This is for documenting definitions and collaries from Do Carmo’s Riemannian geometry, in order to a quick look-up.

Differentiable Manifolds, tangent space

Definition [Differentiable Manifold] A differentiable manifold of dimension $n$ in a set $M$ and a family of injective mappings $x_\alpha:U_\alpha\subset \mathbf{R}^n \rightarrow M$ of open sets $U_\alpha$ of $\mathbf{R}$ into $M$ such that:

$\cup_\alpha x_\alpha(U_\alpha) = M$ .

for any pair $\alpha,\beta$ with $x_\alpha(U_\alpha)\cap x_\beta(U_\beta) = W \neq \varnothing$ , the sets $x_\alpha^{-1}(W)$ and $x_\beta^{-1}(W)$ are open sets in $\mathbf{R}^n$ and the mappings $x^{-1}_\beta\circ x_\alpha$ are differentiable.

The family $\left\{(U_\alpha, x_\alpha)\right\}$ is maximal relative to the condition 1 and 2.

Comments: Condition 3 is purely included for technical reasons.

Definition [Differentiable Structure] A family $\{(U_\alpha, x_\alpha)\}$ satisfying above condition 1 and 2 is called a differentiable structure.

Definition [Differential Mapping] Let $M^n_1$ and $M^m_2$ be differential manifolds. A mapping $\varphi: M_1 \rightarrow M_2$ is differentiable at $p\in M_1$ if given a parametrization $y: V \subset \mathbf{R}^m\rightarrow M_2$ at $\varphi(p)$ , there exists a parametrization $x:U\subset\mathbf{R}^n\rightarrow M_1$ at $p$ such that $\varphi(x(U))\subset y(V)$ and the mapping

$y^{-1}\circ \varphi \circ x :U\subset R^n \rightarrow \mathbf{R}^m$

is differentiable at $x^{-1}(p)$ .

Comment: $\varphi$ is differentiable on an open set of $M_1$ if it is differentiable at all of the points of this open set.

Definition [Tagent Vector] Let $M$ be a differentiable manifold. A differentiable function $\alpha:(-\varepsilon,\varepsilon)\rightarrow M$ is called a (differentiable) curve in $M$ . Suppose that $\alpha(0) = p \in M$ , and let $\mathcal{D}$ be the set of functions on $M$ that are differentiable at $p$ . The tangent vector to the curve $\alpha$ at $t=0$ is a function $\alpha'(0):\mathcal{D}\rightarrow \mathbf{R}$ given by

$\alpha'(0)f = \left.\frac{d(f\circ\alpha)}{dt}\right|_{t=0}=(\sum_ix'_i(0)\frac{\partial}{\partial x_i})f, \quad f\in\mathcal{D}.$

If $\alpha(t)$ has parametrization $\alpha(t)=(x_1(t),\cdots,x_n(t))$ .

Comments:

This means that a tangent vector at $p$ is the tangent vector of some curve that passes $p$ .
[Tangent Space] The set of all tangent vectors to $M$ at $p$ will be indicated by $T_pM$ .
The choice of parametrization $x:U\rightarrow M$ determines an associated basis $\left\{\left(\frac{\partial}{\partial x_1}\right)_0, \cdots,\left(\frac{\partial}{\partial x_n}\right)_0 \right\}$ .

Definition [Differential] Let $M^n_1$ and $M^m_2$ be differentiable manifolds and let $\varphi:M_1\rightarrow M_2$ be a differentiable mapping. For every $p\in M_1$ and for each $v\in T_pM_1$ , choose a differentiable curve $\alpha: (-\varepsilon, \varepsilon)\rightarrow M_1$ with $\alpha(0) = p$ , $\alpha'(0)=v$ . Take $\beta = \varphi \circ \alpha$ . The mapping $d\varphi_p:T_pM_1\rightarrow T_{\varphi(p)M_2}$ given by $d\varphi_p(v) = \beta'(0)$ is called the differential of $\varphi$ at $p$ , which is a linear mapping that does not depend on the choice of $\alpha$ .

Definition [Diffeomorphism] Let $M_1$ and $M_2$ be differentiable manifolds. A mapping $\varphi:M_1\rightarrow M_2$ is a diffeomorphism if it is differentiable, bijective and its inverse $\varphi^{-1}$ is differentiable. $\varphi$ is said to be a local diffeomorphism at $p\in M$ if there exist neigborhoods $U$ of $p$ and $V$ of $\varphi(p)$ such that $\varphi:U \rightarrow V$ is a diffeomorphism.

Comments:

If $\varphi:M_1\rightarrow M_2$ is a diffeomorphism, then $d\varphi_p:T_pM_1\rightarrow T_{\varphi(p)}M_2$ is an isomorphism for all $p\in M_1$ .

Theorem [Local Diffeomorphism] Let $\varphi: M_1^n\rightarrow M^m_2$ be a differentiable mapping and let $p\in M_1$ be such that $d\varphi_p :T_pM_1\rightarrow T_{\varphi(p)}M_2$ is an isomorphism. Then $\varphi$ is a local diffeomorphism at $p$ .

Immersions and Embeddings

In short, immersion + homeomorphism = embedding. When the inclusion map is an embedding, there exists submanifolds.

Definition [Immersion,Embedding] Let $M^m$ and $N^n$ be differentiable manifolds. A differentiable mapping $\varphi: M\rightarrow N$ is said to be an immersion if $d\varphi_p:T_mM\rightarrow T_{\varphi(p)}N$ is injective for all $p\in M$ . If, in addition, $\varphi$ is a homeomorphism onto $\varphi(M)\subset (N)$ , where $\varphi(M)$ has the subspace topology induced from $N$ , we say that $\varphi$ is an embedding. If $M\subset N$ and the inclusion $i:M \subset N$ is an embedding, we say that $M$ is a submanifold of $N$ .

Proposition [Relation between immersion and embedding] Let $\varphi:M_1^n\rightarrow M_2^m$ , $n\leqslant m$ , be an immersion of the differentiable manifold $M_1$ into the differentiable manifold $M_2$ . For every point $p\in M_1$ , there exists a neighborhood $V\subset M_1$ of $p$ such that the restriction $\varphi|V\rightarrow M_2$ is an embedding.

Comments: This shows that every immersion is locally (in a certain sense) an embedding.

Other examples of manifolds

Definitioin [Regular Surface in $\mathbf{R}^n$ ] A subset $M^k\subset \mathbf{R}^n$ is a regular surface of dimension $k$ if for every $p\in M$ there exists a neighborhood $V$ of $p$ in $\mathbf{R}^n$ and a mapping $x:U\subset\mathbf{R}^k \rightarrow M\cap V$ of an open set $U\subset \mathbf{R}^k$ onto $M\cap V$ such that:

$x$ is a differentiable homeomorphism.

$(dx)_q:\mathbf{R}^k\rightarrow \mathbf{R}^n$ is injective for all $q\in U$ .

Definition [Regular and Critical Value] Let $F: U\subset \mathbf{R}^n\rightarrow \mathbf{R}^m$ be a differentiable mapping of an open set $U$ of $\mathbf{R}^n$ . A point $p\in U$ is defined to be a critical point of $F$ if the differential $dF_p:\mathbf{R}^n \rightarrow\mathbf{R}^m$ is not surjective. The image $F(p)$ of a critical point $P$ is called a critical value of $F$ . A point $a\in \mathbf{R}^m$ that is not a critical value is said to be a regular value of $F$ . Note that any point $a\in F(U)$ is trivially a regular value of $F$ and that is there exists a regular value of $F$ in $\mathbf{R^m}$ , then $n\geqslant m$ .

Comments: The inverse image $F^{-1}(a) \subset \mathbf{R}^n$ of a regular value $a \in F(U)$ of $F$ , is a regular surface of dimension $n-m=k$ .

Definition [Orientation] Let $M$ be a diffeerentiable manifold. We say that $M$ is orientable if $M$ admits a differentiable structure $\{(U_\alpha, x_\alpha)\}$ such that:
(i) for every pair $\alpha,\beta$ , with $x_\alpha(U_\alpha)\cap x_\beta(U_\beta) = W \neq \varnothing$ , the differential of the change of coordinates $x^{-1}_\beta\circ x_\alpha$ has positive determinant.

comment: let $M_1$ and $M_2$ be differential manifolds and let $\varphi:M_1 \rightarrow M_2$ be a diffeomorphism, then $M_1$ is orientable if and only if $M_2$ is orientable.

Definition [Action of a Group] A group $G$ acts on a differentiable manifold $M$ if there exists a mapping $\varphi:G\times M\rightarrow M$ such that:

For each $g\in G$ , the mapping $\varphi_g:M\rightarrow M$ given by $\varphi_g(p) = \varphi(g,p), p\in M$ is a diffeomorphism, and $\varphi_e=$ identity.

If $g_1,g_2\in G$ , $\varphi_{g_1,g_2} = \varphi_{g_1}\circ\varphi_{g_2}$ .

Definition [Properly Discontinuous] For $g\in G$ , we say the action is properly discontinuous is every $p\in M$ has a neighborhood $U\subset M$ such that $U\cap g(U) = \varnothing$ for all $g\neq e$ .

Definition [Projection] When $G$ acts on $M$ , the action determines an equivalence relation $\sim$ on $M$ , in which $p_1\sim p_2$ if and only if $p_2 = gp_1$ for some $g\in G$ . Denote the quotient space of $M$ by this equivalence relation by $M/G$ , given by

$\pi(p)=\text{equiv. class of }p=Gp$

will be called the projection of $M$ onto $M/G$ .

Comments: when $G$ is properly discontinuous, then $M/G$ has a differentiable structure with respect to which the projection $\pi :M\rightarrow M/G$ is a local diffeomorphism. (The discountinuous guerantees the injectivity.)

Vector Fields; brackets.

Definition [Vector Field] A vector field $X$ on a differentiable manifold $M$ is a correspondence that associates to each point $p\in M$ a tangent vector $X_p\in T_pM$ . In terms of mappings, $X$ is a mpping of $M$ into the tangent bundle $TM$ . The field is differentiable if the mapping $X: M\rightarrow TM$ is differentiable.

Consider a parametrization on $x:U\in \mathbf{R}^n\rightarrow M$ at $p\in M$ . The vector field $X$ can be written as

$X = \sum_ia_i(p)\frac{\partial}{\partial x_i}.$

where each $a_i$ is a differentiable function on $U$ and $\left\{\frac{\partial}{\partial x_i}\right\}$ is the basis associated to $x$ .

Comments:

It is covenient to think of a vector field as a mapping $X: M\rightarrow \mathcal{D} \rightarrow \mathcal{F}$ from the set $\mathcal{D}$ of differentiable functions on $M$ to the set $\mathcal{F}$ of functions on $M$ , defined in the following way:

$(Xf)(p) = \sum_ia_i(p)\frac{\partial f}{\partial x_i}(p), \quad f\in \mathcal{D}, p\in M.$

where $f$ denotes the expression of $f$ in the parameterization $x$ .

$X$ is differentiable if and only if $X:\mathcal{D}\rightarrow \mathcal{D}$ .
If $\varphi:M\rightarrow M$ is a diffeomorphism, $v\in T_pM$ , and $f$ is a differentiable function in a neighborhood of $\varphi(p)$ , we have

$(d\varphi_p(v)f)\varphi(p) = v(f\circ \varphi)(p).$

Definition [Bracket] Let $X$ and $Y$ be differentiable vector fields on a differentiable manifold $M$ . Then there exists a unique vector field $Z$ such that, for all $f\in\mathcal{D}$ , $Zf = (XY-YX)f$ . This vector field is called the bracket of $X$ and $Y$ and is denoted by $[X,Y]$ .

Proposition [Properties of the bracket] Let $a,b$ be real numbers and $f,g$ bedifferentiable functions. The bracket of vector fields satisfies the following properties:

$[X,Y] = -[Y,X]$ .(Anticommuntativity)

$[aX+bY,Z] = a[X,Z]+b[Y,Z]$ .(Linearity)

$[[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0$ .(Jacobi identity)

$[fX,gY] = fg[X,Y]+fX(g)Y-gY(f)X$ . (Leibniz rule)

Comments: Suppliment difinition. If $f$ is a scalar function on $M$ and $Y$ is a vector field, then $fY$ is a vector field such that $fY(x)=f(x)Y_x$ at $x\in M$ .

Relation with Differential Equations

Let $X$ be a vector field on a differentiable manifold $M$ and let $p\in M$ . Then there exist a neighborhood $U\in M$ of $p$ , an interval $(-\delta,\delta)$ , $\delta>0$ , and a differentiable mapping $\alpha:(-\delta,\delta)\times U\rightarrow M$ such that the curve $t\rightarrow \varphi(t,q)$ , $t\in (-\delta,\delta)$ , $q\in U$ is the unique curve which satisfies $\frac{\partial \varphi}{\partial t} = X(\varphi(t,q))$ and $\varphi(0,q) = q$ . This guarantees the uniqueness of the trajectory and that the mapping obtained depends on $t$ and the initial conditon $q$ .

Definition [Trajectory, Local FLow] A curve $\alpha:(-\delta,\delta)\rightarrow M$ which satisfies the conditions $\alpha'(t)=X(\alpha(t))$ and $\alpha(0)=q$ is called a trajectory of the vector field $X$ that passes through $q$ for $t=0$ . It is common to use the notation $\varphi_t(q)=\varphi(t,q)$ and call $\varphi_t:U\rightarrow M$ the local flow of $X$ .

Proposition Let $X,Y$ be differential vector fields on a fifferentiable manifold $M$ , let $p\in M$ and let $\varphi_t$ be the local flow of $X$ in a nrighboorhood $U$ of $p$ . Then

$[X,Y](p) = \lim_{t\rightarrow 0}\frac{1}{t}[Y-d\varphi_tY](\varphi_t(p)).$

Comments: The bracket $[X,Y]$ can slao be interpretated as a derivation of $Y$ along the “trajectories” of $X$ .

Topology of Manifolds

Definition [Locally Finite] Let $M$ be a differtiable manifold. A family of open sets $\{U_\alpha\}$ is said to be locally finite if for every $p\in M$ there exists a neighborhood $V$ of $p$ such that $V$ intersects only finitely many sets $U_\alpha$ .

Definition [Support] The support of a function $f$ on a differentiable manifold $M$ is the closure of the set of points $p\in M$ such that $f(p)\neq 0$ .

Definition [Differentiable Partition of Unity] We say that a family $\{f_\alpha\}$ of differentiable functions $f_\alpha:M\rightarrow \mathbf{R}$ is a differentiable partition of unity if:

For all $\alpha, f_\alpha\geqslant 0$ and the support of $f_\alpha$ is contained in a coordinate neighborhood $V_\alpha = x_\alpha(U_\alpha)$ of a differentiable structure $\{(U_\alpha, x_\alpha)\}$ of $M$ .

The family $\{V_\alpha\}$ is locally finite.

$\sum_\alpha f_\alpha(p) = 1$ for all $p\in M$ . (This makes sense since for eahc $p$ , only finitely many $f_\alpha$ are non-zero.)

Comments: It is customary to say that the partition of unity $\{f_\alpha\}$ is subordinate to the covering $\{V_\alpha\}$ .

Theorem [Existence of Partition of Unity] A differentiable manifold $M$ has a differentiable partition of unity if and only if every connected component of $M$ is Hausdorff and has a countable basis.

Theorem [Whitney Embedding Theorem] Any differentiable manifold of dimension $n$ can be immersed in $\mathbf{R}^{2n}$ and embedded in $\mathbf{R}^{2n+1}$ (can be refined to $\mathbf{R}^{2n-1}, n>1$ and $\mathbf{R}^{2n}$ , respectively).