### Tangent Spaces

Over the last day I read the first ten pages of Milnor’s famous notes, Topology from the Differentiable Viewpoint. As one might expect, these early pages are spent covering the definitions and basic facts that will be needed further along in the book. The most important of these is, of course, the following:

Definition 1 An topological manifold ${M}$ is an ${n}$-dimensional smooth manifold if, for every ${x \in M}$, there exists a neighbourhood ${U}$ of ${x}$ diffeomorphic to a subset of ${{\mathbb R}^n}$.

I find it quite interesting to contrast Milnor’s exposition with more modern takes, such as Lee’s Introduction to Smooth Manifolds. The biggest difference here for me was in his definition of the tangent space of a smooth manifold. Milnor does this extrinsically, again appealing to an embedding of ${M}$:

Definition 2 Consider some ${x \in M}$, and some smooth parameterisation ${f: U \subset {\mathbb R}^n \rightarrow M}$ with ${f(u) = x}$. We can consider ${M}$ to be a subset of ${{\mathbb R}^m}$ for some ${m}$, and hence we can take the derivative of ${f}$ in the usual way:

$\displaystyle df_u: {\mathbb R}^n \rightarrow {\mathbb R}^m$

Define ${TM_x}$, the tangent space of ${M}$ at ${x}$, to be the image of ${df_u}$.

It is a standard exercise to show that the tangent space is well defined (does not depend on the choice of parameterisation) and is an ${n}$-dimensional real vector space. This definition of a tangent space is very intuitive, and so I feel it’s good from a pedagogical perspective.

The problem for me is two-fold. Firstly, we often construct (smooth) manifolds by gluing together open sets of Euclidean spaces with appropriate maps. Obviously it is true that the result does in fact embed in some ${{\mathbb R}^m}$, but the proof is a bit gross. Because of this, it would be nice not to rely on this fact.

The second reason is generality. It would be nice to extend the notion of a tangent space to objects where the standard analysis tools might not be available. These would be things like locally ringed spaces, algebraic varieties and the like.

To deal with the first issue, one way of thinking about the tangent space is as (obviously) the set of tangent vectors at a point. Intuitively, we should be able to define this using only local data about ${M}$. We can make this precise as follows:

Definition 3 Consider some ${x \in M}$, and a smooth coordinate chart ${\varphi: U \owns x \rightarrow R^n}$. Let ${\Gamma}$ be the set of curves ${\gamma: (-1, 1) \rightarrow M}$ such that ${\gamma(0) = x}$ and ${\varphi \circ \gamma}$ is differentiable at ${0}$ in the usual sense. We can form an equivalence relation ${\sim}$ on ${\Gamma}$ by defining ${\gamma_1}$ and ${\gamma_2}$ to be equivalent if ${(\varphi \circ \gamma_1)'(0) = (\varphi \circ \gamma_2)'(0)}$. Then ${TM_x}$ can be identified with ${\Gamma / \sim}$.

In other words, each element of the tangent space at ${x}$ can be identified with the derivative of some curve through ${x}$. Again, proving that this is independent of the choice of coordinate chart is left to the reader. The chain rule is your friend.

This gets rid of the issue with embeddings. It is still not quite good enough, however – we are still making direct use of analysis in the definition. To do better, we have to get a lot more sneaky.

The idea is to note that the tangent space, being related to derivatives at ${x}$, essentially describes the first order behaviour of lines, surfaces etcetera passing through ${x}$. Taking a hint from algebraic geometry, these surfaces can be viewed as the zeros of real-valued functions of our manifold. Passing to the first order behaviour then amounts to killing off the non-linear parts of these functions.

Firstly, however, what do we mean by a real-valued functions of ${M}$?

Definition 4 Let ${C^\infty(M)}$ be the set of functions ${f: M \rightarrow {\mathbb R}}$ such that, for any smooth coordinate chart ${\varphi: U \rightarrow R^n}$, the composition ${f \circ \varphi^{-1}}$ is smooth. ${C^\infty(M)}$ forms an associative, unital algebra over ${{\mathbb R}}$ in the obvious way.

To define the notion of surfaces passing through ${x}$, let ${I = \{f \in C^\infty(M) : f(x) = 0 \}}$. This set is the kernel of the evaluation homomorphism ${f \mapsto f(x)}$, and so is in fact an ideal of ${C^\infty(M)}$. Furthermore, the evaluation homomorphism is surjective onto a field (consider constant functions), so ${I}$ is a maximal ideal.

We would now like to identify functions in ${I}$ with the same first order behaviour at ${x}$. Morally, this should amount to identifying functions with the same derivative at ${x}$, much like we have done up till now. To get a feel for this, consider the ${1}$-dimensional case. As the functions in ${I}$ are analytic, we can expand them about ${x}$ with Taylor’s theorem:

$\displaystyle f(u) = \sum_{k=1}^{\infty} a_k(u - x)^k$

Now suppose that ${f, g \in I}$ have the same first order behaviour, or derivative, at ${x}$. Then note that:

$\displaystyle (f - g)'(x) = \sum_{k=1}^{\infty}k(a_k - b_k)(u - x)^{k-1} = a_1 - b_1$

So in particular:

$\displaystyle (f - g)(u) = c_2(u - x)^2 + c_3(u - x)^3 + \dots$

Each of the terms on the right hand side is a product of two elements of ${I}$. In other words, functions in ${I}$ with the same first order behaviour differ by some element of the ideal ${I^2}$. To identify them, we need merely quotient!

Definition 5 The cotangent space ${TM^*_x}$ of ${M}$ at ${x}$ is defined as the quotient (vector) space ${I/I^2}$.

You read correctly – this is not quite the tangent space. The difference here is that while the tangent space is a set of vectors, ${TM^*_x}$ as we have defined it is actually a set of linear functionals to ${{\mathbb R}}$. This is nothing but the dual of ${TM_x}$, so we can recover the tangent space by dualising again:

$\displaystyle TM_x \simeq (TM^*_x)^*$

Another way of seeing that ${TM^*_x}$ is not the tangent space is by considering a smooth map ${\varphi: M \rightarrow N}$ between two smooth manifolds. The derivative of this function gives rise to a map ${d\varphi_x: TM_x \rightarrow TN_{\varphi(x)}}$ – for a nice exercise, try to define this!

On the other hand, consider some ${f: N \rightarrow {\mathbb R}}$ in ${TN^*_{\varphi(x)}}$. Pulling back along ${\varphi}$, we get a map ${f \circ \varphi: M \rightarrow {\mathbb R}}$ which vanishes at ${x}$ – in other words, a member of ${TM^*_x}$! Hence ${\varphi}$ gives rise to a map ${d\varphi^*_x: TN^*_{\varphi(x)} \rightarrow TM^*_x}$.

The cool thing about how we’ve defined the cotangent space here is that, once we have an appropriate notion of functions on our space, the rest is just algebra. Thus this generalises nicely – the Zariski tangent space is exactly this idea applied to algebraic varieties, for instance!