Intuitive understanding of inward and outward vector definitions

Question

Let $M$ be a smooth manifold with boundary and $p \in \partial M$. In Lee's Introduction to Smooth Manifolds he gives the following definition:

If $p \in \partial M$, a vector $v \in T_pM \setminus T_p \partial M$ is said to be inward-pointing if for some $\epsilon > 0$ there exists a smooth curve $\gamma: [0, \epsilon) \rightarrow M$ such that $\gamma(0) = p$ and $\gamma'(0) = v$, and it is outward-pointing if there exists such a curve whose domain is $(-\epsilon, 0]$.

Intuitively I understand what an inward and outward pointing vector is from calculus and Euclidean spaces. However, I am having some trouble mapping that intuitive picture to the definition above. In particular, how does the condition $\gamma'(0) = v$ suggest that the vector is pointing inwards, and how does taking the domain to be $(-\epsilon, 0]$ suggest that the vector is pointing outwards?

Answer 1

Before talking of a boundary point, let's look at the more general picture of an interior point $p$. A tangent vector $v \in T_pM$ can be thought of as a vector tangent to a curve $\gamma \colon (-\varepsilon,\varepsilon)$ at $p=\gamma(0)$, with $\gamma'(0)=v$. This means that when passing through $p$, $\gamma$ follows the direction defined by $v$. Actually, this tells two things on the path $\gamma$:

1. $\gamma$ arrives at $p$ in the direction of $v$,
2. $\gamma$ leaves $p$ in the direction of $v$.

Now, imagine that $p$ lies on the boundary of some domain, for instance, the boundary of a flat earth. There are three kinds of tangent vectors at that point:

1. Those that point outward the flat earth: they are tangent to paths coming from the inside of the earth. Following these vectors will make you fall into the void of the universe. Such paths are only defined on $(-\varepsilon,0]$.
2. Those that point inside the flat earth: they are tangent to paths that go inside the earth. Following them will make you stay on the flat earth, and you will be able to survive. Such paths are only defined on $[0,\varepsilon)$.
3. Those that are tangent to the boundary. More things can happen there. For instance, you can follow such vectors and stay along the boundary. It the boundary is convex, you can also follow them and fall into the void. If concave, you will be more likely to go toward the interior of the earth.

The same thing occurs for a more general manifold with boundary. The tangent space $T_pM$ at $p\in \partial M$ splits into three different components $T_pM^{\text{out}}$, $T_pM^{\text{in}}$ and $T_p\partial M$. $T_p\partial M \subset T_pM$ is an hyperplane, and the two other components are the two connected components of $T_pM \setminus T_p\partial M$.