[Tu Differential Geometry] 1. Riemannian Manifolds

 This article is one of Manifold, Differential Geometry, Fibre Bundle.

We start from Riemannian manifold to travel the differential geometry.

1.1 Inner Products on a Vector Space

A point $u$ in $\mathbb{R}^3$ will denote either an ordered triple $(u^1,u^2,u^3)$ of real numbers or a column vector $$\begin{bmatrix}u^1\\ u^2\\ u^3\end{bmatrix}.$$

The Euclidean inner product, or the dot product, on $\mathbb{R}^3$ is defined by $$\left\langle u,v\right\rangle = \sum^3_{i=1} u^iv^i.$$ In terms of this, one can define the length of a vector $$\begin{align}\lVert v\rVert = \sqrt{\left\langle v,v\right\rangle},\end{align}$$ the angle $\theta$ between two nonzero vectors (Figure 1.1) 

$$\begin{align} \cos \theta = \frac{\left\langle u,v\right\rangle}{\lVert u \rVert \lVert v \rVert},\ 0\le \theta \le \pi,\end{align}$$ and the arc length of a parametrized curve $c(t)$ in $\mathbb{R}^3$, $a\le t \le b$: $$s=\int_a^b \lVert c'(t) \rVert dt.$$

Definition 1.1. An inner product on a real vector space $V$ is a positive-definite, symmetric, bilinear form $\left\langle \ ,\ \right\rangle : V\times V \rightarrow \mathbb{R}$. This means that for $u,v,w\in V$ and $a,b\in \mathbb{R}$,

(i) (positive-definiteness) $\left\langle v,v\right\rangle\ge 0$; the equality holds if and only if $v=0$.

(ii) (symmetry) $\left\langle u,v\right\rangle = \left\langle v,u\right\rangle$.

(iii) (bilinearity) $\left\langle au+bv,w\right\rangle = a\left\langle u,w\right\rangle + b\left\langle v,w\right\rangle$.

Proposition 1.2 (Restriction of an inner product to a subspace). Let $\left\langle \ ,\ \right\rangle$ be an inner product on a vector space $V$. If $W$ is a subspace of $V$, then the restriction $$\left\langle \ ,\ \right\rangle_W := \left\langle \ ,\ \right\rangle |_{W\times W}: W\times W \rightarrow \mathbb{R}$$ is an inner product on $W$.

Proposition 1.3 (Nonnegative linear combination of inner products). Let $\left\langle \ ,\ \right\rangle_i$, $i=1,\cdots,r$, be inner products on a vector $V$ and let $a_1,\cdots,a_r$ be nonnegative real numbers with at least one $a_i>0$. Then the linear combination $\left\langle\ ,\ \right\rangle := \sum a_i \left\langle\ ,\ \right\rangle_i$ is again an inner product on $V$.

1.2 Representations of Inner Products by Symmetric Matrices

Let $e_1,\cdots,e_n$ be a basis for a vector space $V$. Relative to this basis we can represent vectors in $V$ as column vectors: $$\sum x^i e_i \leftrightarrow \mathbf{x}=\begin{bmatrix} x^1 \\ \vdots \\ x^n \end{bmatrix},\ \sum y^i e_i \leftrightarrow \mathbf{y} = \begin{bmatrix} y^1 \\ \vdots \\ y^n\end{bmatrix}.$$ By bilinearity, an inner product on $V$ is determined completely by its values on a set of basis vectors. Let $A$ be the $n\times n$ matrix whose entries are $$a_{ij}=\left\langle e_i,e_j\right\rangle.$$ By the symmetry of the inner product, $A$ is a symmetric matrix. In terms of column vectors, $$\left\langle \sum x^ie_i,\sum y^je_j \right\rangle = \sum a_{ij} x^iy^j =\mathbf{x}^T A\mathbf{y}.$$

Definition 1.4. An $n\times n$ symmetric matrix $A$ is said to be positive-definite if

(i) $\mathbf{x}^T A \mathbf{x} \ge 0$ for all $\mathbf{x}$ in $\mathbb{R}^n$, and

(ii) equality holds if and only if $\mathbf{x}=\mathbf{0}$.

Thus, once a basis on $V$ is chosen, an inner product on $V$ determines a positive definite symmetric matrix.

Conversely, if $A$ is an $n\times n$ positive-definite symmetric matrix and $\{e_1,\cdots,e_n\}$ is a basis for $V$, then $$\left\langle \sum x^ie_i,\sum y^ie_i\right\rangle = \sum a_{ij} x^i y^j = \mathbf{x}^T A \mathbf{y}$$ defines an inner product on $V$.

It follows that there is a one-to-one correspondence $$\begin{Bmatrix}\mbox{inner products on a vector}\\ \mbox{space }V\mbox{ of dimension }n\end{Bmatrix} \leftrightarrow \begin{Bmatrix} n\times n \mbox{ positive-definite}\\ \mbox{symmetric matrices}\end{Bmatrix}.$$

The dual space $V^\vee$ pf a vectpr space $V$ is by definition $\mbox{Hom}(V,\mathbb{R})$, the space of all linear maps from $V$ to $\mathbb{R}$. Let $\alpha^1, \cdots,\alpha^n$ be the basis for $V^\vee$ dual to the basis $e_1,\cdots,e_n$ for $V$. If $x=\sum x^ie_i\in V$, then $\alpha^i(x)=x^i$. Thus, with $x=\sum x^i e_i$, $y=\sum \sum y^j e_j$, and $\left\langle e_i, e_j\right\rangle = a_{ij}$, one has $$\left\langle x,y\right\rangle = \sum a_{ij} x^iy^j=\sum a_{ij} \alpha^i(x)\alpha^j(y)=\sum a_{ij} (\alpha^i \otimes \alpha^j )(x,y).$$ So interms of the tensor product, an inner product $\left\langle \ ,\ \right\rangle$ on $V$ may be written as $$\left\langle \ ,\ \right\rangle = \sum a_{ij} \alpha^i \otimes \alpha^j,$$ where $[a_{ij}]$ is an $n\times n$ positive-definite symmetric matrix.

1.3 Riemannian Metrics

Definition 1.5. A Riemannian metric on a manifold $M$ is the assignment to each point $p$ in $M$ of an inner product $\left\langle \ ,\ \right\rangle_p$ on the tangent space $T_pM$; moreover, the assignment $p\mapsto \left\langle\ ,\ \right\rangle_p$ is required to be $C^\infty$ in the following sense: if $X$ and $Y$ are $C^\infty$ vector fields on $M$, then $p\mapsto \left\langle X_p, Y_p\right\rangle_p$ is a $C^\infty$ function on $M$. A Riemannian manifold is a pair $(M,\left\langle\ ,\ \right\rangle )$ consisting of a manifold $M$ together with a Riemannian metric $\left\langle\ ,\ \right\rangle$ on $M$.

The length of a tangent vector $v\in T_pM$ and the angle between two tangent vectors $u,v\in T_pM$ on a Riemannian manifold are defined by the same formulas (1) and (2) as in $\mathbb{R}^3$.

Recall that if $F:N\rightarrow M$ is a $C^\infty$ map of smooth manifolds and $p\in N$ is a point in $N$, then the differential $F_*:T_p N\rightarrow T_{f(p)}M$ is the linear map of tangent spaces given by $$(F_*X_p)g = X_p(g\circ F)$$ for any $X_p\in T_pN$ and any $C^\infty$ function $g$ defined on a neighborhood of $F(p)$ in $M$.

Definition 1.8. A $C^\infty$ map $F:(N,\left\langle\ ,\ \right\rangle')\rightarrow (M,\left\langle\ ,\ \right\rangle)$ of Riemannian manifolds is said to be metric-preserving if for all $p\in N$ and tangent vectors $u,v\in T_pN$, $\left\langle u,v\right\rangle'_p = \left\langle F_*u,F_*v\right\rangle_{F(p)}.$$ An isometry is a metric-preserving diffeomorphism.

1.4 Existence of a Riemannian Metric

A smooth manifold $M$ is locally diffeomorphic to an open subset of a Euclidean space. The local diffeomorphism defines a Riemannian metric on a coordinate open set $(U, x^1, \ldots, x^n)$ by the same formula as for $\mathbb{R}^n$. We will write $\partial_i$ for the coordinate vector field $\partial / \partial x^i$. If $X = \sum a^i \partial_i$ and $Y = \sum b^j \partial_j$, then the formula $$\langle X, Y \rangle = \sum a^i b^j$$  defines a Riemannian metric on $U$.

To construct a Riemannian metric on $M$ one needs to piece together the Riemannian metrics on the various coordinate open sets of an atlas. The standard tool for this is the partition of unity, whose definition we recall now. A collection $\{ \mathcal{S}_\alpha \}$ of subsets of a topological space $S$ is said to be locally finite if every point $p$ in $S$ has a neighborhood $U_p$ that intersects only finitely many of the subsets $\mathcal{S}_\alpha$. The support of a function $f: S \rightarrow \mathbb{R}$ is the closure of the subset of $S$ on which $f \neq 0$: $$\mbox{supp } f = \mbox{cl}\{ x \in S | f(x) \neq 0 \}.$$  Suppose $\{ U_\alpha \}_{\alpha \in A}$ is an open cover of a manifold $M$. A collection of nonnegative $C^\infty$ functions $$\rho_\alpha: M \rightarrow \mathbb{R}, \quad \alpha \in A,$$ is called a $C^\infty$ partition of unity subordinate to $\{U_\alpha\}$ if 

(i) $\mbox{supp} \rho_\alpha \subset U_\alpha$ for all $\alpha$

(ii) the collection of supports, $\{ \mbox{supp} \rho_\alpha \}_{\alpha \in A}$, is locally finite,

(iii) $\sum_{\alpha \in A} \rho_\alpha = 1$.

The local finiteness of the supports guarantees that every point $p$ has a neighborhood $U_p$ over which the sum in (iii) is a finite sum. (For the existence of a $C^\infty$ partition of unity, see [21, Appendix C].)

Theorem 1.12. On every manifold $M$ there is a Riemannian metric.

Let $\{ (U_\alpha, \phi_\alpha) \}$ be an atlas on $M$. Using the coordinates on $U_\alpha$, we define as in (1.4) a Riemannian metric $\langle\ ,\ \rangle_\alpha$ on $U_\alpha$. Let $\{\rho_\alpha\}$ be a partition of unity subordinate to $\{U_\alpha\}$. By the local finiteness of the collection $\{\mbox{supp} \rho_\alpha\}$, every point $p$ has a neighborhood $U_p$ on which only finitely many of the $\rho_\alpha$'s are nonzero. Thus, $\sum \rho_\alpha \langle\ ,\ \rangle_\alpha$ is a finite sum on $U_p$. By Proposition 1.3, at each point $p$ the sum $\sum \rho_\alpha \langle\ ,\ \rangle_\alpha$ is an inner product on $T_pM$.

To show that $\sum\rho_\alpha \langle\ ,\ \rangle_\alpha$ is $C^\infty$, let $X$ and $Y$ be $C^\infty$ vector fields on $M$. Since $\sum \rho_\alpha \langle X,Y\rangle_\alpha$ is a finite sum of $C^\infty$ functions on $U_p$, it is $C^\infty$ on $U_p$. Since $p$ was arbitrary, $\sum \rho_\alpha \langle X,Y\rangle_\alpha$ is $C^\infty$ on $M$.