The idea of this first post is to construct the idea of a tangent space. The intuitive idea comes from surfaces in euclidean space. At each point on a smooth surface you can approximate the surface by a tangent plane defined by the partial derivatives of the function defining the surface. This can’t be held onto because general manifolds don’t exist inside of any ambient space. There isn’t any place for the tangent vectors to point off the surface.
The approach I’ll use in this first post is from Brickell and Clark’s text on Differentiable Manifolds, chapter 4 (referred to as BC from now on). I intend on including other variations over time. This one isn’t particularly modern, but it’s straightforward but rigorous.
First we need to define what it means for a function to be differentiable. As usual with manifolds we just head back to euclidean space. Start with a point and a chart such that part of the domain of f intersects the domain of x. Define a coordinate representative for f to be . Voila, we’re back to euclidean spaces, as . So we define,
Where I’ve used to denote the partial derivative with respect to the i-th coordinate of so as to not confuse those coordinates with the coordinate chart x.
This idea of a derivative with respect to the chart has the usual properties of derivatives, linearity and product rule for example. Onto tangent vectors. We start with a point and all the functions from M to the reals that include p in their domain. We’ll call this . We can equip this set with some structure. We can add functions by choosing the intersection of their domains as the domain of the sum. Products can be defined the same way. BC says this doesn’t have the structure of a vector space because there isn’t always a for each . This isn’t immediately obvious to me. I would think that ‘much’ of the time would be an algebra.
Now we define linear functionals on (though BC calls them operators I prefer to save that for linear transformations that are also endomorphisms). BC proves a couple short results about such things and then makes an important definition
Definition: A derivation on is a linear functional such that . So this is just the product rule for derivatives.
These derivations form a vector space. An example of such is the partial derivative evaluated at . So we know such derivations exist. This vector space is called the tangent space at p and is denoted . You can show that the partial derivatives with respect to the local coordinates forms a basis for the tangent space (so the tangent space has the same dimension as the manifold). Not the only one of course, but one that tends to be used. This will lead to powerful ideas as well shall see.
Of course the point p may belong to more than one coordinate chart, suppose it also belongs to the chart y and so there should be a basis for in terms of that chart. It can be shown,
Exercise: show this (an upcoming post will work this)
Where there is an implied sum on the right hand side over the index i. I do like how BC really focus on the point specific nature of these things. These are derivatives evaluated at a particular value. Relating vectors in different tangent spaces is not easy and either I or Shea will come to that eventually.
Let’s talk about maps between manifolds and what that means for the tangent spaces. Suppose we have a map between manifolds M and N.
Let be in the domain of and so is a point in N. Consider a function from N to the reals.
Then belongs to
Now consider the tangent spaces and .
What I’m going to claim is that induces a map between tangent spaces, . I keep seeing this called the differential but I first learned of it as the pushforward which i prefer since it won’t be long until we need differential forms and the differential of a function is a 1-form. The pushforward map is essentially the Jacobian so there is something to this differential business but I’d like to think it over before using the term.
The method for defining is really clever and pops up fairly often. Given a vector we know it eats functions and spits out vectors. So if maps tangent spaces it must take to a vector in which eats functions like . How does act on f? Like this:
(1) Show that is a derivation on
(2) Show that is a linear transformation between tangent spaces.
I’ve been more than a little lax here. So as this wraps up let me say it a bit more carefully (though still not carefully enough I’d venture). Given a function between manifolds and a point in it’s domain, it induces a linear transformation on the tangent space of that point to the tangent space on the image of the point under . Any function with in it’s domain can then be pulled back via to a function on M. This means that the action of the pushed vector can be defined by pulling back the function.
There is a particularly important example of a tangent space that I really want to understand better: the Lie Algebra of a Lie Group. Given what Shea has been writing about as well as Sean I think it’s time I crack open some books and spend a little time. It occurs to me that this may be another way of assigning topological invariants? Mind my ignorance if I err, but it’s certainly a functor from the category of Lie Groups to the category of Algebras? Lie Algebras? Vector Spaces? (I suppose it depends on how narrow our category should be). Doesn’t this remind you of homotopy, homology, TQFT?