Piecewise smooth path

Definition

Let $M$ be a differential manifold and $p, q \in M$ be two (not necessarily distinct) points. A piecewise smooth path $ω$ from $p$ to $q$ is a map $ω : [0, 1] \to M$ such that:

There exists a partition $0 = t_{0} < t_{1} < \dots < t_{k} = 1$ such that $ω$ restricted to $[t_{i - 1}, t_{i}]$ is smooth.
$ω (0) = p$ and $ω (1) = q$

Notions

Set of all piecewise smooth paths between two points

Further information: Path space of a manifold

The set of all piecewise smooth paths in a differential manifold $M$ between two points $p, q \in M$ is denoted as $Ω (M; p, q)$ , sometimes simply as $Ω (M)$ or simply $Ω$ .

Fill this in later

Tangent space

From the infinite-dimensional manifold definition, the following definition emerges for the tangent space to a piecewise smooth curve with respect to the path space.

The tangent space of a path $ω \in Ω$ is defined as the vector space of all piecewise smooth vector fields $W$ along $ω$ for which $W (0) = W (1) = 0$ . In other words, it is an infinitesimal change in the path, with no change occurring at the endpoints.