# The tangent space of the boundary of a manifold with boundary is a subspace of the tangent space

I was trying to understand the following sentence in some notes I am reading:

Let $M$ be a manifold with boundary.

At any point $p \in {\partial}M$ there is a canonical subspace $T_{p}({\partial}M) \subset T_{p}(M)$; the quotient space is the a real line $\nu_{p}$.

I know of $T_{p}(M)$ as the vector space consisting of operators or derivations $\nu: F(M) \rightarrow \mathbb{R}$ where $F(M)$ is the algebra of smooth functions from $M \rightarrow \mathbb{R}$.

Does this natural subspace involve some sort of imbedding of a $F({\partial}M)$ into $F(M)$?

I apologize if this question is obvious.

• There is a natural map $\rho: F(M) \rightarrow F(\partial M)$ (restriction). This induces a map $Der_p(\partial M) \rightarrow Der_p(M), \delta \mapsto \delta \circ \rho$. – Tim kinsella Oct 22 '14 at 14:49

No, it involves projecting $F(M)\to F(\partial M)$, i.e. observing that a smooth function on $M$ restricts to a smooth function on $\partial M$. This seems to go the wrong directions, but now the derivations come into play: a derivation $F(\partial M)\to\mathbb R$ gives rise to a map $F(M)\to F(\partial M)\to\mathbb R$ (that is also a derivation, as you may check).
• Thanks, I understand, if we have a smooth function $f$ from M to $\mathbb{R}$, we can restrict that map to ${\partial}M$. This allows us to send a derivation in the tangent space of the boundary to a derivation in the tangent space of the manifold, using the composite mentioned in your answer and the comment above. Thanks a lot. That helps. – user135520 Oct 22 '14 at 15:26