Differentiable Manifolds/Pseudo-Riemannian manifolds

Non-degenerate, symmetric bilinear forms and metric tensors

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Definitions 12.1:

Let   be a vector space over   and let   be a bilinear function. We call  

  • symmetric iff  
  • nondegenerate iff  

Theorem 12.2:

Let   be a vector space over  , let   be its dual space and let   be a nondegenerate bilinear form. Then the function

 

is bijective.

Proof:

Definitions 12.3:

Let   be a manifold. A   tensor field   on   is called

  • symmetric iff  
  • nondegenerate iff  

Definition 12.4:

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  .

Theorem 12.5:

Let   be a manifold and   be a metric tensor. Then for each  ,

 

is a symmetric, nondegenerate bilinear form.

Proof: See exercise 1.

Definition 12.6:

A pseudo-Riemannian manifold is a manifold   together with a metric tensor.

Arc length, isometries and Killing vector fields

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Definition 12.7:

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:

 

Definition 12.8:

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

 

Definition 12.9:

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  .

Left and right invariant metric tensors

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Let us repeat, what the left and right multiplication functions were.

Definitions 10.10:

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:

Definitions 12.10:

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.

Exercises

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Sources

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