Riemannian metric

Overview

A Riemannian metric is a generalization of the inner product for a manifold. This is important since each tangent space might not be the “same” for different points on a manifold, so we need a notion of how inner products (and thus angles of linearizations) varies smoothly around the manifold.

Definition

Let $M$ be a smooth manifold. A Riemannian metric (usually denoted $g$) is a smooth assignment of inner products on tangent spaces $T_{p}M$ for $p\in M$.

Formally, $g$ is a covariant 2 tensor field that is

1. Symmetric. $g_p(x,y)=g_p(y,x)$ for $x,y\in T_{p}M$

2. Positive definite $g_p(x,x)>0$ for $x\in T_{p}M$ such that $x\ne 0$.

Yet another way to think about a Riemannian metric is to think of it as a non-degenerate smooth section of the 2nd symmetric power $S^2{T}^{\ast }M$.

With coordinates

Let $(U,x^i)$ be a chart on $M$. Using this chart, we have the coordinate frame $\left\{\frac{\partial }{\partial x^i}\right\}$ on $TM$ such that a vector field $X=X^i\frac{\partial }{\partial x^i}$.

A Riemannian metric is determined by

$$\begin{array}{rl}{g}_{ij}& =⟨\frac{\partial }{\partial x^i},\frac{\partial }{\partial x^j}⟩\\ & =g\left(\frac{\partial }{\partial x^i},\frac{\partial }{\partial x^j}\right)\end{array}$$

where $g_{ij}\in C^{\infty }(M,\mathbb{R})$.

Given Einstein notation, we have

$$g(X,Y)=g_{ij}{X}^i{Y}^j$$

Differences from symplectic form

On a vector space, we can always use the Gram Schmidt process to find an orthonormal basis $\{e_i\}$. This acts as a global chart for the vector space, and the inner product in these coordinates is simply the identity matrix.

For a Riemannian manifold, $(M,g)$, we can find an orthonormal frame $\{e_i\}$ for the tangent bundle, but don’t expect it to be a coordinate frame. In other words, there is no guarantee that we will always find coordinates that will all be orthonormal under $g$.

This is because there can be local invariants in Riemannian manifolds. This is different from symplectic manifolds which by Darboux thoerem says that there are no local invariants and there will always be a local chart that makes the symplectic form into the “standard” symplectic form.

Examples

${\mathbb{R}}^n$

Consider the manifold $({\mathbb{R}}^n,g_{\text{Euc}})$. This is the normal “dot product” or inner product on the vector space.

Therefore,

$$\begin{array}{rl}{g}_{\text{Euc}}& =g_{ij}d{x}^{i}d{x}^j{g}_{ij}={\delta }_j^i\\ & =\sum (d{x}^i{)}^2\end{array}$$

$S^2$

Using spherical coordinates

$$(\phi ,\theta )⟼(\sin \phi \cos \theta ,\sin \phi \sin \theta ,\cos \phi )$$

then

$$g=\left[\begin{array}{cc}1& 0\\ 0& {\sin }^2\phi \end{array}\right]$$

$S^2$ again!

Now using stereographic projection:

todo ADD EQUATIONS HERE…

Using the above formulas, we have

$$\left|\frac{\partial }{\partial u}\right|=\left|\frac{\partial }{\partial v}\right|=1-z\text{and}\frac{\partial }{\partial u}\cdot \frac{\partial }{\partial v}=0$$

Therefore, this gives the metric (after some algebraic manipulation)

$$g_{ij}=\left[\begin{array}{cc}{\left(\frac{2}{1+u^2+v^2}\right)}^2& 0\\ 0& {\left(\frac{2}{1+u^2+v^2}\right)}^2\end{array}\right]$$