Lecture 10 - Metric Manifolds

#math

We establish a structure on a smooth manifold that allows to assign vectors in each tangent space a length (and an angle between vectors in the same tangent space).

From this structure one can then define a notion of length of a curve. Then we can look at shortest curves (geodesics). Requiring then that the shortest curves coincide with the straight curves (wrt to some $\nabla$ ) will result in $\nabla$ being determined by the metric structure.

Metrics

Definition

A metric $g$ on a smooth manifold $(M, O, A)$ is a $(0, 2)$ -tensor field satisfying two conditions:

Symmetric: $g (X, Y) = g (Y, X), \forall X, Y$
Non-degeneracy: for a specific $X$ if $g (X, Y) = 0, \forall Y$ iff $X = 0$ . (and alternative definition: $G_{X} (Y) = G (X, Y)$ , $g$ is non-degenerate iff $G$ is an isomorphism from $Γ (T M) \to Γ (T^{*} M)$ , in other words it is invertible.)

Definition

The $(2, 0)$ -tensor field $g^{- 1}$ w.r.t a metric $g$ is the symmetric:

\begin{aligned} g^{" - 1 "} & : Γ (T^{*} M) \times Γ (T^{*} M) \to C^{\infty} (M) \\ (ω, σ) \to σ (G^{- 1} (σ)) . \end{aligned}

Remark: We will never use the standard use of the metric to pull indices up or down.

Example $(S^{2}, O, A)$ with chart $(U, x)$ . Define the metric:

g_{i j} (x^{- 1} (θ, ϕ)) = {(\begin{matrix} R^{2} & 0 \\ 0 & R^{2} s i n^{2} (θ) \end{matrix})}_{i j}

This is the metric of the round sphere of radius $R$ .

Signature

This is a topic from Linear Algebra, we talk about $A v = λ v$ which could be rewritten to be: $A_{m}^{a} v^{m} = λ v^{a}$ , that is a $(1, 1)$ -tensor has eigenvalues but a $(0, 2)$ -tensor doesn't have eigenvalues, it has signature which take the form of $p$ 1s and $q$ -1s which we notate by $(p, q) .$

Since our $g$ is non-degenerate we have that $p + q = \dim V$ thus we have $\dim V + 1$ choices for the signature of $g$ .

Definition

A metric is called

Riemannian if its signature is $(+ + . . . +)$ .
Lorentizian if its signature is $(+ - . . . -)$ (or $(- + . . . +)$ but we will be using the former notion).

Length of a curve.

Let $γ$ be a smooth curve then we know its velocity $v_{γ, γ (λ)}$ at each $γ (λ)$ , where $λ$ is the parameter of the curve, at each $γ (λ) \in M$

Definition

On a Rimannian metric manifold $(M, O, A, g)$ the speed of a curve at $γ (λ)$ is the number:

(\sqrt{g (v_{γ}, v_{γ})})_{γ (λ)} = s (λ)

Definition

Let $Γ : (0, 1) \to M$ be a smooth curve then the length, $L$ , of $γ$ is the number:

L [M] := \int_{0}^{1} d λ s (λ) = \int_{0}^{1} d λ \sqrt{g (v_{λ}, v_{λ})_{γ (λ)}}

Example

Reconsider the round sphere of radius $R$ .
Consider its equator:
$θ (λ) = (x^{1} \circ γ) (λ) = \frac{π}{2}$
$ϕ (λ) = (x^{2} \circ γ) (λ) = 2 π λ^{3}$

\begin{aligned} L [γ] & = \int_{0}^{1} d λ \sqrt{g_{i j} (x^{- 1} (θ (λ), ϕ (λ)) (x^{i} \circ γ)^{'} (λ) (x^{j} \circ γ)^{'} (λ))} \\ = \int_{0}^{1} d λ \sqrt{R^{2} s i n^{2} (θ (λ)) \cdot 36 π^{2} λ^{4}} \\ = 6 π R \int_{0}^{1} d λ λ^{2} \\ = 2 π R . \end{aligned}

Theorem

$γ : (0, 1) \to M$ smooth, and $σ : (0, 1) \to (0, 1)$ is smooth, bijective and increasing ( $σ$ is a reparametrization), then:

L [γ] = L [γ \circ σ] .

Geodesics

Definition

A curve $γ : (0, 1) \to M$ is called a geodesic on a Riemannian manifold $(M, O, A, g)$ if it is a stationary curve wrt the length functional $L$ .

Theorem

$γ$ is a geodesic iff it satisfies the Euler-Langrange equations for the Langrangian.

Note

The Langrangian, $L$ , we are interested in is a function:

\begin{aligned} L & : T M \to R \\ X \to \sqrt{g (X, X)} \end{aligned}

In a chart, the Euler Langrange equations take the form:

\begin{array}{r} {(\frac{\partial L}{\partial {\dot{x}}^{m}})}^{'} - \frac{\partial L}{\partial x^{m}} = 0 \end{array}

here: $L (γ^{i}, {\dot{γ}}^{i}) = \sqrt{g_{i j} (γ (λ)) {\dot{γ}}^{i} (λ) {\dot{γ}}^{j} (λ)}$

Geodesic equation for $γ$ in a chart:

{\ddot{γ}}^{q} + (g^{- 1})^{q m} \frac{1}{2} (\partial_{i} g_{m j} + \partial_{j} g_{m i} - \partial_{m} g_{i j}) {\dot{γ}}^{i} {\dot{γ}}^{j} = 0

Definition

$Γ^{q}_{i j} = (g^{- 1})^{q m} \frac{1}{2} (\partial_{i} g_{m j} + \partial_{j} g_{m i} - \partial_{m} g_{i j})$ is the Christoffel symbol, they are connection coefficient functions of the so called Levi-Civita connection.

Definition

Some tensors:

The Riemann-Christoffel curvature is defined by

R_{a b c d} = g_{a m} R^{m}_{b c d}

The Ricci tensor: $R_{a b} = R^{m}_{a m b}$
(Ricci) scalar curvature: $R = g^{a b} R_{a b}$
Einstein curvature $G_{a b} = R_{a b} - \frac{1}{2} g_{a b} R$