Topology/Tangent Spaces

Briefly, a tangent is a derivative of a curve. Translated into topology this means that you can effectively remove one dimension from a picture. Typically, when working with space time we can can perform one of two operations: Either remove time as a way to see frozen 3D images or we can remove one spatial dimension and therefore represent space time as a curved surface (the net like drawings so typically used to represent topological surfaces in pictures or such phenomena as black holes which are often drawn as shrinking cones).

Tangent spaces are therefore only a representation of what we understand to be one dimension simpler than the problems of topology. It is a useful tool for visualising space time arguments and positions.

Euclidean prelude

So far we have defined smooth maps on smooth manifolds by requiring the corresponding maps on euclidean space to be smooth. In this section we will generalize the notion of derivative on euclidean space to a notion of the derivative of functions between manifolds.

Recall our definition of the derivative on euclidean space:

Definition 1: Let $f\,:\,\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}$. Then the derivative of $f$ at $p$, if it exists, is a linear map $Df(p)\,:\,\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}$ such that

$\lim _{h\rightarrow 0}{\frac {f(p+h)-f(p)-Df(p)h}{||h||}}=0$

Remark 2: $Df(p)$ is unique if it exists, and can be identified with the jacobian matrix $\left[{\frac {\partial f_{i}}{\partial x_{j}}}\right]_{n\times m}$. This is left as an exercise to the reader. This way of defining the derivative does nt, unfortunately, lend itself to generalization to the manifold level. Instead, we will construct another definition of the derivative on euclidean space.

Definition 3: A smooth curve on $\mathbb {R} ^{m}$ is a smooth function $\gamma \,:\,\mathbb {R} \rightarrow \mathbb {R} ^{m}$. Let $\gamma ,\delta$ be smooth curves on $\mathbb {R} ^{m}$ such that $\gamma (0)=\delta (0)$. Define the equivalence relation $\gamma \sim \delta \,\Leftrightarrow \,\gamma ^{\prime }(0)=\delta ^{\prime }(0)$. Define the tangent space of $\mathbb {R} ^{m}$ at $p$ as the space $T_{p}\mathbb {R} ^{m}$ of all equivalence classes $[\gamma ]$ of smooth curves $\gamma$ on $\mathbb {R} ^{m}$ such that $\gamma (0)=p$.

Remark 4: Note that we only need smooth curves to be defined on an open subset of $\mathbb {R}$ containing $0$.

Lemma 5: $T_{p}\mathbb {R} ^{m}$ is isomorphic to $\mathbb {R} ^{m}$ as a vector space for any $p\in \mathbb {R} ^{m}$.

Proof: Since for any smooth curve $\gamma$ on $\mathbb {R} ^{m}$, $\gamma ^{\prime }(0)$ is a vector in $\mathbb {R} ^{m}$, there is a natural bijection $[\gamma ]\mapsto \gamma ^{\prime }(0)$. Let $\mu$ be this bijection, and give $T_{p}\mathbb {R} ^{m}$ the vector space structure $a[\gamma ]+b[\delta ]=\mu ^{-1}\left(a\mu ([\gamma ])+b\mu ([\delta ])\right)$, and $\mu$ becomes an isomorphism of vector spaces. ∎

Remark 6: Unlike $\mathbb {R} ^{m}$, $T_{p}\mathbb {R} ^{m}$ does not have a natural basis.

Lemma 7: Let $\gamma$ be a smooth curve on $\mathbb {R} ^{m}$ with $\gamma (0)=p$ and $\gamma ^{\prime }(0)=v$. Then $\gamma \sim \delta$ where $\delta (t)=p+vt$.

Proof: First off, note that $\gamma (0)=p=\delta (0)$, so it makes sense to compare them. Secondly, $\delta ^{\prime }(0)=v=\gamma ^{\prime }(0)$, so $\gamma \sim \delta$. ∎

Definition 8: Let $f\,:\,\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}$ be a smooth function. Then the differential of $f$ at $p$ is the map $d_{p}f\,:\,T_{p}\mathbb {R} ^{m}\rightarrow T_{f(p)}\mathbb {R} ^{n}$ given by $d_{p}f([\gamma ])=[f\circ \gamma ]$.

Lemma 9: $d_{p}f$ is well defined.

Proof: Let $\gamma \sim \delta$ where $\gamma (0)=p=\delta (0)$. Then $(f\circ \gamma )^{\prime }(0)=Df(\gamma (0))\gamma ^{\prime }(0)=Df(\delta (0))\delta ^{\prime }(0)=(f\circ \delta )^{\prime }(0)$ by the chain rule and using the usual derivative, therefore $[f\circ \gamma ]=[f\circ \delta ]$ and so $d_{p}f$ is well defined. ∎

Lemma 10: Let $f(p)=g(p)$. Then if $Df(p)=D(g)(p)$, then $d_{p}f=d_{p}g$.

Proof: Let $\gamma$ be any curve at $p$. Then if $Df(p)=Dg(p)$ we have $d_{p}f([\gamma ])=[f\circ \gamma ]=[f(p)+Df(p)\gamma ^{\prime }(0)t]=[g(p)+Dg(p)\gamma ^{\prime }(0)t]=[g\circ \gamma ]=d_{p}g([\gamma ])$. ∎

Thus the differential encodes the information about the derivative. However, it also encodes information about $f(p)$. Unlike the previous definition of the derivative, the differential can, with some slight modifications, be generalized to work on manifolds. That is the topic of the next subsection.

Tangent Spaces

Definition 11: A smooth curve on a manifold $M$ at $p$ is a function $\gamma \,:\,\mathbb {R} \rightarrow M$ such that $\gamma (0)=p$. If $\gamma ,\delta$ are smooth curves on $M$ at $p$, we define the equivalence relation $\gamma \sim \delta$ if and only if there exists a chart $(U,\phi )$ with $p\in U$ such that $(\phi \circ \gamma )^{\prime }(0)=(\phi \circ \delta )^{\prime }(0)$.

Remark 12: We can differentiate $(\phi \circ \gamma )$ since it is a function between euclidean spaces, for which we already have a developed theory of differentiation. Also, the equivalence relation is well defined since if it holds for one chart, it holds for all compatible charts as well.

Definition 13: The tangent space of $M$ at $p$ is the space of all equivalence classes of curves on $M$ at $p$.