Arc-length functional

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Definition

Fix a Riemannian manifold M. 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 \gamma:[0,1] \to M is:

\int_0^1 \left|\frac{d\gamma}{dt}\right| dt

Interestingly, the arc-length is independent of the parametrization of the curve. In other words , if f:[0,1] \to [0,1] is an increasing function, then \gamma \circ f has the same arc-length as \gamma.

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