Differential-Geometry

Tangent Space

Created: September 8, 2022 1:25 PM Date: September 10, 2022 Email: [email protected] Tags: Personal

Cotangent space

Given a differentiable structure on a topological manifold, we can approximate a neighborhood of any point by a linear space.

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Function germs

Suppose ${p}$ is a fixed point of the m-dimension smooth manifold ${M}$. Let ${f}$ be a ${C^\infty}$-function ( ${C^\infty}$ is infinitely differentiable) defined in a neighborhood of ${p}$, ${C_p^\infty}$ represents the set of all of these ${f}$ functions at ${p}$. Two functions ${f}$ and ${g}$ usually have different domains but their addition and multiplication are still well-defined if ${p} \in U \bigcap V$, ${U}$ and ${V}$ are domains of ${f}$ and ${g}$, respectively. If ${f}$ and ${g}$ ${\mathbb\in}$ ${C_p^\infty}$, ${f + g}$ and ${f * g}$ also belong to ${C_p^\infty}$, that is, ${f + g}$ and ${f*g} \in C_p^\infty$. But ${g|_ {U \bigcap V} = f|_ {U \bigcap V}} = 0$, that the zero is not unique, so ${C_p ^\infty}$ can not be defined as a linear space. If defining an equivalence relation ${f\sim g}$ if and only if existing an open neighborhood ${H}$ of ${p}$ such that ${f|_H=g|_H}$. The ${[f]}$ represents ${f}$’s ${\sim}$ equivalence class in ${C_p^\infty}$, the ${[f]}$ is called the function germ of the manifold ${M}$ at ${p}$. Let

$$ {\mathscr{F}_p=C_p^\infty/\sim={[f]|f\in C_p^\infty}}. $$

By introducting addition and scalar multipication into ${\mathscr{F}_p}$, it becomes a linear space, so for ${[f]}$ and ${[g]} \in\mathscr{F}_p$, ${\alpha} \in \mathbb{R}$, define

$$ \begin{cases} [f] + [g] = [f + g],\\ \alpha [f] = [\alpha f]. \end{cases} $$

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Linear space

Suppose ${\gamma (t)}$ is a parameterized curve in manifold ${M}$ through a point ${p}$, ${t \in (-\delta, \delta) \subset \mathbb{R}}$ and ${\gamma (0)=p}$. Denote the set of all these parameterized curves as ${\Gamma_p}$. For any ${\gamma (t) \in \Gamma_p}$, ${[f] \in \mathscr{F}_p}$, Let

$$ \ll \gamma, [f]\gg = \left. \dfrac{d(f \circ \gamma)}{dt}\right|_{t=0} \quad, \quad -\delta \ \lt \ t \ \lt \ \delta . $$

Obviously, the ${\ll \ , \ \gg}$ is linear with respect to the second variable. For arbitrary ${\gamma \in \Gamma_p}$, ${[f]}$ and ${[g] \in \mathscr F_p}$, have

$$ \begin{aligned} \ll \gamma, [f] + [g]\gg &= \ll \gamma, [f]\gg + \ll \gamma, [g]\gg,\\ \ll \gamma, \alpha [f]\gg &= \alpha \ll \gamma, [f]\gg. \end{aligned} $$

Thus, Let ${\ll\gamma, [f]\gg=0}$ can get a linear subspace of ${\mathscr{F}_p}$ as

$$ \mathscr{H}_p={[f] \in \mathscr{F}_p|{\ll\gamma, [f]\gg=0}，\ {\forall} \gamma \in \Gamma_p}. $$

Theorem 1. Suppose ${[f]} \in \mathscr F_p$, for an admissible coordinate chart ${(U,\varphi_U)}$, let

$$ F(x^1, x^2, ..., x^m) = f \circ \varphi_U^{-1}(x^1, x^2, ..., x^m) $$

then, ${[f] \in \mathscr H_p}$ if and only if

$$ \left.\frac{\partial F}{\partial x^i} \right| _{\varphi_U(p)} = 0, \quad \quad 1 \le \ i \ \le \ m. $$

Proof. Suppose ${\gamma (t) \in \Gamma_p}$

$$ {(\varphi_U \circ \gamma (t))^i}=x^i(t),\quad \quad 1 \le \ i \ \le \ m. $$

$$ \begin{aligned} \ll \gamma, [f]\gg &= (\left. \dfrac{df \circ \gamma}{dt}) \right| _{t=0} \\ &= \left.\dfrac{d}{dt}F(x^1,x^2,...,x^m)\right| _{t=0} \\ &= \sum \limits_i^m \left( \frac{\partial f \circ \varphi_U^{-1}}{\partial u^i}\right) _{\varphi_U(p)} \cdot \left.\dfrac{d (\varphi_U \circ \gamma(t))^i}{dt} \right| _{t=0} \\ &= \sum\limits_i^m \left.\frac{\partial F}{\partial x^i} \right| _{\varphi_U(p)}\cdot \left.\frac{dx^i}{dt}\right| _{t=0} \end{aligned} $$

because ${\gamma (t) \in \Gamma_p}$ is an arbitary curve, so $\Large {\frac{dx^i}{dt} \Large|\normalsize_{t=0}}$ is arbitary real value. For arbitary ${\gamma (t) \in \Gamma_p}$ , a necessary and sufficent condition for ${\ll\gamma, [f]\gg=0}$ is

$$ \left.\frac{\partial F}{\partial x^i} \right| _{\varphi_U(p)} = 0, \quad \quad 1 \le \ i \ \le \ m. $$

Theorem 1 indicates that subspace ${\mathscr H_p}$ is exactly the linear space of germs of smooth functions whose partial derivatives with respect to local coordinates all vanish at ${p}$.

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Definition

The quotient space ${\mathscr F_p/\mathscr H_p}$ is called Cotangent Space of ${M}$ at ${p}$, denoted by ${T_p^\ast (M)}$ or ${T _p^\ast}$. The ${\mathscr H_p}$-equivalence class of the function germ ${[f]}$ is denoted by ${[\widetilde{f\ }]}$ or ${(df)_p}$, and is called a cotangent vector on ${M}$ at ${p}$.

${T_p^\ast (M)}$ is a linear space. It has a linear structure induced from the linear space ${\mathscr F_p}$, i.e. for ${[f]}$ and ${[g] \in \mathscr F_p}$, we have

$$ \begin{cases} [\widetilde{f\ }] + [\widetilde{g }] = [\widetilde{f +g\ }],\\ \alpha[\widetilde{f\ }] = \widetilde{\alpha[f\ }].\\ \end{cases} $$

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Theorem 2. Suppose ${f^1}$, ${f^2}$, …, ${f^s} \in C_p^\infty$ and ${F(y^1, y^2, …, y^s)}$ is a smooth function in a neighborhood of ${(f^1(p), f^2(p), ..., f^s(p)) \in \mathbb R^s}$. Then ${f=F(f^1, f^2, …, f^s) \in C_p^\infty}$ and

$$ df_p = \sum \limits_k^s (\frac{\partial F}{\partial f^k})_{(f^1(p), f^2(p), ..., f^s(p))} \cdot (df^k)_p $$

Proof. ${f(p)=F(f^1(p), f^2(p), ..., f^s(p))}$, since ${f=F(f^1, f^2, …, f^s) \in C_p^\infty}$ so ${f(p) \in C_p^\infty}$, let

$$ {a_k = \left( \frac{\partial F}{\partial f^k} \right) _{(f^1(p), f^2(p), ..., f^s(p))}} $$

for an arbitrary ${\gamma(t)}$,

$$ \begin{aligned} \ll \gamma, [f]\gg &= \left. \dfrac{d(f \circ \gamma)}{dt} \right|_{t=0} \\ &= \sum \limits_k^s \left( \frac{\partial F}{\partial f^k}\right) _{(f^1(p), f^2(p), ..., f^s(p))} \cdot \left.\dfrac{d(f^k \circ \gamma(t))}{dt} \right| _{t=0} \\ &= \sum \limits_k^s a_k \cdot \ll \gamma, [f^k] \gg. \end{aligned} $$

Thus,

$$ [f] -\sum \limits_k^s a_k [f^k]=0, $$

i.e.

$$ (df)_p = \sum \limits_k^s a_k (df^k)_p. $$

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Corollary 1. For any ${[f]}$, ${[g] \in C_p^\infty}$, ${a \in \mathbb R}$, have

$$ \begin{aligned} &d(f+g)_p=(df)_p + (dg)_p, \\ &d(af)_p=a \cdot(df)_p, \\ &d(fg)_p=g\cdot(df)_p + f\cdot (dg)_p. \end{aligned} $$

Proof. …

Corollary 2. ${dim (T_p^\ast)=m}$.

Proof. Choose an admissible coordinate chart ${(U, \varphi_U)}$, and define local coordinates ${u^i}$ by

$$ u^i(p) = f^i(p) = (\varphi_U(p))^i $$

as Theorem 2 have

$$ \begin{aligned}f &= F(f^1(p), f^2(p), ...,f^m(p)) \\ &= F(u^1(p), u^2(p), ...,u^m(p)) \end{aligned} $$

then

$$ \begin{aligned}(df)_p &= \sum \limits_i^s \left( \frac{\partial F}{\partial u^i} \right) _{(u^1(p), u^2(p), ..., u^s(p))} \cdot(du^i)_p \\ &= \sum \limits_i^s a_i (du^i)_p \\ where \quad &a_i = \left( \frac{\partial F}{\partial u^i}\right) _{(u^1(p), u^2(p), ..., u^s(p))} \end{aligned} $$

Thus, any ${(df)_p}$ is a linear combination of the ${(du^k)_p}$, ${1 \le \ k \ \le \ m}$, if there have real values ${a_k}$, such that

$$ \sum \limits_i^s a_i (du^i)_p = 0 $$

i.e.

$$ \sum \limits_i^s a_i [u^i] \in \mathscr H_p $$ so for any ${\gamma_(t) \in \Gamma_p}$, have

$$ \ll \gamma(t), \sum \limits_i^s a_i [u^i] \gg \ = \ \sum \limits_i^s a_i \left. \frac{d(u^i \circ \gamma(t))}{dt}\right| _{t=0} \normalsize =0 $$

select ${\lambda_k(t) \in \Gamma_p}$, ${1 \le \ k \ \le \ m}$, let

$$ u^i \circ \lambda_k(t) = u^i(p) + \delta_k^it, $$

where

$$ \quad \delta_k^i =\begin{cases} 1, \quad i=k, \\ 0, \quad i \neq k. \end{cases} $$

then

$$ \quad \quad \left. \dfrac{d(u^i \circ \lambda_k(t))}{dt}\right|_{t=0} = \delta_k^i, $$

so

$$ \begin{aligned} \ll \lambda_k(t), \sum \limits_i^s a_i [u^i] \gg \ &= \ \sum \limits_i^s a_i \left. \dfrac{d(u^i \circ \lambda_k(t))}{dt}\right|_{t=0} \\ &= \sum \limits_i^s a_i \delta_k^i = 0.\end{aligned} $$

hence ${{a_k=0,\ 1 \ \le k \ \le \ m}}$, ${{(du^i)_p,\ 1 \ \le i \ \le \ m}}$ is linearly independent. Therefore it forms a basis for ${T_p^\ast (M)}$ , called the natural basis of ${T_p^* (M)}$ with respect to the local coordinate system ${u^i}$. Thus ${T_p^* (M)}$ is an m-dimensional linear space.

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Tangent space

Definition

${[f] - [g] \in \mathscr H_p}$ if and only if ${\ll \gamma, [f] \gg = \ll \gamma, [g] \gg}$ for any ${\gamma \in \Gamma_p}$, so can define

$$ \ll \gamma, (df)_p \gg \ = \ \ll \gamma, [f] \gg, \quad \gamma \in \Gamma_p, \quad (df)_p \in T_p^\ast. $$

The same problem as linearization of the space of ${f}$ that zero is not unique when linearizing the space of ${\gamma}$, so can define an equivalence relation ${\sim}$ in ${\Gamma_p}$ as follow,

$$ \ll \gamma, (df)_p \gg \ = \ \ll \gamma', (df)_p \gg, \quad \gamma, \gamma' \in \Gamma_p, \quad \forall(df)_p \in T_p^\ast. $$

Denote the equivalence class of ${\gamma}$ as ${[\gamma]}$, so define

$$ <[\gamma], (df)_p > \ = \ \ll \gamma, (df)_p \gg. $$

${{[\gamma]| \gamma \in \Gamma_p}}$ is the dual space of cotangent space ${T_p^*}$. The proof is as follows

Under the local coordinates ${u^i}$, suppose ${\gamma \in \Gamma_p}$, is given by the functions ${u^i = u^i(t) = (\varphi_U \circ \gamma(t))^i, 1 \le \ i \ \le m}$. From Theorem 1, ${f \circ \varphi^{-1}(u^1, u^2, …,u^m) = F(u^1, u^2, …,u^m)}$ , then can derive that

$$ \begin{aligned} <[\gamma], (df)_p > \ &= \ \sum \limits_i^m \left( \frac{\partial f \circ \varphi_U^{-1}}{\partial u^i}\right) _{\varphi_U(p)} \cdot \left .\dfrac{d (\varphi_U \circ \gamma(t))^i}{dt} \right| _{t=0} \\ &= \ \sum \limits_i^m \left(\frac{\partial f \circ \varphi_U^{-1}}{\partial u^i}\right) _{\varphi_U(p)} \cdot \left(\dfrac{d u^i(t)}{dt}\right) _{t=0} \\ &= \ \sum \limits_i^m a_i \xi^i \end{aligned} $$

where

$$ a_i=\left(\frac{\partial f \circ \varphi_U^{-1}}{\partial u^i}\right) _{\varphi_U(p)}, \quad \xi^i=\left(\dfrac{d u^i(t)}{dt}\right) _{t=0} $$

Due to ${\gamma \in \Gamma_p}$ is arbitrary, so ${{\xi^i}}$ can be arbitrary real values such that ${&lt;[\gamma, \cdot&gt;]}$ is all of the linear mappings on ${T_p^}$. The ${[\gamma]}$ forms a dual space of ${T_p^}$, called the tangent space of ${M}$ at ${p}$.

A necessary and sufficient condition for ${[\gamma] = [\gamma']}$ for ${u^i(t) = u'^i(t)}$ is

$$ \left( \dfrac{d u^i(t)}{dt} \right) _{t=0} = \left( \dfrac{d u'^i(t)}{dt} \right) _{t=0} $$

Thus, the geometric meaning of tangent vectors means that these two parametrized curves have the same tangent vector at the point ${p}$.

So far,

$$ \ll X, (df)_p \gg \quad X=[\gamma] \in T_p, \quad (df)_p \in T_p^\ast. $$

is a bilinear mapping of both ${T_p}$ and ${T_p^*}$.

select ${\lambda_k \in \Gamma_p}$ as

$$ u^i \circ \lambda_k(t) = u^i(p) + \delta_k^it, $$

then

$$ <[\lambda_k],(du^i)_p> \ = \ \delta_k^i. $$

so ${{[\lambda_k]| 1 \le \ k \ \le m}}$ can be viewed as the dual baisis of ${{(du^i)_p| 1 \le \ i \ \le m}}$.

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Partial differential operators

Furthermore,

$$ \begin{aligned} <[\lambda_k], (df)_p > \ &= \ \left<[\lambda_k],\sum \limits_i^m \left(\frac{\partial f \circ \varphi_U^{-1} }{\partial u^i}\right) _p \cdot (du^i)_p \right> \\ &= \sum \limits_i^m \left(\frac{\partial f \circ \varphi_U^{-1} }{\partial u^i}\right) _p \ \cdot<[\lambda_k],(du^i)_p> \\ &= \sum \limits_i^m \left(\frac{\partial f \circ \varphi_U^{-1} }{\partial u^i}\right) _p \cdot \delta_k^i \\ &= \left(\frac{\partial f \circ \varphi_U^{-1} }{\partial u^k}\right) _p \end{aligned} $$

Thus, the ${[\lambda_k]}$ are the partial differential operators ${(\partial/\partial u^k)}$ on the function germs ${[f]}$. Then

$$ <[\lambda_k],(du^i)_p> \ = \ \left<\left.\frac{\partial}{\partial u^k}\right| _{p}, (du^i)_p \right> = \delta_k^i. $$

We call the basis of ${{ (\partial/\partial u^i)_p, 1 \le \ i \ \le m}}$ in ${T_p}$ the Natural Basis of the tangent space under the local coordinate system ${u^i}$.

So from ${&lt;[\gamma], (df)_p &gt; = \sum \limits_i^m a_i \xi^i}$ , have

$$ [\gamma] = \sum \limits_i^m \xi^i \left. \frac{\partial}{\partial u^i}\right|_{p} $$

the ${{\xi^i, 1 \le \ i \ \le m}}$ are componets of the tangent vector ${[\gamma]}$ regards to natural basis ${{ (\partial/\partial u^i)_p, 1 \le \ i \ \le m}}$. **

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Tangent map

Under local coordinates ${u^i}$, a tangent vector ${X=[\gamma] \in T _p}$ and a cotangent vector ${a=df \in T _p^\ast}$ have linear representations in terms of natural bases:

$$ X = \sum \limits_i^m \xi^i \frac{\partial}{\partial u^i}, \quad a = \sum \limits_i^s a_i du^i $$

where

$$ \xi^i = \frac{\partial (u^i \circ \gamma)}{\partial u^i}, \quad a _i = \dfrac{df}{du^i} $$

Under another local coordinate system ${u^{'i}}$, if the components of ${X}$ and ${a}$ with respect to the corresponding natural bases are ${\xi^{'}}$ and ${a^{'}}$, respectively, then they satisfy the following transformation rules:

$$ \xi^{'j} = \sum \limits_i^m \xi ^i \dfrac{d u^{'j}}{d u^i}, \quad a _i = \sum \limits_i^m a _j^{'} \cdot \dfrac{du^{'j}}{du^i} $$

where

$$ \dfrac{d u^{'j}}{d u^i} = \frac{\partial (\varphi _U^{'} \circ \varphi _U^{-1})^j}{\partial u^i} $$

Smooth maps between smooth manifolds induce linear maps between tangent spaces and between cotangent spaces. Suppose ${F : M \rightarrow N}$ is a smooth map, ${p \in M}$, and ${q = F(p)}$. Define the map ${F^\ast: T _q^\ast \rightarrow T _p^\ast}$ as follows:

$$ F^\ast (df) = d (d \circ F), \quad \quad df \in T _q^\ast $$

Obviously this is a linear map, called the differential of the map ${F}$.

Consider next the adjoint of ${F^\ast}$, namely the tangent map ${F _\ast: T _p \rightarrow T _q}$ defined for ${X \in T _p^\ast}$, ${\alpha \in T _p}$ as follows:

$$ <F_\ast X,\alpha> = <X, F^\ast \alpha> $$

Suppose ${u ^i}$ and ${v ^ \alpha}$ are local coordinates near ${p}$ and ${q}$, respectively. Then

$$ v^\alpha = F^\alpha(u ^1, u^2, \cdots,u^m) \quad \quad 1 \le \ \alpha \ \le n $$

Thus the action of ${F^*}$ on the natural basis ${{dv ^\alpha, 1 \le \alpha \le n}}$ is given by

$$ \begin{aligned} F^* (dv^\alpha) &= d(v^\alpha \circ F) \\ &= \sum \limits_{i=1}^m \left(\frac{\partial F^\alpha}{\partial u^i} \right) _p \cdot du^i \end{aligned} $$

The matrix representation of ${F^\ast}$ in the natural bases ${dv^\alpha}$ and ${du^i}$ is exactly the Jacobian matrix ${(dF^\alpha/du^i)}$.

$$ \begin{aligned} <F_\ast \left( \frac{\partial}{\partial u^i} \right), (dv^\alpha)> &= <\frac{\partial}{\partial u^i},\ F^\ast (dv^\alpha)> \\ &= <\frac{\partial}{\partial u^i},\ \sum \limits _{j=1}^m \left(\frac{\partial F^\alpha}{\partial u^j} \right) _p \cdot du^j> \\ &= \sum \limits _{j=1}^m <\frac{\partial}{\partial u^i},\ du^j> \cdot \left(\frac{\partial F^\alpha}{\partial u^j} \right) _p \\ &= \sum \limits _{j=1}^m \delta _i^j \cdot \left(\frac{\partial F^\alpha}{\partial u^j} \right) _p \\ &= \left(\frac{\partial F^\alpha}{\partial u^i} \right) _p \\ &= <\sum \limits _{\beta=1}^n \left(\frac{\partial F^\beta}{\partial u^i} \right) _p \cdot \frac{\partial}{\partial v^\beta},\ dv^\alpha> \\ &= \sum \limits _{\beta=1}^n \left(\frac{\partial F^\beta}{\partial u^i} \right) _p \cdot <\frac{\partial}{\partial v^\beta},\ dv^\alpha> \\ &= \sum \limits _{\beta=1}^n \left(\frac{\partial F^\beta}{\partial u^i} \right) _p \cdot \delta _\beta^\alpha \end{aligned} $$

i.e.

$$ F _\ast \left(\frac{\partial}{\partial u^i} \right) = \sum \limits _{\beta=1}^n \left(\frac{\partial F^\beta}{\partial u^i} \right) _p \cdot \frac{\partial}{\partial v^\beta} $$

Hence the matrix representation of the tangent map ${F}$, under the natural bases ${{\partial / \partial u^i}}$ and ${{\partial / \partial v^\alpha}}$ is still the Jacobian matrix ${(\partial F^\alpha / \partial u^i)}.$