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 $\omega$ from $p$ to $q$ is a map $\omega :[0,1]\to M$ such that:

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

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 $\Omega (M;p,q)$, sometimes simply as $\Omega (M)$ or simply $\Omega$.

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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 $\omega \in \Omega$ is defined as the vector space of all piecewise smooth vector fields $W$ along $\omega$ 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.