Chapter 2: Tangent Spaces

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Chapter 2: Tangent Spaces

Contents
Contents
 1.  Tangent spaces
 2.  Computation in coordinates
 3.  Tangent bundles
 4.  Velocity vectors of curves
 5.  Partitions of Unity

The key of differentiation is linearization. We have known a function locally can be approximated by its linear part. For example a curve looks locally like a straight line, and we can use a tangent line to approximate. For a manifold, the corresponding concept is the tangent space. Since there is no ambient space we have to characterize tangent space in an abstract approach.
1. Tangent spaces For a smooth curve in \mathbb R^n defined by \gamma:t\to\gamma(t), the tangent vector on a point \gamma(t) can be given by \gamma'(t) where the derivative is taken componentwise. Then the tangent line is just the vector space spanned by this \gamma'(t). Now, if we identify a vector v in \mathbb R^n with the "directional derivative operator" at a given point a, that is D:f\mapsto D _ v f(a) =\frac{\mathrm d}{\mathrm dt}| _ {t=0}f(a+tv). This operation is linear over \mathbb R and satisfies the product rule: D(fg)=f(a)Dg+g(a)Df. We express v in terms of standard basis by product rule: v=v^ie _ i, and then D(f) can be written as D(f)=v^i\frac{\partial f}{\partial x^i}(a). We can see the computation is also taken derivatives componentwise. Therefore this operator is sufficient for generalization.

It is easy to see that if f is a constant function, then vf=0, and that if f(p)=g(p)=0 then v(fg)=0.

Tangent vectors act locally: If v\in T _ pM and f,g are smooth functions that agree on a neighborhood of p, then vf=vg. It can be seen by v(f-g)=0 (which will be discussed later) and the linearity of v.

Consider a smooth function between Euclidean spaces. The differential at a point is a linear map between them, which is also the best linearization of the function. When it comes to smooth functions between manifolds, we have no the concept of a "linear map" between manifolds. However tangent spaces are vector spaces from which we can construct linear maps.

The linearity of \mathrm dF _ p comes from the linearity of v.

We can check \mathrm dF _ p(v) is a derivation at F(p) indeed. For any smooth function f,g,
\begin{align*}
\mathrm dF _ p(v)(fg) & =v((fg)\circ F)=v((f\circ F)(g\circ F)) \\
& =f(F(p))\mathrm dF _ p(v)(g)+g(F(p))\mathrm dF _ p(v)(f).
\end{align*}

Consider a special case in which F is the identity map: F:M\to M. Then for any tangent vector v\in T _ p M and f\in C^\infty(M), we have \mathrm dF _ p(v)(f)=v(f), indicating \mathrm dF _ p:T _ pM\to T _ pM is the identity map \mathbf 1 _ {T _ pM}.

Let F:M\to N and G:N\to P be smooth maps and p\in M. Then:

The fact that tangent vectors act locally implies the following proposition, which indicates we can identify T _ pU with T _ pM for any point in U

We now prove the previous propositions. Let v\in T _ p M and f be a smooth function on M. The chain rule is obtained from
\begin{gather*}
\mathrm d(G\circ F) _ p(v)(f)=v(f\circ G\circ F) \\
\begin{aligned}
(\mathrm dG _ {F(p)}\circ \mathrm dF _ p)(v)(f) &=\mathrm dG _ {F(p)}(\mathrm dF _ p(v))(f)=\mathrm dF _ p(v)(f\circ G _ {F(p)}) \\
& =v(f\circ G\circ F).
\end{aligned}
\end{gather*}

If F is a diffeomorphism, then F^{-1}:N\to M is a smooth function such that F\circ F^{-1}=\mathbf 1 _ N and F^{-1}\circ F=\mathbf 1 _ M. Taking differentials at F(p) or p we obtain \mathrm dF _ p\circ \mathrm dF^{-1} _ {F(p)}=\mathbf 1 _ {T _ {F(p)N}} and \mathrm dF^{-1} _ {F(p)}\circ\mathrm dF _ p=\mathbf 1 _ {T _ p M}. Thus \mathrm dF _ p is bijective and therefore an isomorphism. The inverse of the differential follows.

To prove the next proposition, use the fact that tangent vectors act locally, and the following extending lemma may be useful.

In this lemma, "f is a smooth function on A" means f has a smooth extension in a neighborhood of each point. Specifically, for every p\in M there is an open subset W\subseteq M containing p and a smooth \tilde f:W\to N whose restriction to W\cap A agrees with f.

Note that the conclusion can be false if A is not closed.

A proof can be given based on partition of unity . Partition of unity is a useful technique that allows one to extend local constructions to the whole space. See Section 5 in which the statement of partitions of unity is provided.

For each p\in A, since f is smooth on A, we can pick for every p\in A an open subset of W _ p\subseteq U and a smooth function f _ p:W _ p\to\mathbb R^k such that p\in W _ p and f _ p| _ {W _ p\cap A}=f. Then the family of sets \{W _ p\mid p\in A\}\cup (M\setminus A) is an open cover of M. The family of sets \{W _ p\mid p\in A\}\cup \{M\setminus A\} is an open cover of M. Let \{\psi _ p\mid p\in A\}\cup\{\psi _ 0\} be a partition of unity subordinate to this cover, with \operatorname{supp}(\psi _ p)\subseteq W _ p and \operatorname{supp}(\psi _ 0)\subseteq M\setminus A.

Now we can define \tilde f:M\to\mathbb R^k by \tilde f(x)=\sum _ {p\in A}\psi _ p(x)f _ p(x). It is clearly a smooth function. For x\in A, whenever \psi _ p(x)\neq0 we have f _ p(x)=f(x) and \psi _ 0(x)=0, so
\[\tilde f(x)=\sum _ {p\in A}\psi _ p(x)f(x)=\Big(\psi _ 0(x)+\sum _ {p\in A}\psi _ p(x)\Big)f(x)=f(x),\]which means \tilde f is an extension of f. In addition, it is easy to see \operatorname{supp}\tilde f\subseteq U. The lemma is proved.

The partition of unity has an another important consequence: Let A be a closed subset of an open U\subseteq M. Then there exists a smooth bump function \phi such that \phi\in C^\infty(M), 0\leqslant\phi\leqslant1 on M, \phi\equiv1 on A and \phi\equiv0 outside U. (Consider open cover \{U,M\setminus A\}.)

We can prove that tangent vectors act locally with this corollary now. If h=f-g takes value zero on a neighborhood of p, we shall prove vh=0. We use a bump function \psi\in C^\infty(M) that is equal to 1 on \operatorname{supp}(h) and supported in M\setminus \{p\}. Then \psi h\equiv h on M, so by h(p)=\psi(p)=0 we have vh=v(\psi h)=0.

Now we return to Proposition 5. On the one hand, to prove injectivity, let v\in T _ p U and \mathrm d\iota _ p(v)=0\in T _ pM. We shall prove v=0 for T _ pU, i.e., for any f\in C^\infty(U), vf=0. Let B be a neighborhood of p such that \overline{B}\subseteq U. We extend this function to \tilde f on the whole M, i.e., \tilde{f}\in C^\infty(M) such that \tilde f| _ {\overline{B}}=f. Therefore
\[
vf=v(\tilde f| _ U)=v(\tilde f\circ\iota)=\mathrm d\iota(v) _ p\tilde f=0.
\]On the other hand, to prove surjectivity, let w\in T _ pM. We shall find a v\in T _ pU such that \mathrm d\iota _ p(v)=w, i.e., for all g\in C^\infty(M) we have \mathrm d\iota _ p(v)g=v(g| _ U)=wg. Define v by setting vf=w\tilde f where \tilde f is a smooth function on M that agrees with f on \overline{B}. It can be checked that this v is well defined and satisfies \mathrm d\iota _ p(v)=w.

The next corollary states that the dimension of the tangent space T _ pM is just \dim M. Let (U,\varphi) be a coordinate chart containing p. Since \varphi is a diffeomorphism, we know \mathrm d\varphi _ p is an isomorphism from T _ p U to T _ {\varphi(p)}\varphi(U), i.e., T _ pU\cong T _ {\varphi(p)}\varphi(U). We also have T _ pM\cong T _ pU and T _ {\varphi(p)}\varphi(U)\cong T _ {\varphi(p)}\mathbb R^n. It can be proved T _ {\varphi(p)}\mathbb R^n has dimension n (which will be discussed soon), so \dim T _ pM=n.
2. Computation in coordinates The development of tangent space theory so far might seem so abstract. It may be helpful if we can work in Euclidean spaces.

Consider the Euclidean space \mathbb R^n. We first explore the structure of its tangent space T _ a\mathbb R^n.

Proof. By definition D _ v is a derivation at a for every v. It can be easily verified that v\mapsto D _ v, so we just need to prove it is bijective.

Suppose v\in\mathbb R^n such that D _ v is zero derivation. We show v=0. In terms of standard basis we write v=v^ie _ i and take f to be the j-th coordinate function x^j, which is a smooth function on \mathbb R^n. Then 0=D _ v(x^j)=v^j. This holds for all j, so v=0.

Now suppose w\in T _ a\mathbb R^n. Let v^i=w(x^i) and v=v^ie _ i. We show w=D _ v. By Taylor's theorem,
\[
f(x)=f(a)+\frac{\partial f}{\partial x^i}(x^i-a^i)+\frac12\frac{\partial^2f}{\partial x^i\partial x^j}\Big| _ {a+\theta(x-a)}(x^i-a^i)(x^j-a^j).
\]We have known for a derivation v at p and smooth functions f,g such that f(p)=g(p)=0, it holds that v(fg)=0. And for a constant function f, Using this fact,
\begin{align*}
wf & =w(f(a))+w\Big(\frac{\partial f}{\partial x^i}(x^i-a^i)\Big)+0 \\
& =0+\frac{\partial f}{\partial x^i}(a)(w(x^i)-w(a^i))=\frac{\partial f}{\partial x^i}(a)v^i=D _ vf.
\end{align*}Hence v\mapsto D _ v is also surjective. The proposition is proved.

The corollary follows since an isomorphism maps bases to bases.

Basis for tangent spaces

With this proposition, we can try to work in local coordinates. Let (U,\varphi) be a chart. Then \varphi is a diffeomorphism from U onto \widehat U=\varphi(U)\subseteq \mathbb R^n. With the development so far, we can see the differential \mathrm d\varphi _ p is an isomorphism, and the derivations \frac{\partial}{\partial x^1}| _ {\varphi(p)},\dots,\frac{\partial}{\partial x^n}| _ {\varphi(p)} form a basis for T _ {\varphi(p)}\mathbb R^n. Their preimages under \mathrm d\varphi _ p are therefore bases for T _ pM. We adopt the notation
\[
\frac{\partial}{\partial x^i}\Big| _ p:=(\mathrm d\varphi _ p)^{-1}\Big(\frac{\partial}{\partial x^i}\Big| _ {\varphi(p)}\Big)=\mathrm d(\varphi)^{-1} _ {\varphi(p)}\Big(\frac{\partial}{\partial x^i}\Big| _ {\varphi(p)}\Big).
\]
Now let f be any smooth function on U, \widehat f=f\circ \varphi^{-1} be the coordinate representation of f and \widehat p=\varphi(p) be the coordinate representation of p. Then
\[
\frac{\partial}{\partial x^i}\Big| _ p f=\frac{\partial}{\partial x^i}\Big| _ {\varphi(p)}(f\circ\varphi^{-1})=\frac{\partial \widehat f}{\partial x^i}(\widehat p),
\]indicating the basis derivation just takes the corresponding partial derivative of f at p.

Differentials in coordinates

We first explore the case in which F:U\to V is a smooth map between open sets in Euclidean spaces: U\subseteq \mathbb R^n and V\subseteq\mathbb R^m. Let p\in M. Denote (x^1,\dots,x^n) the coordinate for U and (y^1,\dots,y^m) the coordinate for V. Then we compute the action of \mathrm dF _ p on a basis tangent vector:
\[
\mathrm dF _ p\Big(\frac{\partial}{\partial x^i}\Big| _ p\Big)f=\frac{\partial}{\partial x^i}\Big| _ p(f\circ F)=\frac{\partial f}{\partial y^j}(F(p))\frac{\partial F^j}{\partial x^i}(p),
\]which implies
\[
\mathrm dF _ p\Big(\frac{\partial}{\partial x^i}\Big| _ p\Big)=\frac{\partial F^j}{\partial x^i}(p)\frac{\partial }{\partial y^j}\Big| _ {F(p)}.
\]Writing in matrix forms, we have
\begin{align*}
& \begin{bmatrix}
\mathrm dF _ p(\frac{\partial}{\partial x^1}| _ p)&\cdots&\mathrm dF _ p(\frac{\partial}{\partial x^n}| _ p)
\end{bmatrix} \\
={} & \begin{bmatrix}
\frac{\partial }{\partial y^1}\Big| _ {F(p)}&\cdots&\frac{\partial }{\partial y^m}\Big| _ {F(p)}
\end{bmatrix}
\begin{bmatrix}
\frac{\partial F^1}{\partial x^1}(p) & \cdots &\frac{\partial F^1}{\partial x^n}(p) \\
\vdots & \ddots & \vdots \\
\frac{\partial F^m}{\partial x^1}(p) & \cdots &\frac{\partial F^m}{\partial x^n}(p)
\end{bmatrix}
\end{align*}
We can see the Jacobian appears here. In this case, the differential \mathrm d F _ p:T _ p\mathbb R^n\to T _ {F(p)}\mathbb R^m corresponds to the usual differential in calculus at p.

Now suppose F:M\to N is a smooth map between manifolds and p\in M. Let (U,\varphi) be a chart in M containing p, (V,\psi) be a chart in N containing F(p), \hat p be F(p), and \widehat F=\psi\circ F\circ\varphi^{-1} be the coordinate representation. We have obtain the result for \mathrm d\widehat{F} _ {\hat{p}}. Then again we compute the action of \mathrm dF _ p on a basis tangent vector:
\[
\mathrm dF _ p\Big(\frac{\partial}{\partial x^i}\Big| _ p\Big)=\mathrm dF _ p\Big[\mathrm d(\varphi^{-1}) _ {\hat p}\Big(\frac{\partial}{\partial x^i}\Big| _ {\hat p}\Big)\Big].
\]By \psi^{-1}\circ \widehat F=F\circ\varphi^{-1} and the chain rule for composite differentials,
\begin{align*}
&\mathrm dF _ p\Big(\frac{\partial}{\partial x^i}\Big| _ p\Big) =\mathrm d(\psi^{-1}) _ {\widehat F(\hat p)}\Big[\mathrm d\widehat F _ {\hat p}\Big(\frac{\partial}{\partial x^i}\Big| _ {\hat p}\Big)\Big] \\
={}&\mathrm d(\psi^{-1}) _ {\widehat F(\hat p)}\Big(\frac{\partial \widehat F^j}{\partial x^i}(\hat p)\frac{\partial}{\partial y^j}\Big| _ {\widehat F(\hat p)}\Big) =\frac{\partial \widehat F^j}{\partial x^i}(\hat p)\frac{\partial}{\partial y^j}\Big| _ { F( p)}.
\end{align*}The result is quite similar with that in the special case of Euclidean spaces.

Change of coordinates

Suppose (U,\varphi) and (V,\psi) are two charts on M and p\in U\cap V. Let coordinates of \varphi be (x _ 1,\dots,x _ n) and coordinates of \psi be (\tilde x _ 1,\dots,\tilde x _ n). If v=v^i\frac{\partial}{\partial x^i}| _ p=\tilde v^i\frac{\partial}{\partial \tilde x^i}| _ p, then it turns out we can compute \tilde v^i given all v^i.

We investigate the transition map \psi\circ\varphi^{-1}. The coordinate of it is \tilde x^i(x), so by the differential in components,
\[
\mathrm d\left(\psi \circ \varphi^{-1}\right) _ {\varphi(p)}\Big(\frac{\partial}{\partial x^{i}}\Big| _ {\varphi(p)}\Big)=\frac{\partial \tilde{x}^{j}}{\partial x^{i}}(\varphi(p)) \frac{\partial}{\partial \tilde{x}^{j}}\Big| _ {\psi(p)} .
\]With this result, we set \hat p=\varphi(p) and have
\begin{align*}
\frac{\partial}{\partial x^{i}}\Big| _ {p} & =\mathrm d(\varphi^{-1}) _ {\varphi(p)}\Big(\frac{\partial}{\partial x^{i}}\Big| _ {\varphi(p)}\Big)
=\mathrm d(\psi^{-1}) _ {\psi(p)} \circ \mathrm d(\psi \circ \varphi^{-1}) _ {\varphi(p)}\Big(\frac{\partial}{\partial x^{i}}\Big| _ {\varphi(p)}\Big) \\
& =\mathrm d(\psi^{-1}) _ {\psi(p)}\Big(\frac{\partial \tilde{x}^{j}}{\partial x^{i}}(\varphi(p)) \frac{\partial}{\partial \tilde{x}^{j}}\Big| _ {\psi(p)}\Big)
=\frac{\partial \tilde{x}^{j}}{\partial x^{i}}(\hat{p}) \frac{\partial}{\partial \tilde{x}^{j}}\Big| _ {p}.
\end{align*}Applying it to v=v^i\frac{\partial}{\partial x^i}| _ p=\tilde v^i\frac{\partial}{\partial \tilde x^i}| _ p we get the transformation rule
\[
\tilde v^j=\frac{\partial\tilde x^j}{\partial x^i}(\hat p)v^i.
\]
3. Tangent bundles If \{A _ i\} _ {i\in I} is a collection of subsets, their disjoint union is defined to be
\[
\coprod _ {i\in I}A _ i:=\bigcup _ {i\in I}(\{i\}\times A _ i).
\]

The proof of this proposition is not covered here but discussed a little bit. Given a chart (U,\varphi), let (x^1,\dots,x^n) be the coordinate functions of \varphi. The natural coordinates on TM is a function \tilde\varphi, defined as follows. Let v=v^i\frac{\partial }{\partial x^i}| _ p be a tangent vector at p\in U. Then
\[
\tilde\varphi(v)=(x^1(p),\dots,x^n(p),v^1,\dots,v^n)\in\mathbb R^{2n}.
\]Clearly, this is a bijection.

4. Velocity vectors of curves If M is a manifold we define a curve on M to be a continuous map \gamma:J\to M where J is an interval on \mathbb R. Suppose \gamma is a smooth curve. The velocity of \gamma at t _ 0 is defined by a tangent vector
\[
\mathrm d\gamma\Big(\frac{ \mathrm d}{\mathrm dt}\Big| _ {t _ 0}\Big)=:\gamma'(t _ 0)\in T _ {\gamma(t _ 0)}M.
\]
For a smooth function f, the defined velocity acts on it by
\[
\gamma'(t _ 0)f=\frac{\mathrm d}{\mathrm dt}\Big| _ {t _ 0}(f\circ\gamma)=(f\circ\gamma)'(t _ 0).
\]Given a chart (U,\varphi), let (x^1,\dots,x^n) be the coordinate functions of \varphi. Suppose the coordinate functions of \gamma is \gamma^i. Then by the coordinate results derived before,
\[
\gamma'(t _ 0)=\frac{\mathrm d\gamma^i}{\mathrm dt}(t _ 0)\frac{\partial}{\partial x^i}\Big| _ {\gamma(t _ 0)}.
\]This is quite similar with the formula in ordinary case. We just take derivatives in every component of \gamma.

A velocity on a curve is a tangent vector at a point. In fact, every tangent vector on a manifold is the velocity of a curve. Let (U,\varphi) be a chart centered at p. We write v=v^i\frac{\partial }{\partial x^i}| _ p in terms of coordinate basis and define locally a curve \gamma the coordinate functions of which are tv^i. Then it can be verified that \gamma(0)=p and \gamma'(0)=v.

We can compute the velocity vector of a composite curve as follows. Let F:M\to N be a smooth map and \gamma :J\to M be a smooth curve. Then it can be easily checked the velocity at t _ 0 of F\circ\gamma is given by (F\circ \gamma)'(t _ 0)=\mathrm dF(\gamma'(t _ 0)). Conversely, we can compute the differential by velocity vector. In order to compute \mathrm dF _ p(v) for a tangent vector v\in T _ pM, we can choose a curve \gamma with a tangent vector v, then the result is obtained by considering the composite curve F\circ \gamma.

5. Partitions of Unity We end this chapter with a short discussion on partitions of unity.

A (smooth) partition of unity subordinate to an open cover (X _ \alpha) _ {\alpha\in A} of M is an indexed family (\psi _ \alpha) _ {\alpha\in A} of smooth functions \psi _ \alpha:M\to\mathbb R with the following properties:

1. For all \alpha\in A and x\in M, 0\leqslant\psi _ \alpha(x)\leqslant1.
   
2. For all \alpha\in A, \operatorname{supp}\psi _ \alpha\subseteq X _ \alpha.
   
3. The family of supports (\operatorname{supp}\psi _ \alpha) _ {\alpha\in A} is locally finite, i.e., every point has a neighborhood that intersects \operatorname{supp}\psi _ \alpha for finitely many values of \alpha.
   
4. \sum _ {\alpha\in A}\psi _ \alpha(x)=1 for all x\in M.

Although this theorem is one of the most important technical tools in the theory of smooth manifolds, its proof is not covered here because it needs a great deal of knowledge and analysis in topology. It is recommended to read Lee's book for a general proof. A simpler version on Euclidean space \mathbb R^n [https://gaomj.cn/partitionofunity/] illustrates the basic ideas of the general proof.

Local finiteness

The property (3) of local finiteness can be characterized by:

1. Let K be a compact subset of M. Then for all but finitely many \alpha we have \psi _ \alpha(x)=0 for all x\in K.

The two statements are equivalent due to the local compactness of manifolds.

On the one hand, assume every point has a neighborhood that intersects \operatorname{supp}\psi _ \alpha for finitely many values of \alpha. For any compact subset K of M, every point in K has such a neighborhood, and by compactness there are finitely many of them whose union covers K. Since each of these finitely many of neighborhood intersects finitely many of the supports, we conclude that K intersects finitely many of the supports.

On the other hand, assume any compact subset K intersects finitely many of the supports. It is a topological property that every topological manifold is locally compact, i.e., every point has a neighborhood contained in a compact subset. This implies that every point has a neighborhood that intersects finitely many supports.

Indices of functions

Usually we use a sequence of functions (\phi _ i) _ {i\in\mathbb N^\ast} instead of a family of functions (\psi _ \alpha) _ {\alpha\in A}, which satisfies the same properties above. If such a sequence (\phi _ i) _ {i\in\mathbb N^\ast} is obtained, we can induce (\psi _ \alpha) _ {\alpha\in A} as follows. For any j, we can choose an index \alpha(j) such that \operatorname{supp}\phi _ j\subseteq X _ {\alpha(j)}.
Let J(\alpha) be the set of indices j such that \alpha(j)=\alpha. Then we define \psi _ \alpha=\sum _ {j\in J(\alpha)}\phi _ j, and if J(\alpha) is empty then \psi _ \alpha is zero function. It can be checked that (\psi _ \alpha) _ {\alpha\in A} has the required properties.