📂Geometry

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.

$$ \ell_{a}^{b}(c) = \int_{a}^{b} g\left( \dfrac{dc}{dt}, \dfrac{dc}{dt} \right)^{1/2}dt = \int_{a}^{b} \left\langle \dfrac{dc}{dt}, \dfrac{dc}{dt} \right\rangle^{1/2}dt $$