First Basic Forms, Riemannian Metrics 📂Geometry

First Basic Forms, Riemannian Metrics

Buildup

Riemannian metric is a concept that comes from the process of calculating the length of curves on a surface, and the process is as follows.

Let’s say $\boldsymbol{\alpha}(t)$ is a regular curve moving on a simple surface $\mathbf{x} : U \to \mathbb{R}^{3}$. Let’s say $(u_{1}, u_{2})$ are the coordinates in $U$. Then, $\boldsymbol{\alpha}$ can be expressed as follows.

$$ \boldsymbol{\alpha}(t) = \mathbf{x}(u_{1}(t), u_{2}(t)) $$

At this point, the length of $\boldsymbol{\alpha}$ at $a \le t \le b$ is defined as follows.

$$ \int_{a}^{b} \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| dt $$

If we resolve the integrand function, it goes as follows.

$$ \begin{align*} \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| =&\ \sqrt{\left\langle \dfrac{d \boldsymbol{\alpha}}{d t} , \dfrac{d \boldsymbol{\alpha}}{d t} \right\rangle} \\ =&\ \sqrt{\left\langle \dfrac{d \mathbf{x}(u_{1}, u_{2})}{d t} , \dfrac{d \mathbf{x}(u_{1}, u_{2})}{d t} \right\rangle} \end{align*} $$

By the chain rule,

$$ \begin{align*} \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| =&\ \sqrt{\left\langle \dfrac{\partial \mathbf{x}}{\partial u_{1}}\dfrac{d u_{1}}{dt} + \dfrac{\partial \mathbf{x}}{\partial u_{2}}\dfrac{d u_{2}}{dt}, \dfrac{\partial \mathbf{x}}{\partial u_{1}}\dfrac{d u_{1}}{dt} + \dfrac{\partial \mathbf{x}}{\partial u_{2}}\dfrac{d u_{2}}{dt} \right\rangle} \\ =&\ \sqrt{\left\langle \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt}, \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt} \right\rangle} \end{align*} $$

At this point, $\mathbf{x}_{1} := \dfrac{\partial \mathbf{x}}{\partial u_{1}}, \mathbf{x}_{2} := \dfrac{\partial \mathbf{x}}{\partial u_{2}}$. Expanding and arranging the inner product,

$$ \begin{align*} & \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| \\ =&\ \sqrt{\left\langle \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt}, \mathbf{x}_{1}\dfrac{d u_{1}}{dt} + \mathbf{x}_{2}\dfrac{d u_{2}}{dt} \right\rangle} \\ =&\ \sqrt{\left( \dfrac{d u_{1}}{dt} \right)^{2} \left\langle \mathbf{x}_{1}, \mathbf{x}_{1} \right\rangle + \dfrac{d u_{1}}{dt}\dfrac{d u_{2}}{dt} \left\langle \mathbf{x}_{1}, \mathbf{x}_{2} \right\rangle + \dfrac{d u_{2}}{dt}\dfrac{d u_{1}}{dt} \left\langle \mathbf{x}_{2}, \mathbf{x}_{1} \right\rangle + \left( \dfrac{d u_{2}}{dt} \right)^{2} \left\langle \mathbf{x}_{2}, \mathbf{x}_{2} \right\rangle} \end{align*} $$

Here, if we denote the dot product above by $g_{ij} = \left\langle \mathbf{x}_{i}, \mathbf{x}_{j} \right\rangle$, and arrange it as $\sum$, it can be expressed as follows.

$$ \begin{align*} \left| \dfrac{d \boldsymbol{\alpha}}{d t} \right| =&\ \sqrt{ \sum \limits_{i=1}^{2}\sum \limits_{j=1}^{2} g_{ij} \dfrac{d u_{i}}{dt}\dfrac{d u_{j}}{dt}} \\ =&\ \sqrt{ g_{ij} \dfrac{d u_{i}}{dt}\dfrac{d u_{j}}{dt}} \end{align*} $$

In the second equality, the summation sign is omitted using Einstein notation.

Definition¹

$g_{ij} = \left\langle \mathbf{x}_{i}, \mathbf{x}_{j} \right\rangle$ is called the coefficient of the Riemannian metric, or the coefficient of the first fundamental form.

Let $M$ be the surface at $\mathbb{R}^{3}$, referred as $p \in M$. Let $\mathbf{X}, \mathbf{Y}$ be the tangent vector at $p$. Then, for the eigenmap $\mathbf{x} : U \to \mathbb{R}^{3}$ of $M$, it is expressed as follows.

$$ \mathbf{X} = X^{1}\mathbf{x}_{1} + X^{2}\mathbf{x}_{2} \quad \text{and} \quad \mathbf{Y} = Y^{1}\mathbf{x}_{1} + Y^{2}\mathbf{x}_{2} $$

The following bilinear form $I$ is defined as the Riemannian metric of the surface $\mathbf{x}$, or the first fundamental form.

$$ I : T_{p}M \times T_{p}M \to \mathbb{R} $$

$$ I (\mathbf{X}, \mathbf{Y}) = \sum \limits_{i=1}^{2} \sum \limits_{j=1}^{2} g_{ij}X^{i}Y^{j} = g_{ij}X^{i}Y^{j} = \begin{bmatrix} X^{1} & X^{2}\end{bmatrix} \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix} \begin{bmatrix} Y^{1} \\ Y^{2}\end{bmatrix} $$

The determinant of the matrix of coefficients $\left[ g_{ij} \right]$ is denoted as $g$.

$$ g := \det (\left[ g_{ij} \right]) = \begin{vmatrix} g_{11} & g_{12} \\ g_{21} & g_{22}\end{vmatrix} = g_{11}g_{22} - g_{12}g_{21} $$

The $(k,l)$ component of the inverse matrix of the matrix $\left[ g_{ij} \right]$ is denoted as $g^{kl}$.

$$ \begin{align*} \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} ^{-1} =&\ \dfrac{1}{\det \left[ g_{ij} \right]} \begin{pmatrix} g_{22} & - g_{21} \\ g_{12} & g_{22} \end{pmatrix} = \dfrac{1}{g} \begin{pmatrix} g_{22} & - g_{21} \\ g_{12} & g_{22} \end{pmatrix} \\[1em] =&\ \begin{pmatrix}\dfrac{g_{22}}{g} & - \dfrac{g_{21}}{g} \\[1em] -\dfrac{g_{12}}{g} & \dfrac{g_{11}}{g} \end{pmatrix} \\[1em] =&\ \begin{pmatrix} g^{11} & g^{12} \\[1em] g^{21} & g^{22} \end{pmatrix} \end{align*} $$

Explanation

These days, the term “first fundamental form” is hardly used and mostly, the term “Riemannian metric” is used. The name “metric” is used because, as seen in the buildup, it is used for measuring the length of curves on the surface.

Notations like $E = g_{11}$, $F=g_{21}=g_{12}$, $G=g_{22}$ are also widely used.

The reason the concept of Riemannian metric, which did not appear in curve theory, arises is because the basis of the tangent space $\left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\}$ is not generally an orthonormal basis. If it were orthonormal, $g_{ij} = \delta_{ij}$, therefore it would be meaningless. Here, $\delta$ is the Kronecker delta. Using the Riemannian metric and Einstein notation, the length of the curve $\boldsymbol{\alpha}$ on the surface can be expressed as follows.

$$ \begin{align*} L (\boldsymbol{\alpha}) =&\ \text{length of } \boldsymbol{\alpha} \\ =&\ \int_{a}^{b} \sqrt{ g_{ij} \dfrac{d u_{i}}{dt}\dfrac{d u_{j}}{dt}} dt \\ =&\ \int_{a}^{b} \sqrt{ g_{ij} \alpha_{i}^{\prime} \alpha_{j}^{\prime} } dt \\ =&\ \int_{a}^{b} \sqrt{ E\left( \dfrac{d u_{1}}{dt} \right)^{2} + 2F\dfrac{d u_{1}}{dt}\dfrac{d u_{2}}{dt} + G\left( \dfrac{d u_{2}}{dt} \right)^{2}} dt \end{align*} $$

The area of the surface is also defined by the integral of the Riemannian metric.

For a certain region $R$ on the simple surface $\mathbf{x}$, let it be $Q = \mathbf{x}^{-1}(R)$. In other words, $Q \subset U \subset \R^{2}$. Then, the area of $R$ is as follows.

$$ \text{area of } R = \iint _{Q} \sqrt{g} du_{1}du_{2} = \iint _{Q} \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right| du_{1}du_{2} = \iint _{Q} \sqrt{EG-F^{2}} du_{1}du_{2} $$

Properties

For a simple surface $\mathbf{x} : U \to \mathbb{R}^{3}$,

(a) $g = \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|^{2}$

(b) $g^{11} = \dfrac{g_{22}}{g} \quad \text{and} \quad g^{12} = g^{21} = -\dfrac{g_{12}}{g} \quad \text{and} \quad g^{22} = \dfrac{g_{11}}{g}$

(c) $\forall i,j$, $\sum \limits_{k=1}^{2} g_{ik}g^{kj} = {\delta_{i}}^{j}$

Here, $\delta$ is the Kronecker delta.

Proof

(a)

Due to the properties of the cross product and the definition of Riemannian metric, the following holds.

$$ \begin{align*} \left| \mathbf{x}_{1} \times \mathbf{x}_{2} \right|^{2} =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2} \sin ^{2} \theta \\ =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2}\left(1- \cos ^{2} \theta \right) \\ =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2}\left(1- \dfrac{\mathbf{x}_{1} \cdot \mathbf{x}_{2}}{\left| \mathbf{x}_{1} \right| \left| \mathbf{x}_{2} \right| } \right) \\ =&\ \left| \mathbf{x}_{1} \right|^{2} \left| \mathbf{x}_{2} \right|^{2} - \left( \mathbf{x}_{1} \cdot \mathbf{x}_{2} \right)^{2} \\ =&\ g_{11}g_{22} - g_{12}g_{21} \\ =&\ \det( [g_{ij}] ) \\ =&\ g \end{align*} $$

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(b)

It’s according to the definition.

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(c)

Since $[g^{kl}]$ is the inverse matrix of $[g_{ij}]$, it naturally holds.

$$ \begin{align*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} =&\ \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} \begin{pmatrix} g^{11} & g^{12} \\ g^{21} & g^{22} \end{pmatrix} \\[1em] =&\ \begin{pmatrix} g_{11}g^{11}+g_{12}g^{21} & g_{11}g^{12} + g_{12}g^{22} \\[1em] g_{21}g^{11} + g_{22}g^{21} & g_{21}g^{12} + g_{22}g^{22} \end{pmatrix} \end{align*} $$

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