Differentiable Manifolds/Pseudo-Riemannian manifolds

Non-degenerate, symmetric bilinear forms and metric tensors

Definitions 12.1:

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

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

Theorem 12.2:

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

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

is bijective.

Proof:

Definitions 12.3:

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

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

Definition 12.4:

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

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

$\langle \cdot ,\cdot \rangle (p)$

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

Theorem 12.5:

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

$\langle \cdot ,\cdot \rangle (p)$

is a symmetric, nondegenerate bilinear form.

Proof: See exercise 1.

Definition 12.6:

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

Arc length, isometries and Killing vector fields

Definition 12.7:

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

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

Definition 12.8:

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

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

Definition 12.9:

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

Left and right invariant metric tensors

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

Definitions 10.10:

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

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

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

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

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

Definitions 12.10:

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

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

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