Piecewise smooth path

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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.

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 \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.