# Piecewise smooth path

From Diffgeom

## Contents

## Definition

Let be a differential manifold and be two (not necessarily distinct) points. A **piecewise smooth path** from to is a map such that:

- There exists a partition such that restricted to is smooth.
- and

## 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 between two points is denoted as , sometimes simply as or simply .

*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 is defined as the vector space of all piecewise smooth vector fields along for which . In other words, it is an infinitesimal change in the path, with no change occurring at the endpoints.