Vector Fields Along Curves on Differential Manifolds
Definition1
Let $M$ be a differentiable manifold. A $c : I\subset \mathbb{R} \to M$ that is a (parameterized) curve is called a differentiable function.
A differentiable $V$ that satisfies the following is called a vector field along the curve $c : I \to M$. Being differentiable means that for a differentiable function $f$ on $M$, the function $t \mapsto V(t)f$ is differentiable on $I$.
$$ V : I \to T_{c(t)}M \quad \text{ by } \quad t \mapsto V(t) $$
Explanation
Given the definition of a curve, aside from being differentiable, there are no other restrictions, so intersections as well as edges are allowed.
A vector field $dc(\frac{d}{dt})$ is simply denoted as $\dfrac{dc}{dt}$, and it is called a velocity field or a tangent vector field.
A segment is a contraction mapping from the closed interval $c$ to $[a, b] \subset I$. If $M$ is a Riemannian manifold, the length can be measured with metric $g$, and the length of a segment is defined as follows.
$$ \ell_{a}^{b}(c) = \int_{a}^{b} g\left( \dfrac{dc}{dt}, \dfrac{dc}{dt} \right)^{1/2}dt = \int_{a}^{b} \left\langle \dfrac{dc}{dt}, \dfrac{dc}{dt} \right\rangle^{1/2}dt $$
Manfredo P. Do Carmo, Riemannian Geometry (Eng Edition, 1992), p42-43 ↩︎