Basic Concepts of Manifolds

Topological Manifold

Definition

Definition. An $n$-dimensional topological manifold is a Hausdorff topological space which is locally homeomorphic to an $n$-dimensional Euclidean space.

The statement “locally locally homeomorphic to an $n$-dimensional Euclidean space” means that given any point $p$ in the topological space $M$, there exists a neighborhood of $p$, $N(p)$, such that $N(p)$ is locally homeomorphic to the Euclidean space $\mathbb{R}^n$.

Coordinate Chart

Definition. If $U_\alpha$ is an open ser in $\mathbb{R}^n$,

$$\mathbf{x}_\alpha:U_\alpha\to \mathbf{x}_\alpha (U_\alpha)\subset M$$

is a homeomorphism and $\mathbf{x}_\alpha (U_\alpha)$ contains $p$, then the two-tuple $(U_\alpha,\mathbf{x}_\alpha)$ is said to be a coordinate chart of $M$ at the point $p$.

Given a coordinate chart, $\mathbf{x}^{-1}_{\alpha}$ is often called the local coordinate mappingat thr point $p$. The image of the point $p$ under $\mathbf{x}^{-1}_{\alpha}$, namely $\mathbf{x}^{-1}_\alpha(p)$ is called the local coordinate of $p$. Coordinate charts can transform the local part of a topological manifold into the coordinates in the familiar $\mathbb{R}^n$ , so they are sometimes called parameterization.

Topological manifold $M$ has at least one coordinate card at any point $p$. In fact, let $\eta$ be a homeomorphism from $N(p)$ to $\mathbb{R}^n$. According to the definition of neighborhood, there is an open set $W$ in $M$ such that $p\in W\subset N(p)$. Therefore, if $\eta$ is limited to $W$ and $\mathbf{x}=(\left.\eta\right|_{W})^{-1}$, then

$$\mathbf{x}:U\to \mathbf{x}(U)$$

is also a homeomorphism. Hence, $(U,\mathbf{x})$ is a coordinate chart at the point $p$.

Atlas

Definition. If $\Sigma=\{(U_\alpha,\mathbf{x}_\alpha)\}_{\alpha\in A}$ is a set of coordinate charts of $M$abd $\{\mathbf{x}_\alpha(U_\alpha)\}_{\alpha\in A}$ is a cover of $M$, then $\Sigma$ is said to be an atlas of $M$.

The topological manifold $M$ can be regarded as a topological space obtained by gluing together many open subsets $U_\alpha (\alpha \in A)$ of Euclidean spaces.

Given an atlas $\Sigma$ of the topological manifold $M$, where $\Sigma=\{(U_\alpha,\mathbf{x}_\alpha)\}_{\alpha\in A}$, some regions covered by coordinate charts in the atlas may overlap. In other words, some points in $M$ may be covered by multiple coordinate charts. Suppose the regions covered by $(U_\alpha,\mathbf{x}_\alpha)$ and $(U_\beta,\mathbf{x}_\beta)$ overlap, that is,

$$\mathbf{x}_\alpha(U_\alpha)\cap \mathbf{x}_\beta(U_\beta)\ne\varnothing,$$

we can define the transition function

$$\mathbf{x}^{-1}_\beta\circ \mathbf{x}_{\alpha}: \mathbf{x}^{-1}_\alpha(\mathbf{x}_\alpha(U_\alpha)\cap \mathbf{x}_\beta(U_\beta)) \longrightarrow \mathbf{x}^{-1}_\beta(\mathbf{x}_\alpha(U_\alpha)\cap \mathbf{x}_\beta(U_\beta))$$

and

$$\mathbf{x}^{-1}_{\alpha}\circ \mathbf{x}_\beta: \mathbf{x}^{-1}_\beta(\mathbf{x}_\alpha(U_\alpha)\cap \mathbf{x}_\beta(U_\beta))\longrightarrow \mathbf{x}^{-1}_\alpha(\mathbf{x}_\alpha(U_\alpha)\cap \mathbf{x}_\beta(U_\beta)).$$

Because the composition of homeomorphisms is still a homeomorphism, the two transition functions mentioned above are both homeomorphisms. It has been mentioned earlier that a point $p$ in $M$ may be covered by multiple coordinate charts, meaning that point $p$ may have multiple local coordinates. With the help of transition functions, we can switch between the multiple local coordinates of point $p$.

Differential Manifold

Motivation: Define Differentiable Functions

A topological manifold is a type of topological space that locally “resembles” Euclidean space. Here, “resembles” refers to having a consistent topological structure. For topological manifolds, we have a Euclidean topology locally, which allows us to use techniques developed for Euclidean topologies to handle topological manifolds. However, Euclidean spaces also possess structures more refined than just topology; for example, we can define differentiable functions and compute derivatives in Euclidean spaces. If a topological manifold locally inherits this more refined structure, similarly, we can define differentiable functions and compute derivatives on the manifold. Next, we will provide a formal definition of this more refined structure.

Suppose $M$ is an $n$-dimensional topological manifold, and $\Sigma=\{(U_\alpha,\mathbf{x}_\alpha)\}_{\alpha\in A}$ is an atlas of $M$. Our goal is to define differentiable functions $f:M\to \mathbb{R}^m$ correctly on the topological manifold. A natural idea is to use the local coordinate chart $(U_\alpha,\mathbf{x}_\alpha)$ to pull back the domain from a subset of $M$ to $U_\alpha \subset \mathbb{R}^n$. Specifically, we can propose that the function $f:M\to \mathbb{R}^m$ is differentiable at point $p$ if and only if for any coordinate chart $(U_\alpha,\mathbf{x}_\alpha)$ covering point $p$, the composed function

$$f\circ \mathbf{x}_\alpha:U_\alpha\longrightarrow \mathbb{R}^m$$

is differentiable at the point $\mathbf{x}^{-1}_\alpha(p)$.

Compatible Atlas

While this definition is theoretically sound, it can be cumbersome in practice. To prove the differentiability of $f$ at point $p$, we would need to verify it for all coordinate charts covering point $p$. However, if we consider so-called compatible atlases, the situation simplifies significantly. In this case, we only need to verify it for one coordinate chart covering point $p$.

Two coordinate charts $(U_\alpha,\mathbf{x}_\alpha),(U_\beta,\mathbf{x}_\beta)$ are termed compatible if

$$U_\alpha\cap U_\beta\ne\varnothing\implies\mathbf{x}^{-1}_\beta\circ \mathbf{x}_{\alpha},\mathbf{x}^{-1}_{\alpha}\circ \mathbf{x}_\beta \text{ are differentiable functions}.$$

If any two coordinate charts in an atlas are compatible, then the atlas is termed compatible. Let $(U_\alpha,\mathbf{x}_\alpha),(U_\beta,\mathbf{x}_\beta)$ be two coordinate charts covering point $p$. Notably,

$$\begin{aligned} f\circ \mathbf{x}_\beta &= (f\circ \mathbf{x}_\alpha)\circ(\mathbf{x}^{-1}_\alpha\circ \mathbf{x}_{\beta}),\\ f\circ \mathbf{x}_\alpha &= (f\circ \mathbf{x}_\beta)\circ(\mathbf{x}^{-1}_\beta\circ \mathbf{x}_{\alpha}), \end{aligned}$$

we have that $f\circ \mathbf{x}_\beta$ is differentiable if and only if $f\circ \mathbf{x}_\alpha$ is differentiable. Consequently, if the given atlas is compatible, we can define differentiable functions as follows: the function $f:M\to \mathbb{R}^m$ is differentiable at point $p$ if and only if there exists a coordinate chart $(U_\alpha,\mathbf{x}_\alpha)$ covering point $p$, such that the composed function

$$f\circ \mathbf{x}_\alpha:U_\alpha\longrightarrow \mathbb{R}^m$$

is differentiable at point $\mathbf{x}^{-1}_\alpha(p)$.

Differential Structure and Differential Manifold

Compatible atlases can be maximized. Let $\Sigma=\{(U_\alpha,\mathbf{x}_\alpha)\}_{\alpha\in A}$ be a compatible atlas of the topological manifold $(M,\tau)$. Then,

$$\Sigma_{\max}=\{(U_\alpha,\mathbf{x}_\alpha)|(U_\alpha,\mathbf{x}_\alpha) \text{ is a coordinate chart of } M, \text{ and is compatible with every coordinate chart in } \Sigma\}$$

is termed the maximalization atlas of $\Sigma$. A maximalization compatible atlas $\Sigma_{\max}$ on the topological manifold $(M,\tau)$ is termed a differential structure of $(M,\tau)$. Equipped with a differential structure, a topological manifold is called a differential manifold, denoted as the triple $(M,\tau,\Sigma_{\max})$. If not specifically stated otherwise, when discussing coordinate charts of a differential manifold in the future, it is always assumed that these coordinate charts are from the atlas inherent to the differential manifold.

If we replace “smooth” with $C^r$ differentiable or “smooth” throughout, we can similarly define $C^r$ compatible atlases, smooth compatible atlases, $C^r$ differential structures, smooth structures, $C^r$ differential manifolds, and smooth manifolds. In the following discussions, we will mainly focus on smooth manifolds, but the concepts are entirely transferable to differential manifolds and $C^r$ differential manifolds.

Smooth Mapping

Definition

One of the motivations for defining smooth (differentiable) manifolds is to define smooth (differentiable) functions. Let $(M,\tau,\Sigma)$ be a smooth manifold. A function $f:M\to \mathbb{R}^n$ is smooth at point $p$ if and only if there exists a coordinate chart $(U_\alpha,\mathbf{x}_\alpha)\in\Sigma$ covering point $p$, such that the composed function

$$f\circ \mathbf{x}_\alpha:U_\alpha\longrightarrow \mathbb{R}^m$$

is smooth at point $\mathbf{x}^{-1}_\alpha(p)$.

Furthermore, we can define smooth mappings between smooth manifolds $(M_1,\tau_1,\Sigma_1), (M_2,\tau_2,\Sigma_2)$. To define the smoothness of a mapping $f:M_1\to M_2$ at point $p$, besides pulling back the domain from a subset of $M_1$ to $U_\alpha\subset\mathbb{R}^n$, we also need to push forward the codomain from $M_2$ to $V_\beta\subset\mathbb{R}^m$.

A mapping $f:M_1\to M_2$ is smooth at point $p$ if there exists a coordinate chart $(U_\alpha,\mathbf{x}_\alpha)\in\Sigma_1$ covering point $p$, and a coordinate chart $(V_\beta,\mathbf{y}_\beta)\in\Sigma_2$ covering point $f(p)$, such that the composed mapping

$$\mathbf{y}^{-1}_\beta \circ f\circ \mathbf{x}_\alpha:U_\alpha\longrightarrow V_\beta$$

is smooth at point $\mathbf{x}^{-1}_\alpha(p)$.

Note that

$$f = \mathbf{y}_\beta\circ(\mathbf{y}^{-1}_\beta \circ f\circ \mathbf{x}_\alpha)\circ\mathbf{x}^{\hspace{2pt}-1}_\alpha$$

so the smoothness of $f$ at point $p$ implies continuity of $f$ at point $p$.

Also note that

$$\tilde{\mathbf{y}}^{-1}_\beta \circ f\circ \tilde{\mathbf{x}}_\alpha=(\tilde{\mathbf{y}}^{-1}_\beta\circ\mathbf{y}_\beta) \circ(\mathbf{y}^{-1}_\beta \circ f\circ \mathbf{x}_\alpha)\circ(\mathbf{x}^{-1}_\alpha\circ \tilde{\mathbf{x}}_\alpha),$$

thus the smoothness of $f:M_1\to M_2$ at point $p$ does not depend on the choice of coordinate charts.

If the mapping $f:M_1\to M_2$ is smooth at every point in an open set $U\subset M_1$, we say $f$ is smooth on $U$. If $f$ is smooth at every point in $M_1$, we say $f:M_1\to M_2$ is a smooth mapping.

Now that we have defined smooth mappings, another motivation for defining smooth (differentiable) manifolds is to compute differentials. Following the idea of defining smooth mappings, if we use coordinate charts to convert smooth mappings into smooth functions from $U_\alpha\in\mathbb{R}^n$ to $\mathbb{R}^m$, we find that with different choices of coordinate charts, we obtain different smooth functions and naturally different differentials. A simplest example is a smooth real-valued function $f:\mathbb{R}\to\mathbb{R}$. If we choose different coordinate charts $(U_\alpha,x_\alpha),(V_\beta,y_\beta)$ at point $p$, differentials of the form as shown below

$$d(y^{-1}_\beta \circ f\circ x_\alpha)|_p:\mathbb{R}\to\mathbb{R}$$

will inevitably differ. However, these different differentials cannot be entirely unrelated because their differences lie only in the differentials of the transition functions. Similar situations occur in linear algebra. With different choices of bases, a linear transformation induces different transformations from coordinate space $\mathbb{R}^n$ to coordinate space $\mathbb{R}^m$. The differences between these transformations lie only in the change of basis. Hence, we can imagine that the true “differential” of a smooth mapping might also be a linear mapping between abstract linear spaces. Only with specific choices of coordinate charts do we obtain linear mappings between Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^m$. In fact, the so-called “abstract linear spaces” here are precisely what we are about to define as tangent spaces.

Tangent Vector and Tangent Space

The First Definition: Equivalent Class of Curve

Consider an intuitive example, a $2$-dimensional smooth surface in $\mathbb{R}^3$. At any point $p$ on the surface, we can construct a tangent plane, and any vector starting at point $p$ and ending at any point within the tangent plane is a tangent vector. Thus, the tangent plane at point $p$ of the surface can be seen as a linear space $\mathbb{R}^2$ composed of tangent vectors.

For any tangent vector, it can be regarded as the tangent vector of a smooth curve $\gamma$ passing through point $p$ on the surface at point $p$. If the parameterization of the curve $\gamma$ is given by $\gamma(t)=(x(t),y(t),z(t))$, and satisfies $\gamma(t_0)=p$, then its derivative at $p$

$$\gamma'(t_0)=\left(\left.\frac{dx}{dt}\right|_{t_0},\left.\frac{dy}{dt}\right|_{t_0}, \left.\frac{dz}{dt}\right|_{t_0}\right)$$

is a tangent vector at that point. Suppose there is another curve $\theta$ on the surface passing through point $p$. If $\theta$ and $\gamma$ have the same derivative at point $p$, i.e., $\theta’(t_0)=\gamma’(t_0)$, then these two curves actually correspond to the same tangent vector. Therefore, a tangent vector can also be viewed as an equivalence class of curves, where all curves in the class have the same derivative at point $p$. If we can generalize the concepts of curves and curve derivatives to general smooth manifolds, we can define tangent vectors and tangent spaces for smooth manifolds in the same way.

Given an $n$-dimensional smooth manifold $(M,\tau,\Sigma)$ and a point $p$ on it. If a smooth mapping $\gamma:(-\epsilon,\epsilon)\to M$ satisfies $\gamma(0)=p$, then $\gamma$ is termed a smooth curve on $M$ passing through point $p$. If we want to “take derivatives” of curves, we must first parameterize them, i.e., choose a coordinate chart $(U,\mathbf{x})\in\Sigma$ at point $p$, and then take derivatives of the composed function

$$\mathbf{x}^{-1} \circ \gamma:(-\epsilon,\epsilon)\to \mathbb{R}^n$$

The result of taking derivatives is $\left.\frac{d(\mathbf{x} \circ \gamma)}{dt}\right|_0$. If the curve $\theta$ and $\gamma$ have the same derivative under a certain coordinate chart $(U_1,\mathbf{x}_1)$ at point $p$, i.e.,

$$\left.\frac{d(\mathbf{x}_1^{-1} \circ \theta)}{dt}\right|_0=\left.\frac{d(\mathbf{x}_1^{-1} \circ \gamma)}{dt}\right|_0,$$

then under any other coordinate chart $(U_2,\mathbf{x}_2)$ at point $p$, the curves $\theta$ and $\gamma$ will also have the same derivative. To prove this, it suffices to note that according to the chain rule,

$$\left.\frac{d(\mathbf{x}_2^{-1} \circ \theta)}{dt}\right|_0=\left.\frac{d((\mathbf{x}_2 \circ \mathbf{x}_1)\circ(\mathbf{x}_1^{-1} \circ \theta))}{dt}\right|_0=\left.\frac{d(\mathbf{x}_2^{-1} \circ \mathbf{x}_1)}{dx}\right|_p\left.\frac{d(\mathbf{x}_1^{-1} \circ \theta)}{dt}\right|_0.$$

Therefore, we can define: curves $\theta$ and $\gamma$ are equivalent at point $p$ if and only if there exists a coordinate chart $(U,\mathbf{x})$ at point $p$, such that

$$\left.\frac{d(\mathbf{x}^{-1} \circ \theta)}{dt}\right|_0=\left.\frac{d(\mathbf{x}^{-1} \circ \gamma)}{dt}\right|_0.$$

Let the equivalence class of the curve $\gamma$ be denoted as $[\gamma]_*$. The quotient set of all curves on $M$ passing through point $p$ under this equivalence relation is the tangent space of $M$ at point $p$, denoted as

$$T_pM=\{[\gamma]_*:\gamma\text{ is a smooth curve on } M \text{ passing through point } p\}.$$

The Second Definition: Curve Produces functional

By considering tangent vectors as equivalence classes of curves, we have successfully extended the notion of tangent vectors of smooth surfaces in $\mathbb{R}^n$ to general smooth manifolds. Although this definition of tangent vectors is intuitive, to demonstrate that the tangent space consisting of all tangent vectors is a linear space, we need to define appropriate addition and scalar multiplication operations for equivalence classes of curves. The most straightforward idea is to define the sum of equivalence classes of curves as the equivalence class of the sum of curves. However, by a simple attempt

$$\gamma(0)=\theta(0)=p\implies (\gamma+\theta)(0)=2p\ne p,$$

we can discover that the set of all smooth curves passing through point $p$ on $M$ does not form a linear space under ordinary addition. Therefore, unfortunately, this approach is not feasible. Of course, we can try to define a completely new addition, but this is not very convenient or natural. A better approach may be to reconsider our definition of tangent vectors.

The core of the definition of tangent vectors is the derivative

$$\left.\frac{d(\mathbf{x}^{-1} \circ \gamma)}{dt}\right|_0,$$

which is an $n$-dimensional vector in $\mathbb{R}^n$. Let

$$\begin{aligned} p_i:\mathbb{R}^n&\longrightarrow \mathbb{R}\\ (x_1,x_2,\cdots,x_n)&\longmapsto x_i \end{aligned}$$

be the projection function in $\mathbb{R}^n$,

$$\begin{aligned} c_i=p_i\circ \mathbf{x}^{-1}:M&\longrightarrow \mathbb{R}\\ p&\longmapsto x_i \end{aligned}$$

be the coordinate function of the smooth manifold $M$ at point $p$ given a coordinate chart $(U,\mathbf{x})$, we have

$$\left(p_i\circ\left.\frac{d(\mathbf{x}^{-1} \circ \gamma)}{dt}\right)\right|_0=\left.\frac{d(p_i\circ\mathbf{x}^{-1} \circ \gamma)}{dt}\right|_0=\left.\frac{d(c_i \circ \gamma)}{dt}\right|_0.$$

So, we can expand the derivative as

$$\left.\frac{d(\mathbf{x}^{-1} \circ \gamma)}{dt}\right|_0=\left(\left.\frac{d(c_1 \circ \gamma)}{dt}\right|_0,\left.\frac{d(c_2 \circ \gamma)}{dt}\right|_0,\cdots,\left.\frac{d(c_n \circ \gamma)}{dt}\right|_0\right).$$

Let $\mathcal{D}_p(M)$ denote the set of all functions defined on the smooth manifold $M$ that are smooth at point $p$. It is easy to see that $c_1, c_2, \cdots, c_n$ all belong to $\mathcal{D}_p(M)$. For each determined derivative value $\left.\frac{d}{dt}(\mathbf{x}^{-1} \circ \gamma)\right|_0$, we also determine a functional

$$\begin{aligned} v_\gamma:=\left.\frac{d(\cdot \circ \gamma)}{dt}\right|_0:\mathcal{D}_p(M)&\longrightarrow \mathbb{R}\\ f&\longmapsto \left.\frac{d(f \circ \gamma)}{dt}\right|_0 \end{aligned}$$

taking values at $c_1, c_2, \cdots, c_n$. In fact, the values of the functional at $c_1, c_2, \cdots, c_n$ determine its values over the entire $\mathcal{D}_p(M)$. To see this, we just need to pull back and push forward $f$ and $\gamma$ with respect to the coordinate chart $(U,\mathbf{x})$, and apply the chain rule:

$$\left.\frac{d(f \circ\gamma)}{dt}\right|_0=\left.\frac{d((f \circ \mathbf{x})\circ (\mathbf{x}^{-1}\circ \gamma))}{dt}\right|_0=\sum_{i=1}^n\left.\frac{\partial (f \circ \mathbf{x})}{\partial x_i}\right|_{p}\left.\frac{d(c_i \circ \gamma)}{dt}\right|_0.$$

Therefore, for any curve $\gamma$ on $M$ passing through point $p$, there exists a one-to-one correspondence between the equivalence class $[\gamma]_*$ and the functional $\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0$ induced by $\gamma$. Hence, the functional $\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0$ can be viewed as an alternative definition of a tangent vector.

We can also verify that $v_\gamma=\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0$ is a linear functional, i.e.,

$$\left.\frac{d((f_1+f_2) \circ \gamma)}{dt}\right|_0=\left.\frac{d(f_1\circ \gamma+f_2\circ \gamma )}{dt}\right|_0=\left.\frac{d(f_1\circ \gamma)}{dt}\right|_0+\left.\frac{d(f_2\circ \gamma)}{dt}\right|_0,$$ $$\left.\frac{d((kf) \circ \gamma)}{dt}\right|_0=\left.\frac{d(k(f \circ \gamma))}{dt}\right|_0=k\left.\frac{d(f \circ \gamma)}{dt}\right|_0.$$

Let $\mathcal{D}_p(M)^*$ denote the dual space of $\mathcal{D}_p(M)$, i.e., the space of all linear functionals on $\mathcal{D}_p(M)$. Furthermore, we can give the definition of the tangent space accordingly:

$$T_pM=\left\{\left.\frac{d(\cdot \circ \gamma)}{dt}\right|_0\in\mathcal{D}_p(M)^*:\gamma\text{ is a smooth curve on }M\text{ passing through point }p\right\}.$$

Let $(U,\mathbf{x})$ be a coordinate chart at point $p$. A curve

$$\begin{aligned} \delta_i:(-\epsilon,\epsilon)&\longrightarrow M\\ t&\longmapsto\mathbf{x}(c_1(p),\cdots,c_i(p)+t,\cdots,c_n(p)) \end{aligned}$$

is called the $x_i$-curve associated with the coordinate chart $(U,\mathbf{x})$.

Using the formula above, we can determine the functional derived from the $x_i$-curve

$$\left.\frac{d(f \circ\delta_i)}{dt}\right|_0=\left.\frac{d((f \circ \mathbf{x})\circ (\mathbf{x}^{-1}\circ \delta_i))}{dt}\right|_0=\sum_{k=1}^n\left.\frac{\partial (f \circ \mathbf{x})}{\partial x_k}\right|_{p}\left.\frac{d(c_k \circ \delta_k)}{dt}\right|_0=\left.\frac{\partial (f \circ \mathbf{x})}{\partial x_i}\right|_{p}.$$

These $n$ vectors

$$(v_{\delta_1},v_{\delta_2},\cdots,v_{\delta_n})=\left(\left.\frac{\partial (\cdot \circ \mathbf{x})}{\partial x_1}\right|_{p},\left.\frac{\partial (\cdot \circ \mathbf{x})}{\partial x_2}\right|_{p},\cdots,\left.\frac{\partial (\cdot \circ \mathbf{x})}{\partial x_n}\right|_{p}\right)$$

are called the coordinate tangent vectors of the coordinate chart $(U,\mathbf{x})$.

The linear space spanned by the coordinate tangent vectors is denoted as $\mathop{\mathrm{span}}\limits_{1\le i \le n}(v_{\delta_i})$. It is easy to see that any tangent vector in $T_pM$ can be expressed as a linear combination of coordinate tangent vectors

$$v_\gamma(f)=\left.\frac{d(f \circ\gamma)}{dt}\right|_0=\sum_{i=1}^n\left.\frac{d(\tilde{x}_i \circ \gamma)}{dt}\right|_0\left.\frac{d(f \circ\varepsilon_i)}{dt}\right|_0=\sum_{i=1}^n\left.\frac{d(\tilde{x}_i \circ \gamma)}{dt}\right|_0v_{\varepsilon_i}(f),$$

that is,

$$v_\gamma = \sum_{i=1}^n\left.\frac{d(c_i \circ \gamma)}{dt}\right|_0v_{\delta_i}.$$

Therefore, $T_pM\subset \mathop{\mathrm{span}}\limits_{1\le i \le n}(v_{\delta_i})$. Conversely, for any linear combination of coordinate tangent vectors $\sum_{i=1}^n\lambda_iv_{\delta_i}$, let the curve $\delta$ be

$$\begin{aligned} \delta:(-\epsilon,\epsilon)&\longrightarrow M\\ t&\longmapsto\mathbf{x}(c_1(p)+\lambda_1t,c_2(p)+\lambda_2t,\cdots,c_n(p)+\lambda_nt), \end{aligned}$$

then we have

$$v_\delta(f)=\sum_{i=1}^n\left.\frac{d(c_i \circ \delta)}{dt}\right|_0\left.\frac{d(f \circ\delta_i)}{dt}\right|_0=\sum_{i=1}^n\lambda_iv_{\delta_i}(f),$$

which implies $v_\delta=\sum_{i=1}^n\lambda_iv_{\delta_i}$. Therefore, $\mathop{\mathrm{span}}\limits_{1\le i \le n}(v_{\delta_i})\subset T_pM$. Thus, $T_pM=\mathop{\mathrm{span}}\limits_{1\le i \le n}(v_{\delta_i})$, which proves that $T_pM$ is a linear space.

Furthermore, we can prove that $v_{\delta_1},v_{\delta_2},\cdots,v_{\delta_n}$ are linearly independent. Let

$$\sum_{i=1}^n\lambda_iv_{\delta_i}=0,$$

acts on both sides of the equation on $c_i(1\le i\le n)$ and we get

$$\sum_{i=1}^n\lambda_iv_{\delta_i}(c_i)=\sum_{i=1}^n\lambda_i\left.\frac{d(c_i\circ\delta_i)}{dt}\right|_0=\lambda_i=0,$$

which proves the linear independence. Thus, we know that, given a coordinate chart $(U,\mathbf{x})$, the derived coordinate tangent vectors $v_{\delta_1},v_{\delta_2},\cdots,v_{\delta_n}$ are a basis of the linear space $T_pM$, and this basis is called the natural basis of the tangent space $T_pM$ under the coordinate chart $(U,\mathbf{x})$. The dimension of the tangent space $T_pM$ is $n$.

The Third Definition: Sheaf Theory

We can also define tangent vectors and tangent spaces using the language of sheaf theory. Sheaf theory is a more abstract theory, but it applies not only to differential geometry but also to more general geometry.

A presheaf is essentially a contravariant functor. A set presheaf is a contravariant functor $\mathcal{F}: \mathsf{C}^{\mathrm{op}} \to \mathsf{Set}$ from a category $\mathsf{C}$ to the category of sets $\mathsf{Set}$. An abelian group presheaf is a contravariant functor $\mathcal{F}: \mathsf{C}^{\mathrm{op}} \to \mathsf{Ab}$ from a category $\mathsf{C}$ to the category of abelian groups $\mathsf{Ab}$. For any object $X$ in $\mathsf{C}$, an element of $\mathcal{F}(X)$ is called a section of $\mathcal{F}$ over $X$.

When studying a smooth manifold $M$, the category $\mathsf{C}$ we choose consists of the open sets of $M$ as objects and the inclusion maps between these open sets as morphisms. The presheaf we consider is the set of all smooth real-valued functions on an open set $X$, denoted $C^\infty(X; M \to \mathbb{R})$, or simply $C^\infty(X)$ when there is no risk of confusion. $C^\infty(\cdot)$ can be a presheaf of real vector spaces, a presheaf of rings, or a presheaf of commutative real algebras.

Given a point $p$ in $M$, the set of smooth real-valued functions defined on open sets containing the point $p$ is denoted by

$$\widetilde{\mathcal{C}}_p = \{(\varphi, X) : p \in X, \varphi \in C^\infty(X)\}.$$

The set $\widetilde{\mathcal{C}}_p$ modulo the equivalence relation

$$(\varphi_1, X_1) \overset{p}\sim (\varphi_2, X_2) \iff \text{there exists an open set } X_3 \subset X_1 \cap X_2 \text{ such that } \varphi_1|_{X_3} = \varphi_2|_{X_3}$$

is denoted by $C^\infty_p$, and is called the stalk of smooth functions at the point $p$. It is not difficult to verify that the stalk of smooth functions is a commutative real algebra. The equivalence class of a smooth function $\varphi$ is denoted by $[\varphi]_p$ and is called the germ of $\varphi$ at the point $p$.

Since all smooth functions within a germ have exactly the same local properties at point $p$, it is natural to combine them and ignore the distinctions between them. With the concept of function germs, we can first rewrite the second definition. The functional $\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0$ was previously applied to functions $f$ that are smooth at point $p$ and defined on the entire manifold. However, the behavior of $f$ outside a neighborhood of $p$ is irrelevant and it doesn’t matter if it is undefined. Therefore, we can assume that $f$ is defined in some open neighborhood of $p$, and then let $\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0$ act directly on the germ of $f$ at $p$, i.e., we define

$$\begin{aligned} \left.\frac{d(\cdot \circ \gamma)}{dt}\right|_0 : C^\infty_p &\longrightarrow \mathbb{R}\\ [f]_p &\longmapsto \left.\frac{d(f \circ \gamma)}{dt}\right|_0 \end{aligned}$$

and

$$T_pM = \left\{ \left.\frac{d(\cdot \circ \gamma)}{dt}\right|_0 \in (C^\infty_p)^* : \gamma \text{ is a smooth curve on } M \text{ passing through } p \right\}.$$

Then we can give a third definition of tangent vectors in a more axiomatic manner without using curves: a tangent vector of the manifold $M$ at the point $p$ is a linear functional defined on $C^\infty_p$ that satisfies the Leibniz rule. Formally,

Definition. Given a smooth manifold $M$ and a point $p$ on it. If the function $\nu: C^\infty_p \to \mathbb{R}$ satisfies: for any $\lambda_1, \lambda_2 \in \mathbb{R}$, and any $[f]_p, [g]_p \in C^\infty_p$,

1. Linearity: $\nu\left(\lambda_1 [f]_p + \lambda_2 [g]_p\right) = \lambda_1 \nu([f]_p) + \lambda_2 \nu([g]_p)$,
2. Leibniz rule: $\nu\left([f]_p [g]_p\right) = [f]_p \, \nu\left([g]_p\right) + \nu([f]_p) [g]_p$,

then $\nu$ is called a tangent vector of the manifold $M$ at the point $p$. The set of all tangent vectors of the manifold $M$ at the point $p$ forms a vector space, denoted by $T_pM$, and is called the tangent space of $M$ at the point $p$.

Tagent Mapping

After defining the tangent space, we can finally define an analogous concept of differentiation for smooth mappings—the tangent map.

Consider a smooth mapping $f: M_1 \to M_2$ between smooth manifolds $(M_1, \tau_1, \Sigma_1)$ and $(M_2, \tau_2, \Sigma_2)$ that is smooth at point $p$. Suppose $\gamma$ is a curve on $M_1$ passing through point $p$, and its corresponding tangent vector is $\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0$. By applying $f$ to $\gamma$, we obtain another curve $f(\gamma)$ on $M_2$ passing through point $f(p)$, and its corresponding tangent vector is $\left.\frac{d}{dt}(\cdot \circ f(\gamma))\right|_0$. The mapping that sends the tangent vector $\left.\frac{d}{dt}(\cdot \circ \gamma)\right|_0 \in T_p M_1$ to

$$\left.\frac{d}{dt}(\cdot \circ f(\gamma))\right|_0 \in T_{f(p)} M_2$$

is called the tangent map, denoted by $(df)_p$. To ensure this is a well-defined map, we need to prove that for any two curves $\gamma_1$ and $\gamma_2$ passing through point $p$, if $\left.\frac{d}{dt}(\cdot \circ \gamma_1)\right|_0 = \left.\frac{d}{dt}(\cdot \circ \gamma_2)\right|_0$, then

$$\left.\frac{d(\cdot \circ f(\gamma_1))}{dt}\right|_0 = \left.\frac{d(\cdot \circ f(\gamma_2))}{dt}\right|_0.$$

We choose coordinate charts $(U, \mathbf{x}) \in \Sigma_1$ at $p$ and $(V, \mathbf{y}) \in \Sigma_2$ at $f(p)$, with the corresponding coordinate functions $c_j = p_j \circ \mathbf{x}^{-1}$ and $\tilde{c}_i = p_i \circ \mathbf{y}^{-1}$. The natural bases corresponding to these coordinate charts are $v_{\delta_1}, v_{\delta_2}, \cdots, v_{\delta_n}$ and $v_{\eta_1}, v_{\eta_2}, \cdots, v_{\eta_m}$, respectively. Note that the expression for $v_{f(\gamma)}$ in the natural basis is given by

$$\begin{aligned} v_{f(\gamma)}&=\left.\frac{d(\cdot \circ f(\gamma))}{dt}\right|_{0}\\ &=\sum_{i=1}^m\left.\frac{d(\tilde{c}_i \circ f(\gamma))}{dt}\right|_0v_{\eta_i}\\ &=\sum_{i=1}^m\left.\frac{d((\tilde{c}_i \circ f\circ\mathbf{x})\circ(\mathbf{x}^{-1}\circ\gamma))}{dt}\right|_0v_{\eta_i}\\ &=\sum_{i=1}^m\left.\frac{d(\tilde{c}_i \circ f\circ\mathbf{x})}{dx}\right|_{\mathbf{x}^{-1}(p)}\left.\frac{d(\mathbf{x}^{-1}\circ\gamma)}{dt}\right|_0v_{\eta_i}\\ &=\sum_{i=1}^m\sum_{j=1}^n\left.\frac{\partial(\tilde{c}_i \circ f\circ\mathbf{x})}{\partial x_j}\right|_{\mathbf{x}^{-1}(p)}\left.\frac{d(c_j\circ\gamma)}{dt}\right|_0v_{\eta_i} \end{aligned}$$

That means the tangent map is well-defined. Moreover, from the above expression, we can see that the tangent map is a linear map.

Note that

$$\begin{aligned} v_{f(\delta_k)}&=\left.\frac{d(\cdot \circ f(\delta_k))}{dt}\right|_{0}\\ &=\sum_{i=1}^m\sum_{j=1}^n\left.\frac{\partial(\tilde{c}_i \circ f\circ\mathbf{x})}{\partial x_j}\right|_{\mathbf{x}^{-1}(p)}\left.\frac{d(c_j\circ\delta_k)}{dt}\right|_0v_{\eta_i}\\ &=\sum_{i=1}^m\left.\frac{\partial(\tilde{c}_i \circ f\circ\mathbf{x})}{\partial x_k}\right|_{\mathbf{x}^{-1}(p)}v_{\eta_i}\\ &=\begin{pmatrix} v_{\eta_1}&v_{\eta_2}&\cdots&v_{\eta_m} \end{pmatrix} \begin{pmatrix} \left.\frac{\partial(\tilde{c}_1 \circ f\circ\mathbf{x})}{\partial x_k}\right|_{\mathbf{x}^{-1}(p)}\\ \left.\frac{\partial(\tilde{c}_2 \circ f\circ\mathbf{x})}{\partial x_k}\right|_{\mathbf{x}^{-1}(p)}\\ \vdots\\ \left.\frac{\partial(\tilde{c}_m \circ f\circ\mathbf{x})}{\partial x_k}\right|_{\mathbf{x}^{-1}(p)} \end{pmatrix}. \end{aligned}$$

We can further write the matrix representation of the tangent map $(df)_p$ in the natural basis:

$$\begin{aligned} &\;\quad (df)_p \begin{pmatrix} v_{\delta_1}&v_{\delta_2}&\cdots&v_{\delta_n} \end{pmatrix}\\ &= \begin{pmatrix} (df)_pv_{\delta_1}&(df)_pv_{\delta_2}&\cdots&(df)_pv_{\delta_n} \end{pmatrix}\\ &= \begin{pmatrix} v_{f(\delta_1)}&v_{f(\delta_2)}&\cdots&v_{f(\delta_n)} \end{pmatrix}\\ &= \begin{pmatrix} v_{\eta_1}&v_{\eta_2}&\cdots&v_{\eta_m} \end{pmatrix} \left.\begin{pmatrix} \frac{\partial(\tilde{c}_1 \circ f\circ\mathbf{x})}{\partial x_1}&\frac{\partial(\tilde{c}_1 \circ f\circ\mathbf{x})}{\partial x_2}&\cdots&\frac{\partial(\tilde{c}_1 \circ f\circ\mathbf{x})}{\partial x_n}\\ \frac{\partial(\tilde{c}_2 \circ f\circ\mathbf{x})}{\partial x_1}&\frac{\partial(\tilde{c}_2 \circ f\circ\mathbf{x})}{\partial x_2}&\cdots&\frac{\partial(\tilde{c}_2 \circ f\circ\mathbf{x})}{\partial x_n}\\ \vdots&\vdots&&\vdots\\ \frac{\partial(\tilde{c}_m \circ f\circ\mathbf{x})}{\partial x_1}&\frac{\partial(\tilde{c}_m \circ f\circ\mathbf{x})}{\partial x_2}&\cdots&\frac{\partial(\tilde{c}_m \circ f\circ\mathbf{x})}{\partial x_n} \end{pmatrix}\right|_{\mathbf{x}^{-1}(p)}\\ &= \begin{pmatrix} v_{\eta_1}&v_{\eta_2}&\cdots&v_{\eta_m} \end{pmatrix} \begin{pmatrix} \left.\frac{\partial(\tilde{c}_i \circ f\circ\mathbf{x})}{\partial x_j}\right|_{\mathbf{x}^{-1}(p)} \end{pmatrix}_{m\times n}. \end{aligned}$$

We find that the matrix representation of the tangent map $(df)_p$ in the natural basis is

$$J_pf:= \begin{pmatrix} \left.\frac{\partial(\tilde{c}_i \circ f\circ\mathbf{x})}{\partial x_j}\right|_{\mathbf{x}^{-1}(p)} \end{pmatrix}_{m\times n}$$

which is exactly the Jacobian matrix of the mapping $\mathbf{y}^{-1}\circ f\circ\mathbf{x}$ at $\mathbf{x}^{-1}(p)$.

Cotangent vector and cotangent space

The cotangent space at point $p$ of a manifold is the dual space of the tangent space $T_pM$, denoted as $T_p^*M$. Vectors in the cotangent space are called cotangent vectors.

Suppose we choose a coordinate chart $(U, \mathbf{x})$ at point $p$, and the corresponding natural basis $v_{\delta_1}, v_{\delta_2}, \cdots, v_{\delta_n}$ is a basis for the tangent space $T_pM$. The following is a standard construction from linear algebra.

We construct the mapping

$$\begin{aligned} v_{\delta_j}^*: T_pM & \longrightarrow \mathbb{R} \\ \sum_{i=1}^n \lambda_i v_{\delta_i} & \longmapsto \sum_{i=1}^n \lambda_i v_{\delta_i}(c_j) \end{aligned}$$

We have $v_{\delta_j}^* \in T_p^*M$ and

$$v_{\delta_j}^*(v_{\delta_i}) = \begin{cases} 0, & i \ne j, \\ 1, & i = j. \end{cases}$$

This shows that $v_{\delta_1}^*, v_{\delta_2}^*, \cdots, v_{\delta_n}^*$ is the dual basis of $v_{\delta_1}, v_{\delta_2}, \cdots, v_{\delta_n}$ in $(T_pM)^*$. Next, we construct the mapping

$$\begin{aligned} v_{\delta_j}^{**}: T_p^*M & \longrightarrow \mathbb{R} \\ \sum_{i=1}^n \lambda_i v^*_{\delta_i} & \longmapsto \sum_{i=1}^n \lambda_i v^*_{\delta_i}(v_{\delta_j}) \end{aligned}$$

We have $v^{**}_{\delta_j} \in T_pM$ and

$$v_{\delta_j}^{**}(v^*_{\delta_i}) = \begin{cases} 0, & i \ne j, \\ 1, & i = j. \end{cases}$$

This shows that $v_{\delta_1}^{**}, v_{\delta_2}^{**}, \cdots, v_{\delta_n}^{**}$ is the dual basis of $v^*_{\delta_1}, v^*_{\delta_2}, \cdots, v^*_{\delta_n}$ in $(T_pM)^{**}$. The mapping

$$\begin{aligned} **: T_pM & \longrightarrow (T_pM)^{**} \\ \sum_{i=1}^n \lambda_i v_{\delta_i} & \longmapsto \left( \sum_{i=1}^n \lambda_i v_{\delta_i} \right)^{**} := \sum_{i=1}^n \lambda_i v_{\delta_i}^{**} \end{aligned}$$

is a canonical isomorphism, and

$$\forall v_\gamma\in T_pM, w\in T_p^*M,\;v_\gamma^{**}(w)=w(v_\gamma).$$

Define the mapping

$$\begin{aligned} \langle\cdot,\cdot\rangle:T_pM\times T_p^*M&\longrightarrow\mathbb{R}\\ (v_\gamma,w)&\longmapsto\langle v_\gamma,w\rangle:=w(v_\gamma). \end{aligned}$$