# Differentiable Manifolds/Pseudo-Riemannian manifolds

## Non-degenerate, symmetric bilinear forms and metric tensors

Definitions 12.1:

Let ${\displaystyle V}$  be a vector space over ${\displaystyle \mathbb {R} }$  and let ${\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to \mathbb {R} }$  be a bilinear function. We call ${\displaystyle \langle \cdot ,\cdot \rangle }$

• symmetric iff ${\displaystyle \forall \mathbf {v} ,\mathbf {w} \in V:\langle \mathbf {v} ,\mathbf {w} \rangle =\langle \mathbf {w} ,\mathbf {v} \rangle }$
• nondegenerate iff ${\displaystyle \forall \mathbf {v} \in V:\left((\forall \mathbf {w} \in V:\langle \mathbf {v} ,\mathbf {w} \rangle =0)\Rightarrow \mathbf {v} =0\right)}$

Theorem 12.2:

Let ${\displaystyle V}$  be a vector space over ${\displaystyle \mathbb {R} }$ , let ${\displaystyle V^{*}}$  be its dual space and let ${\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to \mathbb {R} }$  be a nondegenerate bilinear form. Then the function

${\displaystyle J:V\to V^{*},J(\mathbf {v} ):=\langle \mathbf {v} ,\cdot \rangle }$

is bijective.

Proof:

Definitions 12.3:

Let ${\displaystyle M}$  be a manifold. A ${\displaystyle (0,2)}$  tensor field ${\displaystyle \mathbf {T} }$  on ${\displaystyle M}$  is called

• symmetric iff ${\displaystyle \forall p\in M:\left(\forall \mathbf {v} _{p},\mathbf {w} _{p}\in T_{p}M:\mathbf {T} (\mathbf {v} _{p},\mathbf {w} _{p})=\mathbf {T} (\mathbf {w} _{p},\mathbf {v} _{p})\right)}$
• nondegenerate iff ${\displaystyle \forall p\in M:\left(\mathbf {v} _{p}\in T_{p}M:\left((\forall \mathbf {w} _{p}\in T_{p}M:T(\mathbf {v} ,\mathbf {w} )=0)\Rightarrow \mathbf {v} _{p}=0\right)\right)}$

Definition 12.4:

Let ${\displaystyle M}$  be a manifold of class ${\displaystyle {\mathcal {C}}^{n}}$ . By the term metric tensor on ${\displaystyle M}$  we mean symmetric and nondegenerate ${\displaystyle (0,2)}$  tensor field on ${\displaystyle M}$  of class ${\displaystyle {\mathcal {C}}^{n}}$ .

In the following, we shall denote a metric tensor by ${\displaystyle \langle \cdot ,\cdot \rangle (\cdot )}$ . Let's explain this notation a bit further: A ${\displaystyle (0,2)}$  tensor field on ${\displaystyle M}$  is a function on ${\displaystyle M}$  which maps every point ${\displaystyle p\in M}$  to a ${\displaystyle (0,2)}$  tensor with respect to ${\displaystyle T_{p}M}$ . At each point ${\displaystyle p\in M}$  now, our metric tensor takes the value of the ${\displaystyle (0,2)}$  tensor

${\displaystyle \langle \cdot ,\cdot \rangle (p)}$

, where the two ${\displaystyle \cdot }$ s denote the two inputs for elements of ${\displaystyle T_{p}M}$ .

Theorem 12.5:

Let ${\displaystyle M}$  be a manifold and ${\displaystyle \langle \cdot ,\cdot \rangle }$  be a metric tensor. Then for each ${\displaystyle p\in M}$ ,

${\displaystyle \langle \cdot ,\cdot \rangle (p)}$

is a symmetric, nondegenerate bilinear form.

Proof: See exercise 1.

Definition 12.6:

A pseudo-Riemannian manifold is a manifold ${\displaystyle M}$  together with a metric tensor.

## Arc length, isometries and Killing vector fields

Definition 12.7:

Let ${\displaystyle M}$  be a pseudo-Riemannian manifold with metric tensor ${\displaystyle \langle \cdot ,\cdot \rangle }$ , let ${\displaystyle I\subseteq \mathbb {R} }$  be an interval and let ${\displaystyle \gamma :I\to M}$  be a curve. The length of ${\displaystyle \gamma }$ , denoted by ${\displaystyle l(\gamma )}$ , is defined as follows:

${\displaystyle l(\gamma ):=\int _{I}{\sqrt {\langle \gamma '_{x},\gamma '_{x}\rangle (\gamma (x))}}dx}$

Definition 12.8:

Let ${\displaystyle M}$  and ${\displaystyle N}$  be two pseudo-Riemannian manifolds of class ${\displaystyle {\mathcal {C}}^{n}}$ , where ${\displaystyle \langle \cdot ,\cdot \rangle _{M}}$  is the metric tensor of ${\displaystyle M}$  and ${\displaystyle \langle \cdot ,\cdot \rangle _{N}}$ . By an isometry between ${\displaystyle M}$  and ${\displaystyle N}$ , we mean a diffeomorphism ${\displaystyle \psi :M\to N}$  of class ${\displaystyle {\mathcal {C}}^{n}}$  such that for each curve ${\displaystyle \rho :I\to M}$  defined on a finite interval ${\displaystyle I\subset \mathbb {R} }$ , we have

${\displaystyle l(\rho )=l(\psi _{*}\rho )}$

Definition 12.9:

Let ${\displaystyle M}$  be a manifold. We call ${\displaystyle \mathbf {V} \in {\mathfrak {X}}(M)}$  a Killing vector field (named after Wilhelm Killing; this has nothing to do with killing) iff for each ${\displaystyle x\in \mathbb {R} }$ , ${\displaystyle \Phi _{x}}$  is an isometry between ${\displaystyle M}$  and ${\displaystyle M}$  for all ${\displaystyle x\in \mathbb {R} }$  such that the domain of ${\displaystyle \Phi _{x}}$  is equal to the whole ${\displaystyle M}$ .

## Left and right invariant metric tensors

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

Definitions 10.10:

Let ${\displaystyle G}$  be a Lie group with group operation ${\displaystyle *}$ , and let ${\displaystyle g\in G}$ . The left multiplication function with respect to ${\displaystyle g}$ , denoted by ${\displaystyle L_{g}}$ , is defined to be the function

${\displaystyle L_{g}:G\to G,L_{g}(h):=g*h}$

The right multiplication function with respect to ${\displaystyle g}$ , denoted by ${\displaystyle R_{g}}$ , is defined to be the function

${\displaystyle R_{g}:G\to G,R_{g}(h):=h*g}$

Now we are ready to define left and right invariant metric tensors:

Definitions 12.10:

Let ${\displaystyle G}$  be a Lie group. A metric tensor of ${\displaystyle G}$  is called left invariant iff for all ${\displaystyle g\in G}$ , the function ${\displaystyle L_{g}}$  is an isometry between ${\displaystyle G}$  and ${\displaystyle G}$ .

A metric tensor of ${\displaystyle G}$  is called right invariant iff for all ${\displaystyle g\in G}$ , the function ${\displaystyle R_{g}}$  is an isometry between ${\displaystyle G}$  and ${\displaystyle G}$

We have already seen in chapter 10, that both ${\displaystyle L_{g}}$  and ${\displaystyle R_{g}}$  are diffeomorphisms of the class of the Lie group. Therefore, if we want to check if a metric tensor of ${\displaystyle G}$  is left or right invariant, we only have to check if ${\displaystyle L_{g}}$  or ${\displaystyle R_{g}}$  preserves the length of curves.