Riemannian Geometry/Parametrisation of curves

regular curve Let be a Riemannian manifold. A curve is called regular if and only if for all , we have . existence of parametrisation by arc-length of regular curves Let be a regular, continuously differentiable curve, where is an open, connected interval. Then there exists a different open, connected interval and a continuously differentiable bijective function , such that . By the chain rule, we have for all . Hence, the equation becomes equivalent to , and the existence of a solution to the latter equation implies the existence of a solution to the former. Moreover, it suffices to solve , because the right hand side will always be positive. From Peano's theorem, we may infer the existence of a satisfying the given identity on a neighbourhood of , where we impose the initial condition , where is arbitrary. Then we may extend the solution to the maximum interval, existence of regular modification Let be a curve which is twice continuously differentiable. Then there exists a modified curve , defined on an interval , where is some positive number, such that . Define ; by continuity of , this is an open subset of . It decomposes into at most countably many connected components , where for all ; when there are only finitely many connected components, say many, we will set for by convention. Now fix an such that . We would like to modify such that for all points in the domain of the modification (whatever that may be), the absolute value of the derivative of the modification equals . By induction on , we assume that we have already found a modification that traverses parametrisation by arc-length characterisation of parametrisation by arc-length