# Tangent vectors at boundary points

It is well known that we can define the tangent space of a manifold $$M$$ at a point $$p\in M$$ as the set of speeds, at time $$0$$, of curves $$\alpha : (-\varepsilon, \varepsilon) \to M$$ such that $$\alpha(0) = p$$. What if $$M$$ is a manifold with boundary and $$p$$ is a boundary point? Is it correct to say that

$$T_p M = \{ \alpha'(0^+) : \alpha : (-\varepsilon, \varepsilon) \to M \text{ is smooth and } \alpha(0) = p \} \cup \{ \alpha'(0^-) : \alpha : (-\varepsilon, \varepsilon) \to M \text{ is smooth and } \alpha(0) = p \}$$

where $$\alpha(0^+)$$ and $$\alpha(0^-)$$ are directional derivatives?

• you might be interested to take a look at Abraham, Marsden, Ratiu's book Manifolds, Tensor Analysis and Applications, in particular section 8.2. Here, they define manifolds with boundary and also its tangent space at a boundary point. (I'm still in the process of reading so I'm afraid I don't understand it well enough to explain it to someone else) – peek-a-boo Jul 20 at 10:19