The duality of tangent and cotangent spaces

Some intuitive help

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I always struggled to get an intuitive picture of the relationship between the tangent and cotangent spaces (/ bundles). The definition of the tangent space was given to me in terms of equivalence classes of curves, and when the cotangent space wasn’t just defined directly in terms of the tangent space it was defined in terms of a particular function space from the manifold to R. I could do the proof, of course, that the dual space to the former was naturally isomorphic to the latter; but it never really ‘clicked’ until I thought about it from the perspective outlined below. I make no claim to originality, this is all implicit in the way a lot of textbooks describe the relationship, but I have found it very helpful to be explicit about it and I wanted to share it. The following will not be the most mathematically rigourous post in the world, but that’s kind of the point—I’m sharing intuitions that helped me build up to the rigourous stuff.

Let’s start intuitively. A tangent vector, essentially, represents infinitesimal displacement on the manifold. If you have a curve γ: R → M that represents a path of successive finite displacements on the manifold, its derivative γ′(a) at any given a∈R is a vector that represents the infinitesimal displacement between γ(a) and γ(a + dx).

So if you have two curves α and β that both pass through p∈M (let’s say, w.l.o.g., α(0)=β(0)=p), they will be associated with the same tangent vector at point p just in case their derivatives are identical: α′(0)=β′(0). So if we define an equivalence relation on the set of curves γ such that γ(0)=p, treating two curves as equivalent just in case they have the same derivative at 0, we can know that each equivalence class can be associated with one (and only one) tangent vector at p.

All of this should be intuitive; there’s nothing fancy here, this is just normal stuff. But it already gives us enough material to make a statement about the tangent space:

The tangent space at p∈M is the quotient of F = {f: R → M | f is smooth and f(0) = p} under f ~ g iff f′(0) = g′(0).

This statement is, in fact, true! It is not an adequate definition of the tangent space, because it relies on the idea of the derivative of a function R → M, but (as we’ve seen) the derivative of such a function would be an element of the tangent space. To eliminate this circularity and turn it into a workable definition, we will have to break this statement down further. But for now we should stick with it, because it leads us to the cotangent space.

The category theorists are always telling us that duality emerges from ‘turning arrows around’. In this case, we can and should take them literally. If we just turn the arrows around in the above statement about the tangent space, we get:

The cotangent space at p∈M is the quotient of G = {g: M → R | g is smooth and g(p) = 0} under f ~ g iff f′(p) = g′(p).

This is also a true statement about the cotangent space!¹ Again, it’s not an adequate definition (for the same reason as before); but it has two major virtues.

1. It gives us a useful sense of what the cotangent space is supposed to be, what its elements represent.

2. It makes the relationship of these two spaces obvious and immediately apparent. If we ignore the problem of defining the derivative, it is incredibly easy to prove that these two spaces are vector-space duals.²

In terms of the former virtue, we can see that the cotangent space at p is concerned with scalar fields on M. In particular, it is concerned with their local behaviour around p. If we have two scalar fields f and g such that f - g is approximately constant everywhere in a neighbourhood of p (in a particular sense of ‘approximately’), then f and g are associated with the same cotangent vector at p. This is a meaningful thing to be concerned with. It also helps you see why we integrate cotangent vector fields (‘1–differential forms’) along curves and not tangent vector fields: cotangent vectors basically represent the local behaviour of scalar fields, so integrating cotangent vectors along a path gives you a scalar. And it’s immediately apparent how these concerns are dual (in both the specific linear algebra sense and also the more general sense) to the concern with tangents to curves on the manifolds: in one case we’re looking at the local behaviour of functions R → M, in the other with the local behaviour of functions M → R.³

OK, so now it’s time to turn these statements into proper definitions. Let’s start with the tangent space. The nice thing about smooth manifolds is that they come equipped with charts, which are very useful to work with. If φ is a chart for a neighbourhood of p, then for any γ: R → M, φ∘γ is a curve in a real vector space. The derivative of such a curve is well-defined—you learned the definition in secondary school—so we can bypass the need to define derivatives on the manifold by setting α ~ β iff (φ∘α)′(0) = (φ∘β)′(0).

Of course, we don’t want our definition to depend on the particular choice of chart we use. But you can use the chain rule to convince yourself that we should have (φ∘α)′(0) = (φ∘β)′(0) if and only if (ψ∘α)′(0) = (ψ∘β)′(0) for any two charts φ and ψ; and in fact, this result holds even without assuming the applicability of the chain rule. (You may have been asked to prove this at some point.) So we have the following definition:

The tangent space at p∈M is the quotient of F = {f: R → M | f is smooth and f(0) = p} under f ~ g iff (φ∘ f)′(0) = (φ∘ g)′(0), with vector space structure defined by φ.

This is a good definition. (Indeed, it is the standard definition.) You need to check that any two charts φ and ψ define the same linear structure, but this isn’t too hard to do, and with that checked we can guarantee that the tangent space is part of the intrinsic structure of the manifold. Great!

Now onto the cotangent space. We could do a similar thing, using the inverse chart to turn all our functions into maps between R-linear spaces. But we have an advantage in this case, because G = {g: M → R | g is smooth and g(p) = 0} is already a real vector space. If we can find a subspace H containing all and only the functions h with h′(p) = 0, then G / H will be exactly the space we want, no fussing about with charts needed: we will have f ~ g iff f = g + h for some h∈H, and differentiating both sides at p we see that this is the same as f′(p) = g′(p).

It might seem like we will need charts to define H, since we’re (again) relying on the derivative to pick it out. But if we assume that we are entitled to use Taylor’s theorem (which we aren’t yet, but stick with it), then by definition the elements of H will be functions that have zero for the constant and linear terms in their Taylor expansion. We are left only with the higher-order terms: the function is a linear combination of products of the function (x - p). Obviously (x - p) is an element of G, because it is equal to zero at p.

And if we feel entitled to use the product rule, we can see that all linear combinations of functions of the form f(x)g(x) with f, g∈G will have derivative 0 at p. We now know that this condition is both necessary (by Taylor’s theorem) and sufficient (by the product rule) for a scalar field in G to have derivative 0 at p.

So, if we assume the basics of calculus should apply in some form, we now have a purely algebraic definition of H: the set of linear combinations of scalar fields of the form f(x)g(x) with f(p)=g(p)=0. We can drop calculus completely from our definition of the cotangent space. Since H is (by definition) closed under linear operations, it forms a subspace of G. A rigourous definition of the cotangent space follows immediately:

The cotangent space at p∈M is the vector space quotient G / H, where G is the real vector space {g: M → R | g is smooth and g(p) = 0} and H is the subspace generated by {f ⋅ g | f, g ∈ G}.

Again, this is the standard definition you’ll find in differential geometry textbooks, minus a little ring theory that mathematicians typically use to package the definition of H a little more compactly.⁴

At the end of the process, we have two adequate definitions that look very different from each other. One uses notions from calculus and imports the vector space structure using charts; the other sticks in abstract algebra and has no truck with charts. If you were to start from these definitions, you could prove (and thousands of differential geometry students have proven) that the cotangent space is canonically isomorphic to the space of linear forms on the tangent space. But this duality might be (as it was for me) a little unintuitive, hard to grasp and perhaps hard to see the utility and relevance of.

But if you remember where we got both the definitions, it becomes much easier. In both cases, we’re concerning ourselves with the local behaviour of functions, considering two functions to be equivalent if they have the same behaviour around p. The differences are twofold. First, in the case of the tangent space we’re concerned with functions R → M, whereas with the cotangent space we are looking at functions M → R; secondly, we end up defining ‘the same local behaviour’ very differently in both cases. But this latter consideration is essentially just a technicality, and once we are able to define the derivative for all functions on or to the the manifold, we can drop the technicalities and return to the initial intuitive statements.

I found this whole line of thinking extremely illuminating once I finally figured it out for myself, and I hope it’s helpful to some other people.

1. Well, normally the derivate in this case would be denoted df instead of f′, since its domain is a multi-dimensional space and so the derivative has to be multi-dimensional (in particular, it turns out to be analogous to the gradient). But that’s a purely notational difference, especially since we’ve not actually defined the derivative in this context yet.

2. We can associate every scalar field g: M → R with a map L that takes curves f: R → M to real numbers, defined as L(f) = (g∘f)′(0). The proof that this association respects the relevant equivalences and gives rise to an isomorphism falls out naturally.

3. As in the case of the tangent space, the limitation in the case of the cotangent space to only considering scalar fields with g(p) = 0 is w.l.o.g.; it just means you don’t have to add an arbitrary constant field to everything.

4. Indeed, you find this definition in algebraic geometry textbooks too: when you’re dealing with varieties that don’t come equipped with charts, the definition of the tangent space is useless, but you can use a version of this definition of the cotangent space and then just define the tangent space as its dual.