Topics in Differential Geometry

 Algebraic Description of Tangent/Cotangent Space via Derivations and Ideals of Taylor Series

Algebraic Description of Tangent/Cotangent Space via Derivations and Ideals of Taylor Series

Cotangent SpaceThe definitionCotangent space as a functorTangent SpaceTangent Space Functor

 

Cotangent Space

The definition

Let $\mathbb{R}[X_1,...,X_n]$$\mathbb R[X_1,...,X_n]$ be the polynomial ring and $\mathbb{R}[[X_1,...,X_n]]$$\mathbb R[[X_1,...,X_n]]$​​ be the formal power series ring.

Let $I_k:(X_1^{{\alpha }_1}...X_n^{{\alpha }_n}),l(\alpha )={\alpha }_1+...+{\alpha }_n=k$$I_k:(X_1^{\alpha_1}... X_n^{\alpha_n}),l(\alpha)=\alpha_1+...+\alpha_n=k$. Easy to see that $I_k=I_1^k$$I_k=I_1^k$. This ideal will appear at different rings, so distinguish it according to the context.

Let $C^{\mathrm{\infty }}(M{)}_p$$C^{\infty}(M)_p$ be the stalk at $p$$p$ and the chart $(U,\phi ),p\in U$$(U,\varphi),p\in U$​ induce a isomorphism

$$\begin{array}{}\text{(1)}& C^{\mathrm{\infty }}(U)\cong C^{\mathrm{\infty }}(\phi (U)),f⟼f\circ {\phi }^{-1}⟹C^{\mathrm{\infty }}(M{)}_p\cong C^{\mathrm{\infty }}({\mathbb{R}}^n{)}_{\phi (p)}\end{array}$$

and consider the Taylor series ${\tau }_k:C^{\mathrm{\infty }}(M{)}_p\to \mathbb{R}[X_1,...,X_n]/I_k$$\tau_k: C^{\infty}(M)_p\to\mathbb R[X_1,...,X_n]/I_k$, which is a $\mathbb{R}$$\mathbb R$ algbra homomorphism.

We denote the kernel of ${\tau }_k$$\tau_k$ as $M_k$$M_k$ and kenrel of $\tau :C^{\mathrm{\infty }}(M{)}_p\to \mathbb{R}[[X_1,...,X_n]]$$\tau:C^{\infty}(M)_p\to\mathbb R[[X_1,...,X_n]]$ as $M_{\mathrm{\infty }}$$M_{\infty}$.

Notice that $M_{\mathrm{\infty }}$$M_\infty$ is nontrivial since $C^{\mathrm{\infty }}(M{)}_p$$C^{\infty}(M)_p$ is not a intergal domain hence zero is not prime ideal but $\mathbb{R}[[X_1,...,X_n]]$$\mathbb R[[X_1,...,X_n]]$ is, hence $M_{\mathrm{\infty }}$$M_\infty$ is a prime ideal and $M_{\mathrm{\infty }}\ne (0)$$M_{\infty}\ne (0)$.

The inverse limit of

$$\begin{array}{}\text{(2)}& ...\mathbb{R}[X_1,...,X_n]/I_{k+1}\stackrel{{\pi }_k}{\to }\mathbb{R}[X_1,...,X_n]/I_k...\end{array}$$

is $\mathbb{R}[[X_1,...,X_n]]$$\mathbb R[[X_1,...,X_n]]$ and kenrel of ${\pi }_k$$\pi_k$ is $I_k$$I_k$. From the diagram we see that:

$$\begin{array}{}\text{(3)}& {\tau }_k={\pi }_k\circ {\tau }_{k+1}\end{array}$$

Thus

$$\begin{array}{}\text{(4)}& M_k={\tau }_k^{-1}(0)={\tau }_{k+1}^{-1}({\pi }_k^{-1}(0))\supset {\tau }_{k+1}^{-1}(0)=M_{k+1}\end{array}$$

Also, let $p_k$$p_k$ be the canonical map from the limit, $p_k:\mathbb{R}[[X_1,...,X_n]]\to \mathbb{R}[X_1,...,X_n]/I_k$$p_k:\mathbb R[[X_1,...,X_n]]\to \mathbb R[X_1,...,X_n]/I_k$.

$$\begin{array}{}\text{(5)}& {\tau }_k=p_k\circ {\tau }_{\mathrm{\infty }}\end{array}$$

Hence

$$\begin{array}{}\text{(6)}& M_k={\tau }_k^{-1}(0)={\tau }_{\mathrm{\infty }}^{-1}(p_k^{-1}(0))={\tau }_{\mathrm{\infty }}^{-1}(I_k)={\tau }_{\mathrm{\infty }}^{-1}(I_1^k)=({\tau }_{\mathrm{\infty }}^{-1}(I_1){)}^k=M_1^k={\mathfrak{m}}_p^k\end{array}$$

The equation

$$\begin{array}{}\text{(7)}& {\tau }_{\mathrm{\infty }}^{-1}(I_1^k)=({\tau }_{\mathrm{\infty }}^{-1}(I_1){)}^k\end{array}$$

hold since $I_1\subset \mathrm{Im}({\tau }_{\mathrm{\infty }})$$I_1\subset\mathrm{Im}(\tau_{\infty})$​, and contraction and extension of ideal is a pair of inverse when we consider surjective. Click here. Let $e(I)$$e(I)$ be the extension of $I$$I$ from $C^{\mathrm{\infty }}(M{)}_p$$C^{\infty}(M)_p$ to $\mathbb{R}[[X_1,...,X_n]]$$\mathbb R[[X_1,...,X_n]]$ and $c(I)$$c(I)$ be the contraction.

$$\begin{array}{}\text{(8)}& {\tau }_{\mathrm{\infty }}^{-1}(I_1^k)=c(I^k)=c(e(c(I){)}^k)=c(e((c(I){)}^k))=(c(I){)}^k=({\tau }_{\mathrm{\infty }}^{-1}(I_1){)}^k\end{array}$$

From $(3)$$(3)$ we see that $M_k\supset M_{k+1}$$M_k\supset M_{k+1}$, hence we have $\begin{array}{c}{M}_{\mathrm{\infty }}=\bigcap _{i=1}^{\mathrm{\infty }}{M}_i=\bigcap _{i=1}^{\mathrm{\infty }}{\mathfrak{m}}_p^i\end{array}$$M_\infty=\bigcap_{i=1}^{\infty} M_i=\bigcap_{i=1}^{\infty} \mathfrak m_p^i\\$.

Then we define the cotangent space $T_p^{\ast }(M)$$T_p^*(M)$ as ${\mathfrak{m}}_p/{\mathfrak{m}}_p^2$$\mathfrak m_p/\mathfrak m_p^2$​​​​.

This consturction could be generalized to locally ringed space, for example, affine scheme. For ${\mathfrak{p}}_x\in \mathrm{Spec}(R)$$\mathfrak p_x\in\mathrm{Spec}(R)$ consider the stalk at $x$$x$, $R_{{\mathfrak{p}}_{\mathfrak{x}}}$$R_\mathfrak{p_x}$ whcih is a local ring as well, then consider ${\mathfrak{p}}_x/{\mathfrak{p}}_x^2$$\mathfrak p_x/\mathfrak p_x^2$, whcih is a $R_{{\mathfrak{p}}_{\mathfrak{x}}}/{\mathfrak{p}}_x$$R_\mathfrak{p_x}/\mathfrak p_x$ vector space.

Cotangent space as a functor

Let ${\mathrm{Man}}_{\ast }$$\mathrm{Man}_*$ be the category of smooth manifold with base point, then we could define the cotangent space functor:

$$\begin{array}{}\text{(9)}& T_p^{\ast }(-):{\mathrm{Man}}_{\ast }^{\mathrm{op}}\to \mathbb{R}-\mathrm{Mod}\end{array}$$

Let $g:M_p\to N_{g(p)}$$g:M_p\to N_{g(p)}$ be a smooth function, then we induce a ring homomorphism

$$\begin{array}{}\text{(10)}& g^{\ast }:C^{\mathrm{\infty }}(N{)}_{g(p)}\to C^{\mathrm{\infty }}(M{)}_p\end{array}$$

And

$$\begin{array}{}\text{(11)}& g^{\ast }:{\mathfrak{m}}_{g(p)}\to {\mathfrak{m}}_p,{\mathfrak{m}}_{g(p)}^2\to {\mathfrak{m}}_p^2⟹g^{\ast }:T_{g(p)}^{\ast }(N)\to T_p^{\ast }(M)\end{array}$$

Tangent Space

Proposition.

$$\begin{array}{}\text{(12)}& {\mathrm{Der}}_{\mathbb{R}}(C^{\mathrm{\infty }}(M{)}_p,-)\cong {\mathrm{Hom}}_{\mathbb{R}-\mathrm{Mod}}({\mathfrak{m}}_p/{\mathfrak{m}}_p^2,-)\end{array}$$

Proof. Let ${\mathrm{Der}}_{\mathbb{R}}(C^{\mathrm{\infty }}(M{)}_p,V)$$\mathrm{Der}_{\mathbb R}(C^{\infty}(M)_p,V)$ be the set such that $D$$D$ is $\mathbb{R}$$\R$ linear and $D(fg)=f(p)D(g)+D(f)g(p)$$D(fg)=f(p)D(g)+D(f) g(p)$.

Let $D:{\mathfrak{m}}_p\to V$$D:\mathfrak m_p\to V$ be a linear map and $\mathrm{Ker}D\supseteq {\mathfrak{m}}_p^2.$$\mathrm{Ker} D\supseteq\mathfrak m_p^2.$ Then for $f,g\in C^{\mathrm{\infty }}(M{)}_p$$f,g\in C^{\infty}(M)_p$, define $D(f)=D(f-f(p))$$D(f)=D(f-f(p))$​.

Notice that

$$\begin{array}{}\text{(13)}& fg=(f(p)+(f-f(p)))(g(p)+(g-g(p)))\end{array}$$

Simplify it we get

$$\begin{array}{}\text{(14)}& f(p)g(p)+f(p)(g-g(p))+g(p)(f-f(p))+(f-f(p))(g-g(p))\end{array}$$

then we have

$$\begin{array}{}\text{(15)}& D(fg)=D(f(p)g(p)+f(p)(g-g(p))+g(p)(f-f(p))+(f-f(p))(g-g(p)))\end{array}$$

The last term in ${\mathfrak{m}}_p^2$$\mathfrak m_p^2$, hence we have

$$\begin{array}{}\text{(16)}& D(fg)=f(p)D(g)+D(f)g(p)\end{array}$$

Conversely, let $\mathcal{D}$$\mathcal D$ be a derivation, then restrict $\mathcal{D}$$\mathcal D$ to ${\mathfrak{m}}_p$$\mathfrak m_p$ we have

$$\begin{array}{}\text{(17)}& \mathcal{D}(fg)=f(p)\mathcal{D}(g)+\mathcal{D}(f)g(p)=0\end{array}$$

Then $\mathrm{Ker}\mathcal{D}\supseteq {\mathfrak{m}}_p^2$$\mathrm{Ker}\mathcal D\supseteq\mathfrak m_p^2$. Hence we have a bijection between derivation and linear map from ${\mathfrak{m}}_p$$\mathfrak m_p$ to sth and $\mathrm{Ker}D\supseteq {\mathfrak{m}}_p^2.$$\mathrm{Ker} D\supseteq\mathfrak m_p^2.$

The following theorem establish the bijection

between linear map from ${\mathfrak{m}}_p$$\mathfrak m_p$ to sth and $\mathrm{Ker}D\supseteq {\mathfrak{m}}_p^2$$\mathrm{Ker} D\supseteq\mathfrak m_p^2$ to ${\mathrm{Hom}}_{\mathbb{R}-\mathrm{Mod}}({\mathfrak{m}}_p/{\mathfrak{m}}_p^2,-)$$\mathrm{Hom}_{\mathbb R-\mathrm{Mod}}(\mathfrak m_p/\mathfrak m_p^2,-)$.

The naturalness leaves to readers.

Corollary. ${\mathrm{Der}}_{\mathbb{R}}(C^{\mathrm{\infty }}(M{)}_p,\mathbb{R})\cong {\mathrm{Hom}}_{\mathbb{R}-\mathrm{Mod}}({\mathfrak{m}}_p/{\mathfrak{m}}_p^2,\mathbb{R})$$\mathrm{Der}_{\mathbb R}(C^{\infty}(M)_p,\mathbb R)\cong\mathrm{Hom}_{\mathbb R-\mathrm{Mod}}(\mathfrak m_p/\mathfrak m_p^2,\mathbb R)$. Hence $T_p(M)\cong {\mathrm{Der}}_{\mathbb{R}}(C^{\mathrm{\infty }}(M{)}_p,\mathbb{R})$$T_p(M)\cong \mathrm{Der}_{\mathbb R}(C^{\infty}(M)_p,\mathbb R)$.

The basis of ${\mathfrak{m}}_p/{\mathfrak{m}}_p^2$$\mathfrak m_p/\mathfrak m_p^2$ is $x_1-x_1(p),...,x_n-x_n(p)$$x_1-x_1(p),...,x_n-x_n(p)$ since ${\tau }_2^{-1}(0)={\mathfrak{m}}_p^2$$\tau_2^{-1}(0)=\mathfrak m_p^2$, ${\tau }_2:{\mathfrak{m}}_p\to \mathrm{Im}({\tau }_2)\cong {\mathfrak{m}}_p/{\mathfrak{m}}_p^2$$\tau_2:\mathfrak m_p\to \mathrm{Im}(\tau_2)\cong\mathfrak m_p/\mathfrak m_p^2$.

Easy to see that the dual basis of $x_i-x_i(p)$$x_i-x_i(p)$ is ${\displaystyle \frac{\partial }{\partial x_i}}{|}_p$$\dfrac{\partial}{\partial x_i}|_p$.

Usually we denote the elements of ${\mathfrak{m}}_p/{\mathfrak{m}}_p^2$$\mathfrak m_p/\mathfrak m_p^2$ as $d{x}_1,...,d{x}_n$$dx_1,...,dx_n$, $d{x}_i\left({\displaystyle \frac{\partial }{\partial x_j}}{|}_p\right)={\displaystyle \frac{\partial }{\partial x_j}}{|}_p{x}_i={\delta }_{i,j}$$dx_i\left(\dfrac{\partial}{\partial x_j}|_p\right)=\dfrac{\partial}{\partial x_j}|_px_i=\delta_{i,j}$.

Tangent Space Functor

As we have seen, $T_p(-)$$T_p(-)$ should be defined as ${\mathrm{Hom}}_{\mathbb{R}-\mathrm{Mod}}(T_p^{\ast }(-),\mathbb{R})$$\mathrm{Hom}_{\mathbb R-\mathrm{Mod}}(T^*_p(-),\mathbb R)$.

Let $(x_1,...,x_m)$$(x_1,...,x_m)$ be the coordinate around $p$$p$ and $(y_1,...,y_n)$$(y_1,...,y_n)$ be the coordinate around $g(p)$$g(p)$.

For a smooth function $g:M_p\to N_{g(p)}$$g:M_p\to N_{g(p)}$,

$$\begin{array}{}\text{(18)}& T_p(g)\left(\frac{\partial }{\partial x_j}\right)=(g^{\ast }{)}^{\ast }\left({\displaystyle \frac{\partial }{\partial x_j}}\right)={\displaystyle \frac{\partial }{\partial x_j}}(g^{\ast }(-))={\displaystyle \frac{\partial }{\partial x_j}}(-\circ g)=\sum _{i=1}^n\frac{\partial g^i}{\partial x_j}\frac{\partial }{\partial y_i}.\end{array}$$