# Arc-length functional

From Diffgeom

## Definition

Fix a Riemannian manifold . The **arc-length functional** is a map from the space of piecewise smooth curves in the manifold, to real numbers, defined as follows. The arc-length of a curve is:

Interestingly, the arc-length is independent of the parametrization of the curve. In other words , if is an increasing function, then has the same arc-length as .