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 $T_p\partial\mathbb{H}^2\cong \mathbb{R}$?


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 $\partial M \hookrightarrow M$.


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