Subjects: Differential Geometry
Links: The Tensor Bundles, Correlations, Musical Isomorphisms, Embedded Smooth Submanifolds, Local and Global Sections of Vector Bundles

Def: Let $M$ be a smooth manifold with or without boundary. A Riemannian metric on $M$ is a smooth symmetric covariant $2$-tensor field on $M$ that is positive definite at each point. A Riemannian manifold is a pair $(M,g)$, where $M$ is a smooth manifold and $g$ is a Riemannian metric on $M$. A Riemannian manifold with boundary is defined similarly.

If $g$ is a Riemannian metric on $M$, then for each $p\in M$, the $2$-tensor $g_p$ is an inner product on $T_{p}M$. Because of this we use the notation $⟨v,w{⟩}_g$ to denote the real number $g_p(v,w)$ for $v,w\in T_{p}M$.

In any smooth local coordinates $(x^i)$, a Riemannian metric can be written as $$g = g_{ij} dx^i \otimes dx^j,$$but we can make use of the symmetric product to get $$g = g_{ij} dx^i \otimes dx^j = g_{ij} dx^i dx^j. $$
Def: The simplex example of a Riemannian metric is the Euclidean metric $\overline{g}$ on ${\mathbb{R}}^n$, given in standard coordinates by $$\bar g = \delta_{ij} dx^i dx^j $$

Def: If $(M_1,g_1)$ and $(M_2,g_2)$ are Riemannian manifolds, then product $M_1\times M_2$ has a natural Riemannian metric $g=g_1\oplus g_2$, called the product metric, defined by $$g(X_1 + X_2, Y_1 + Y_2) := g_1(X_1, Y_1) + g_2(X_2, Y_2),$$where $X_i,Y_i\in T_{p_i}{M}_i$, under the natural identification $T_{(p_1,p_2)}(M_1\times M_2)=T_{p_1}{M}_1\times T_{p_2}{M}_2$.

Obs: Any local coordinates $(x^1,\dots ,x^n)$ for $M_1$ and $(x^{n+1},\dots ,x^{n+m})$ for $M_2$ give coordinates $(x^1,\dots x^{n+m})$ for $M_1\times M_2$. In terms of these coordinates the product metric has the local expression $g=g_{ij}d{x}^{i}d{x}^j$, where $g_{ij}$ is a block diagonal matrix $$(g_{ij}) = \begin{pmatrix}
(g_1){ij} & 0 \
0 & (g_2)
\end{pmatrix}$$
Existence of Riemannian Metrics: Every smooth manifold with or without boundary admits a Riemannian metric.

We have a few of the geometric constructions that can be defined on a Riemannian manifold $(M,g)$ with or without boundary.

• The length or norm of a tangent vector $v\in T_{p}M$ is defined to be $$|v|_g := \langle v, v\rangle_g^{1/2} = g_p(v, v)^{1/2}. $$
• The angle between two nonzero tangent vectors $v,w\in T_{p}M$ is the unique $\theta \in [0,\pi ]$ satisfying $$\cos\theta = \frac{\langle v,w\rangle_g}{|v|_g |w|_g}. $$
• Tangent vectors $v,w\in T_{p}M$ are said to be orthogonal if $⟨v,w{⟩}_g=0$.

Def: Let $(M,g)$ be a an $n$-dimensional Riemannian manifold with or without boundary. We say that a local frame $(E_1,\dots ,E_n)$ for $M$ on an open subset $U\subseteq M$ is an orthonormal frame if the vectors $(E_1{|}_p,\dots ,E_n{|}_p)$ from an orthonormal basis for $T_{p}M$ at each point $p\in U$, or equivalently if $⟨E_i,E_j⟩={\delta }_{ij}$.

Prop: Suppose $(M,g)$ is a Riemannian manifold with or without boundary, and $(X_j)$ is a smooth local frame for $M$ over an open subset $U\subseteq M$. Then there is a smooth orthonormal frame $(E_i)$ over $U$ such that $\text{span}(E_1{|}_p,\dots ,E_j{|}_p)=\text{span}(X_1{|}_p,\dots ,X_j{|}_p)$ for each $j=1,\dots ,n$ and each $p\in U$.

Existence of Local Orthonormal Frames: Let $(M,g)$ be a Riemannian manifold with or without boundary. For each $p\in M$, there is a smooth orthonormal frame on a neighbourhood of $p$.

Pullback Metrics

Def: Suppose $M$, $N$ are smooth manifolds with or without boundary, $g$ is a Riemannian metric on $N$, and $F:M\to N$ is smooth. The pullback $F^{\ast }g$ is a smooth $2$-tensor field on $M$. If it is positive definite, it is a Riemannian metric on $M$, calle the pullback metric determined by $F$.

Prop: Suppose $F:M\to N$ is a smooth map and $g$ is a Riemannian metric on $N$. Then $F^{\ast }g$ is a Riemannian metric on $M$ iff $F$ is a smooth immersion.

Def: If $(M,g)$ and $(\tilde{M},\tilde{g})$ are both Riemannian manifolds, a smooth map $F:M\to \tilde{M}$ is called a Riemannian isometry if it is a diffeomorphism that satisfies $F^{\ast }\tilde{g}=g$. More generally, $F$ is called a local isometry if every point $p\in M$ has a neighbourhood $U$ such that $F{|}_U$ is an isometry of $U$ onto an open subset of $\tilde{M}$.

If there exists a Riemannian isometry between $(M,g)$ and $(\tilde{M},\tilde{g})$, we say that they are isometric as Riemannian manifolds. If each point of $M$ has a neighbourhood that is isometruc to an open subset of $(\tilde{M},\tilde{g})$, then we say that $(M,g)$ is locally isometric to $(\tilde{M},\tilde{g})$. The study of properties of Riemannian manifolds that are invariant under (local or global) isometries is called Riemannian geometry.

A Riemannian $n$-manifold $(M,g)$ is said to be a flat Riemannian manifold, and $g$ is a flat metric, if $(M,g)$ is locally isometric $({\mathbb{R}}^n,\overline{g})$.

Def: An isometry $\phi :(M,g)\to (M,g)$ is called an isometry of $M$. the set of isometries of $M$ is a group, called the isometry group of $M$; it is denoted $\mathcal{I}(M)$.

Th: Let $(M,g)$ be a Riemmanian manifold. The following statements are equivalent:

• $g$ is flat.
• Each point of $M$ is contained in the domain of a smooth coordinate chart in which $g$ has the coordinate representation $g={\delta }_{ij}d{x}^{i}d{x}^j$.
• Each point of $M$ is contained in the domain of a smooth coordinate chart in which the coordinate frame is orthonormal.
• Each point of $M$ is contained in the domain of a commuting orthonormal frame.

Lemma: Suppose $U,V\subseteq {\mathbb{R}}^n$ are connected open sets, $\phi ,\psi :U\to V$ are Riemannian isometries, and for some $p\in U$ they satisfy $\phi (p)=\psi (p)$ and $d{\phi }_p=d{\psi }_p$, then $\phi =\psi$.

Prop: The set of maps from ${\mathbb{R}}^n$ to itself given by the action of $\text{E}(n)$, the Euclidean Group, on ${\mathbb{R}}^n$ is the full group of Riemannian isometries of $({\mathbb{R}}^n,\overline{g})$.

Def: Let $(M,g)$ be a Riemannian manifold. A smooth vector field $V$ on $M$ is called a Killing vector field for $g$, named after the late nineteenth/early twentieth-century German mathematician Wilhelm Killing, if the flow of $V$ acts by isometries of $g$.

Obs: We see that a smooth vector field is a Killing vector field iff ${\mathcal{L}}_{v}g=0$.

Prop: The set of all Killing vectors on $M$ constitutes a Lie subalgebra of $\mathfrak{X}(M)$.

Prop: Let $(M,g)$ be a Riemannian manifold, and $V$ is a smooth vector field on $M$. $V$ is a Killing vector field iff it satisfies the following equation in each smooth local coordinate chart$$V^k \frac{\partial g_{ij}}{\partial x^k} + g_{jk} \frac{\partial V^k}{\partial x^i} + g_{ik} \frac{\partial V^k}{\partial x^j} = 0$$
Prop: Let $K\subseteq \mathfrak{X}({\mathbb{R}}^n)$ denote the Lie algebra of Killing vector fields with respect to the Euclidean metric. Then we see that a vector field is a Killing vector field iff $$\frac{\partial V^i}{\partial x^ j}+ \frac{\partial V^j}{\partial x^i} = 0.$$This means that the following vector fields form a basis for $K$: $$\frac{\partial}{\partial x^i}, \quad 1 \le i \le n; \qquad x^i\frac{\partial}{\partial x^j}- x^j\frac{\partial}{\partial x^i}, \quad 1\le i < j \le n. $$These vector fields represent translation, and rotations. If we consider $K_0\subseteq K$ denote the subspace consisting of fields that vanish at the origin. Then the map $$V \mapsto \left(\frac{\partial V^i}{\partial x^j}(0)\right) $$is an injective linear map from $K_0$ to $\mathfrak{so}(n)$.

Submanifolds

Pullback metrics are particularly important for submanifolds.

Def: If $(M,g)$ is a Riemannian manifold with or without boundary, every submanifold $S\subseteq M$ automatically inherits a pullback metric ${\iota }^{\ast }g$, where $\iota :S\hookrightarrow M$ is inclusion. We know that ${\iota }^{\ast }g$ is just the restriction of $g$ to pairs of vectors tangent to $S$. With this metric, $S$ is called a Riemannian submanifold (with or without boundary) of $M$.

Example: The metric $\dot{g}={\iota }^{\ast }\overline{g}$ induced on ${\mathbb{S}}^n$ by the usual inclusion $\iota :{\mathbb{S}}^n\hookrightarrow {\mathbb{R}}^{n+1}$ is called the round metric, or the standard metric, on the sphere.

Computations on a submanifold are usually most conveniently carried out in terms a local parametrization: this is an embedding of an open subset $U\subseteq {\mathbb{R}}^n$ into $N$, whose image is an open subset of $M$.

Example: If $X:U\to {\mathbb{R}}^n$ is a parametrization of a submanifold $M\subseteq {\mathbb{R}}^n$ with the induced metric, the induced metric in standard coordinates $(u^1,\dots ,u^n)$ on $U$ is just $$g = \sum_{i = 1}^n (dX^i)^2 = \sum_{i = 1}^n \left(\frac{\partial X^i}{\partial u^j} du^j\right)^2.$$
Induced Metrics on Surfaces of Revolution: Let $C$ be an embedded $1$-dimensional submanifold of the half plane $\{(r,z)\mid r>0\}$, and let $S_C$ be the surface of revolution generated by $C$ is given by $$S_C := \left{(x, y, z) \in\Bbb R^3 ;\left\rvert; \left(\sqrt{x^2+ y^2}, z\right)\right.\in C\right}. $$To compute the induced metric on $S_C$, we can choose a smooth local parametrization $\gamma (t):=(a(t),b(t))$ for $XC$, and note that the map $X(t,\theta )=(a(t)\cos \theta ,a(t)\sin \theta ,b(t))$ yields a smooth local parametrization of $S_C$, provided $(t,\theta )$ is restricted to a sufficiently small open subset of the plane. Then $$\begin{align*}
X^* g &= d((a'(t) \cos\theta ,dt - a(t) \sin\theta ,d\theta)^2 + d((a'(t) \sin\theta dt + a(t) \cos\theta d\theta)^2 + (b'(t) dt)^2 \
&= (a'(t)^2 + b'(t)^2) dt^2+ a(t)^2d\theta^2.
\end

Existence of Adapated Orthonormal Frames: Let $S\subseteq M$ be an embedded Riemannian submanifold of a Riemannian manifold $(M,g)$. For each $p\in S$, there is a smooth adapted orthonormal frame on a neighbourhood $p$ in $M$.

Prop: Suppose $(M,g)$ is an Riemannian manifold and $S\subseteq M$ an immersed $k$-dimensional manifold. The ambient tangent bundle $TM{|}_S$ is isomorphic to the Whitney sum $TS\oplus NS,$ where $NS\to S$ is the normal bundle.

Def: Suppose $\pi :N\to M$ is a smooth covering map. A covering transformation, or deck transformation is a smooth map $\phi :N\to N$ such that $\pi \circ \phi =\pi$. If $g$ is Riemannian metric on $M$, then $h:={\pi }^{\ast }g$ is a Riemannian metric on $N$ that is invariant under all covering transformations. In this case $h$ is called a covering metric, and $\pi$ is called Riemannian covering.

Prop: If $(M,g)$ is a Riemannian $n$-manifold with or without boundary, let $UM\subseteq TM$ be the subset $\{(x,v)\in TM\mid |v{|}_g=1\}$, called the unit tangent bundle of $M$. Then $UM$ is a smooth fibre bundle over $M$ with model fibre ${\mathbb{S}}^{n-1}$.