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Curves are functions from an interval of the real numbers to a differentiable manifold.

Given a metric on the manifold, arc length is a property of the image of the curves, not of the curves itself. In other word, it is independent of the parametrization of the curve. In the case of the Euclidean space, arc length can be defined without using any parametrization of the curve, namely by the limit of the length of the approximating polygons. My question is whether could we give similar definition of the arc length in the case of manifolds? The standard definition is [tex]l(\gamma)=\int_{0}^{1} {|\dot\gamma(t)|} dt\ ,[/tex] and unlike in the case of the definition with polygons, this definition involves the parametrization of the curve, in spite that all parametrizations give the same result. How could we eliminate the unnecessary parametrization from the definition?

Given a metric on the manifold, arc length is a property of the image of the curves, not of the curves itself. In other word, it is independent of the parametrization of the curve. In the case of the Euclidean space, arc length can be defined without using any parametrization of the curve, namely by the limit of the length of the approximating polygons. My question is whether could we give similar definition of the arc length in the case of manifolds? The standard definition is [tex]l(\gamma)=\int_{0}^{1} {|\dot\gamma(t)|} dt\ ,[/tex] and unlike in the case of the definition with polygons, this definition involves the parametrization of the curve, in spite that all parametrizations give the same result. How could we eliminate the unnecessary parametrization from the definition?

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