📂Geometry

Riemann Metric and Riemann Manifolds

Definition¹

A Riemannian metric $g$ on a $n$-dimensional differentiable manifold $M$ is a function that maps each point $p \in M$ to $g_{p}$. Here, $g_{p}$ is an inner product defined in the tangent space $p$ over $T_{p}M$.

$$ \begin{align*} g : M &\to \left\{ \text{all inner products on tangent space } T_{p}M \right\} \\ p &\mapsto g_{p}=\left\langle \cdot, \cdot \right\rangle_{p} \end{align*} $$

$$ \begin{align*} g_{p} : T_{p}M \times T_{p}M &\to \mathbb{R} \\ (\mathbf{X}_{p}, \mathbf{Y}_{p}) &\mapsto g_{p}(\mathbf{X}_{p}, \mathbf{Y}_{p})=\left\langle \mathbf{X}_{p}, \mathbf{Y}_{p} \right\rangle_{p} \end{align*} $$

In this case, $g_{p}$ should be differentiable in the following sense:

Let $\mathbf{x} : U \subset \mathbb{R}^{n} \to M$ be a coordinate system around $p$, and say that $\mathbf{x}(x_{1}, \dots, x_{n}) = p$. Then, the following $g_{ij} : \mathbb{R}^{n} \to \mathbb{R}$ must be differentiable:

$$ g_{ij} (x_{1}, \dots, x_{n}) = \left\langle \left. \dfrac{\partial }{\partial x_{i}} \right|_{p}, \left. \dfrac{\partial }{\partial x_{j}} \right|_{p}\right\rangle_{p} $$

$g_{ij}$ is called the local representation of the Riemannian metric. A differentiable manifold $(M, g)$ given a Riemannian metric is called a Riemannian manifold.

Explanation

• Studying a Riemannian manifold $(M, g)$ is referred to as Riemannian geometry. $g$ is also referred to as the Riemannian structure. $g_{ij}$ is learned in differential geometry as the coefficient of the first fundamental form.

• Let’s say $\mathbf{X}_{p}, \mathbf{Y}_{p} \in T_{p}M$. Then, since $T_{p}M$ is a $n$-dimensional vector space with basis $\left\{ \left. \dfrac{\partial }{\partial x_{i}} \right|_{p} \right\}$,

  $$ \mathbf{X}_{p} = X^{i}(p)\left. \dfrac{\partial }{\partial x_{i}} \right|_{p} \text{ and } \mathbf{Y}_{p} = Y^{j}(p)\left. \dfrac{\partial }{\partial x_{j}} \right|_{p} $$

  Therefore,

  $$ g_{p}(\mathbf{X}_{p}, \mathbf{Y}_{p}) = \left\langle \mathbf{X}_{p}, \mathbf{Y}_{p} \right\rangle = X^{i}(p)Y^{j}(p) \left\langle \left. \dfrac{\partial }{\partial x_{i}} \right|_{p}, \left. \dfrac{\partial }{\partial x_{j}} \right|_{p} \right\rangle = X^{i}(p)Y^{j}(p)g_{ij}(p) $$

  Generalizing from $p$,

  $$ g(\mathbf{X}, \mathbf{Y}) = X^{i}Y^{j} \left\langle \dfrac{\partial }{\partial x_{i}}, \dfrac{\partial }{\partial x_{j}} \right\rangle = X^{i}Y^{j}g_{ij} $$

• Another expression for the condition of differentiability is as follows:

  A function $g(\mathbf{X}, \mathbf{Y}) : M \to \mathbb{R}$ defined on the manifold is differentiable.

  $$ g(\mathbf{X}, \mathbf{Y}) (p) = g_{p} (\mathbf{X}_{p}, \mathbf{Y}_{p}), \quad \mathbf{X}_{p}, \mathbf{Y}_{p} \in T_{p}M $$

  In this case, it is also denoted as $g(\mathbf{X}, \mathbf{Y})_{p} = g(\mathbf{X}, \mathbf{Y}) (p)$.

• It is known that “all differentiable manifolds have a Riemannian metric.” Thus, the direction of research is not ‘conditions under which a manifold $M$ has a Riemannian metric’, but rather ‘what kind of good Riemannian metric can be given to a manifold $M$’.

Induced Metric

Given an immersion $f : M \to N$ between differentiable manifolds $M, N$, suppose $(N, h)$ is a Riemannian manifold. The Riemannian metric $g$ on $M$ defined as follows is called the induced metric on $M$ by $f$.

$$ g_{p}(v, w) := h_{f(p)}(df_{p}(v), df_{p}(w)) $$

Here, $df_{p}$ is the derivative of $f$ at point $p$.

Euclidean Space

Consider Euclidean space as a differentiable manifold $M = \mathbb{R}^{n}$. Then, $T_{p}\mathbb{R}^{n} \approx \mathbb{R}^{n}$ and the basis $\left\{ \dfrac{\partial }{\partial x}_{i} \right\}$ is the same as the standard basis $\left\{ e_{i} = (0, \dots, 1, \dots, 0) \right\}$ of Euclidean space. Therefore, the metric coefficients are as follows.

$$ g_{ij} = \left\langle e_{i}, e_{j} \right\rangle = \delta_{ij} $$

Hence, the Riemannian metric of Euclidean space is the inner product defined as standard in Euclidean space itself. Studying $(\mathbb{R}^{n}, g)$ is referred to as Euclidean geometry.