0
$\begingroup$

I know that if $M$ is a smooth $n$-dimensional manifold with boundary, then $\partial M$ is a smooth $(n-1)$-dimensional manifold. So for $p\in\partial M$, we have $T_p\partial M\cong\mathbb{R}^{n-1}\subset\mathbb{R}^n\cong T_pM$. But I'm confused as to what elements of $T_p\partial M$ "look like". I imagine it is something like taking a vector in $\mathbb{R}^n$ and setting its last entry to $0$. So for the half plane $\mathbb{H}^2$ and $p\in\partial\mathbb{H}^2$, would $T_p\partial\mathbb{H}^2\cong \mathbb{R}$?

$\endgroup$
2
$\begingroup$

Assuming $p\in \partial M$, $T_p \partial M$ should be understood as the subspace of $T_p M$ of vectors which point in a direction which is tangent to $\partial M$. There are several different ways to understand what this means, precisely.

One way is to recall that every tangent vector in $T_p M$ is the velocity vector at $p$ of some smooth curve in $M$. Then $T_p \partial M$ is the subspace of $T_p M$ which consists of velocity vectors at $p$ of smooth curves which lie entirely in $\partial M$. This is a great, geometrically intuitive way to think about it.

Another solution is to take a coordinate chart centered at $p$. Let $U \subset M$ be the coordinate neighborhood and let $\phi: U \to \Bbb{H}^n$ be the coordinate map. Then $\phi$ is a diffeomorphism of $U$ onto a neighborhood of $\Bbb{H}^n$, and therefore provides us with linear isomorphisms $T_p M \simeq T_0 \Bbb{H}^n$ and $T_p \partial M \simeq T_0 \partial \Bbb{H}^n$. Thus the problem is reduced to the case in which $M = \Bbb{H}^n$. In this case $T_0 \Bbb{H}^n$ is the vector space spanned by $\{\partial/\partial x_1, \ldots, \partial /\partial x_n\}$ (all evaluating at $0$, and the $\partial /\partial x_n$ is thought of as a one-sided derivative). Then $T_0 \partial \Bbb{H}^n$ is the subspace spanned by $\{\partial /\partial x_1, \ldots, \partial/\partial x_{n-1}\}$.

To see why any of the above is correct, you need to consider something much more general: An immersion $\psi : M \to N$ (e.g. an embedding or an inclusion of a submanifold) induces an injective linear map $d \psi_p: T_p M \to T_{\psi(p)} N$. We often identify $T_p M$ with its image under $d \psi_p$. The above is just the special case of this where $\psi$ is the inclusion $\partial M \hookrightarrow M$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.