If we have a manifold with boundary , then at all the interior points 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 ?
Well, if is a chart around with , then we see that the part of the boundary within — — is the surface . The point has a perfectly good tangent space as a point in : . We will consider this to be the tangent space of at zero, even though half of its vectors “point outside” the space itself.
We can use this to define the tangent space . Indeed, the function goes from to and takes the point to ; it only makes sense to define as .
This is all well and good algebraically, but geometrically it seems that we’re letting tangent vectors spill “off the edge” of . But remember our geometric characterization of tangent vectors as equivalence classes of curves — of “directions” that curves can go through . Indeed, a curve could well run up to the edge of at the point 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 be the function we get by restricting to the boundary . The function sends the boundary to the boundary — at least locally — and there is an inclusion . On the other hand, there is an inclusion , which then sends to — again, at least locally. That is, we have the equation
Taking the derivative, we see that
But must be the inclusion of the subspace into the tangent space . That is, the tangent vectors to the boundary manifold are exactly those tangent vectors on the boundary that sends to tangent vectors in whose th component is zero.