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?

  • $\begingroup$ 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) $\endgroup$ – peek-a-boo Jul 20 at 10:19

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