Non-degenerate, symmetric bilinear forms and metric tensorsedit
Let be a vector space over and let be a bilinear function. We call
Let be a vector space over , let be its dual space and let be a nondegenerate bilinear form. Then the function
Let be a manifold. A tensor field on is called
Let be a manifold of class . By the term metric tensor on we mean symmetric and nondegenerate tensor field on of class .
In the following, we shall denote a metric tensor by . Let's explain this notation a bit further: A tensor field on is a function on which maps every point to a tensor with respect to . At each point now, our metric tensor takes the value of the tensor
, where the two s denote the two inputs for elements of .
Let be a manifold and be a metric tensor. Then for each ,
is a symmetric, nondegenerate bilinear form.
Proof: See exercise 1.
A pseudo-Riemannian manifold is a manifold together with a metric tensor.
Arc length, isometries and Killing vector fieldsedit
Let be a pseudo-Riemannian manifold with metric tensor , let be an interval and let be a curve. The length of , denoted by , is defined as follows:
Let and be two pseudo-Riemannian manifolds of class , where is the metric tensor of and . By an isometry between and , we mean a diffeomorphism of class such that for each curve defined on a finite interval , we have
Let be a manifold. We call a Killing vector field (named after Wilhelm Killing; this has nothing to do with killing) iff for each , is an isometry between and for all such that the domain of is equal to the whole .
Let us repeat, what the left and right multiplication functions were.
Let be a Lie group with group operation , and let . The left multiplication function with respect to , denoted by , is defined to be the function
The right multiplication function with respect to , denoted by , is defined to be the function
Now we are ready to define left and right invariant metric tensors:
Let be a Lie group. A metric tensor of is called left invariant iff for all , the function is an isometry between and .
A metric tensor of is called right invariant iff for all , the function is an isometry between and
We have already seen in chapter 10, that both and are diffeomorphisms of the class of the Lie group. Therefore, if we want to check if a metric tensor of is left or right invariant, we only have to check if or preserves the length of curves.