7.1 Riemannian manifolds and pseudo-Riemannian manifolds

7.1.1 Metric tensors

In elementary geometry, the inner product between two vectors $\mathbf{U}$ and $\mathbf{V}$ is defined by $\mathbf{U}\cdot \mathbf{V}={\sum }_{i=1}^m{U}_i{V}_i$ where $U_i$ and $V_i$ are the components of the vectors in ${\mathbb{R}}^m$ . On a manifold, an inner product is defined at each tangent space $T_{p}M$ .

Definition 7.1

Let $M$ be a differentiable manifold. A Riemannian metric $g$ on $M$ is a type (0, 2) tensor field on $M$ which satisfies the following axioms at each point $p\in M$ :

(i) $g_p(U,V)=g_p(V,U)$
(ii) $g_p(U,U)\ge 0$ , where the equality holds only when $U=0$

Here $U,V\in T_{p}M$ and $g_p=g{|}_p$ . In short, $g_p$ is a symmetric positive-definite bilinear form.

A tensor field $g$ of type (0, 2) is a pseudo-Riemannian metric if it satisfies (i) and

(ii’) if $g_p(U,V)=0$ for any $U\in T_{p}M$ , then $V=0$ .

In chapter 5, we have defined the inner product between a vector $V\in T_M$ and a dual vector $\omega \in T_p^{\ast }M$ as a map $⟨,⟩:T_p^{\ast }M\times T_{p}M\to \mathbb{R}$ . If there exists a metric $g$ , we define an inner product between two vectors $U,V\in T_{p}M$ by $g_p(U,V)$ . Since $g_p$ is a map $T_{p}M\otimes T_{p}M\to \mathbb{R}$ we may define a linear map $g_p(U,\cdot ):T_{p}M\to \mathbb{R}$ by $V\mapsto g_p(U,V)$ . Then $g_p(U,\cdot )$ is identified with a one-form ${\omega }_U\in T_p^{\ast }M$ . Similarly, $\omega \in T_p^{\ast }M$ induces $V_{\omega }\in T_{p}M$ by $⟨\omega ,U⟩=g(V_{\omega },U)$ . Thus, the metric $g_p$ gives rise to an isomorphism between $T_{p}M$ and $T_p^{\ast }M$ .

Let $(U,\phi )$ be a chart in $M$ and $\{x^{\mu }\}$ the coordinates. Since $g\in {\mathcal{T}}_2^0(M)$ , it is expanded in terms of $\mathrm{d}{x}^{\mu }\otimes \mathrm{d}{x}^{\nu }$ as

$$g_p=g_{\mu v}(p)\mathrm{d}{x}^{\mu }\otimes \mathrm{d}{x}^v.\text{\hspace{1em}(7.1a)}$$

It is easily checked that

$$g_{\mu \nu }(p)=g_p\left(\frac{\partial }{\partial x^{\mu }},\frac{\partial }{\partial x^{\nu }}\right)=g_{\nu \mu }(p)(p\in M).\text{\hspace{1em}(7.1b)}$$

We usually omit $p$ in $g_{\mu \nu }$ unless it may cause confusion. It is common to regard $(g_{\mu \nu })$ as a matrix whose $(\mu ,\nu )$ th entry is $g_{\mu \nu }$ . Since $(g_{\mu \nu })$ has the maximal rank, it has an inverse denoted by $(g^{\mu \nu })$ according to the tradition: $g_{\mu \nu }{g}^{\nu \lambda }=g^{\lambda \nu }{g}_{\nu \mu }={\delta }_{\mu }^{\lambda }$ . The determinant $\det (g_{\mu \nu })$ is denoted by $g$ . Clearly $\det (g^{\mu \nu })=g^{-1}$ . The isomorphism between $T_{p}M$ and $T_p^{\ast }M$ is now expressed as

$${\omega }_{\mu }=g_{\mu v}{U}^v,U^{\mu }=g^{\mu v}{\omega }_v.\text{\hspace{1em}(7.2)}$$

From (7.1a) and (7.1b) we recover the ‘old-fashioned’ definition of the metric as an infinitesimal distance squared. Take an infinitesimal displacement ${\mathrm{dx}}^{\mu }\partial /\partial x^{\mu }\in T_{p}M$ and plug it into $g$ to find

$$\begin{array}{rl}\mathrm{d}{s}^2& =g\left(\mathrm{d}{x}^{\mu }\frac{\partial }{\partial x^{\mu }},\mathrm{d}{x}^{\nu }\frac{\partial }{\partial x^{\nu }}\right)=\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }g\left(\frac{\partial }{\partial x^{\mu }},\frac{\partial }{\partial x^{\nu }}\right)\\ & =g_{\mu \nu }\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }.\end{array}\text{\hspace{1em}(7.3)}$$

We also call the quantity ${\mathrm{ds}}^2=g_{\mu \nu }{\mathrm{dx}}^{\mu }{\mathrm{dx}}^{\nu }$ a metric, although in a strict sense the metric is a tensor $g=g_{\mu \nu }{\mathrm{dx}}^{\mu }\otimes {\mathrm{dx}}^{\nu }$

Since $(g^{\mu \nu })$ is a symmetric matrix, the eigenvalues are real. If $g$ is Riemannian, all the eigenvalues are strictly positive and if $g$ is pseudo-Riemannian, some of them may be negative. If there are $i$ positive and $j$ negative eigenvalues, the pair $(i,j)$ is called the index of the metric. If $j=1$ , the metric is called a Lorentz metric. Once a metric is diagonalized by an appropriate orthogonal matrix, it is easy to reduce all the diagonal elements to $\pm 1$ by a suitable scaling of the basis vectors with positive numbers. If we start with a Riemannian metric we end up with the Euclidean metric $\delta =\mathrm{diag}(1,\dots ,1)$ and if we start with a Lorentz metric, the Minkowski metric $\eta =\mathrm{diag}(-1,1,\dots ,1)$ .

If $(M,g)$ is Lorentzian, the elements of $T_{p}M$ are divided into three classes as follows,

(i) $g(U,U)>0⟶U$ is spacelike,
(ii) $g(U,U)=0⟶U$ is lightlike (or null), (7.4)
(iii) $g(U,U)<0⟶U$ is timelike.

Exercise 7.1

Diagonalize the metric

$$(g_{\mu \nu })=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

to show that it reduces to the Minkowski metric. The frame on which the metric takes this form is known as the light cone frame. Let $\{e_0,e_1,e_2,e_3\}$ be the basis of the Minkowski frame in which the metric is $g_{\mu \nu }={\eta }_{\mu \nu }$ . Show that $\{e_{+},e_{-},e_2,e_3\}$ are the basis vectors in the light cone frame, where $e_{\pm }\equiv (e_1\pm e_0)/\sqrt{2}$ . Let $V=(V^{+},V^{-},V^2,V^3)$ be components of a vector $V$ . Find the components of the corresponding one-form.

If a smooth manifold $M$ admits a Riemannian metric $g$ , the pair $(M,g)$ is called a Riemannian manifold. If $g$ is a pseudo-Riemannian metric, $(M,g)$ is called a pseudo-Riemannian manifold. If $g$ is Lorentzian, $(M,g)$ is called a Lorentz manifold. Lorentz manifolds are of special interest in the theory of relativity. For example, an $m$ -dimensional Euclidean space $({\mathbb{R}}^m,\delta )$ is a Riemannian manifold and an $m$ -dimensional Minkowski space $({\mathbb{R}}^m,\eta )$ is a Lorentz manifold.

7.1.2 Induced metric

Let $M$ be an $m$ -dimensional submanifold of an $n$ -dimensional Riemannian manifold $N$ with the metric $g_N$ . If $f:M\to N$ is the embedding which induces the submanifold structure of $M$ (see section 5.2), the pullback map $f^{\ast }$ induces the natural metric $g_M=f^{\ast }{g}_N$ on $M$ . The components of $g_M$ are given by

$$g_{M\mu \nu }(x)=g_{N\alpha \beta }(f(x))\frac{\partial f^{\alpha }}{\partial x^{\mu }}\frac{\partial f^{\beta }}{\partial x^{\nu }}\text{\hspace{1em}(7.5)}$$

where $f^{\alpha }$ denote the coordinates of $f(x)$ . For example, consider the metric of the unit sphere embedded in $({\mathbb{R}}^3,\delta )$ . Let $(\theta ,\varphi )$ be the polar coordinates of $S^2$ and define $f$ by the usual inclusion

$$f:(\theta ,\varphi )\mapsto (\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$$

from which we obtain the induced metric

$$\begin{array}{rl}{g}_{\mu \nu }\mathrm{d}{x}^{\mu }\otimes \mathrm{d}{x}^{\nu }& ={\delta }_{\alpha \beta }\frac{\partial f^{\alpha }}{\partial x^{\mu }}\frac{\partial f^{\beta }}{\partial x^{\nu }}\mathrm{d}{x}^{\mu }\otimes \mathrm{d}{x}^{\nu }\\ & =\mathrm{d}\theta \otimes \mathrm{d}\theta +{\sin }^2\theta \mathrm{d}\varphi \otimes \mathrm{d}\varphi .\end{array}\text{\hspace{1em}(7.6)}$$

Exercise 7.2

Let $f:T^2\to {\mathbb{R}}^3$ be an embedding of the torus into $({\mathbb{R}}^3,\delta )$ defined by

$$f:(\theta ,\varphi )\mapsto ((R+r\cos \theta )\cos \varphi ,(R+r\cos \theta )\sin \varphi ,r\sin \theta )$$

where $R>r$ . Show that the induced metric on $T^2$ is

$$g=r^2\mathrm{d}\theta \otimes \mathrm{d}\theta +(R+r\cos \theta {)}^2\mathrm{d}\varphi \otimes \mathrm{d}\varphi .\text{\hspace{1em}(7.7)}$$

When a manifold $N$ is pseudo-Riemannian, its submanifold $f:M\to N$ need not have a metric $f^{\ast }{g}_N$ . The tensor $f^{\ast }{g}_N$ is a metric only when it has a fixed index on $M$ .

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