The Unapologetic Mathematician

Mathematics for the interested outsider

The Tangent Space at the Boundary

If we have a manifold with boundary M, then at all the interior points M\setminus\partial M it looks just like a regular manifold, and so the tangent space is just the same as ever. But what happens when we consider a point p\in\partial M?

Well, if (U,x) is a chart around p with x(p)=0\in H^n, then we see that the part of the boundary within U — U\cap\partial M — is the surface \{q\in U\vert x^n(q)=0\}. The point 0\in H^n\subseteq\mathbb{R}^n has a perfectly good tangent space as a point in \mathbb{R}^n: \mathcal{T}_0\mathbb{R}^n. We will consider this to be the tangent space of H^n at zero, even though half of its vectors “point outside” the space itself.

We can use this to define the tangent space \mathcal{T}_pM. Indeed, the function x^{-1} goes from H^n to M and takes the point 0 to p; it only makes sense to define \mathcal{T}_pM as (x^{-1})_{*0}\left(\mathcal{T}_0\mathbb{R}^n\right).

This is all well and good algebraically, but geometrically it seems that we’re letting tangent vectors spill “off the edge” of M. But remember our geometric characterization of tangent vectors as equivalence classes of curves — of “directions” that curves can go through p. Indeed, a curve could well run up to the edge of M at the point p in any direction that — if continued — would leave the manifold through its boundary. The geometric definition makes it clear that this is indeed the proper notion of the tangent space at a boundary point.

Now, let y be the function we get by restricting x to the boundary \partial M. The function y^{-1} sends the boundary \partial H^n\cong\mathbb{R}^{n-1}\times\{0\} to the boundary \partial M — at least locally — and there is an inclusion i:\partial M\to M. On the other hand, there is an inclusion j:\mathbb{R}^{n-1}\times\{0\}\to\mathbb{R}^n, which x^{-1} then sends to U — again, at least locally. That is, we have the equation

\displaystyle i\circ y^{-1}=x^{-1}\circ j

Taking the derivative, we see that

\displaystyle i_*\left((y^{-1})_*\left(\mathcal{T}_0\mathbb{R}^{n-1}\right)\right)=(x^{-1})_*\left(j_*\left(\mathcal{T}_0\mathbb{R}^{n-1}\right)\right)

But i_* must be the inclusion of the subspace \mathcal{T}_p(\partial M) into the tangent space \mathcal{T}_pM. That is, the tangent vectors to the boundary manifold are exactly those tangent vectors on the boundary that x_* sends to tangent vectors in \mathcal{T}_0\mathbb{R}^n whose nth component is zero.

September 15, 2011 Posted by | Differential Topology, Topology | 1 Comment