of curves in are necessarily comparable to the usual lengths in . The result now follows.
An introduction in terms of synthetic differential geometry is in