Differentiable Manifolds/Pseudo-Riemannian manifolds

Non-degenerate, symmetric bilinear forms and metric tensors

Theorem 12.2:

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

$J:V\to V^{*},J(𝐯):=⟨𝐯,\cdot ⟩$

is bijective.

Proof:

In the following, we shall denote a metric tensor by $⟨\cdot ,\cdot ⟩(\cdot )$. 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

$⟨\cdot ,\cdot ⟩(p)$

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

Theorem 12.5:

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

$⟨\cdot ,\cdot ⟩(p)$

is a symmetric, nondegenerate bilinear form.

Proof: See exercise 1.

Left and right invariant metric tensors

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

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

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.