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Riemannian Geometry/Parametrisation of curves

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Definition (regular curve):

Let be a Riemannian manifold. A curve is called regular if and only if for all , we have

.

Proposition (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

.

Proof: 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,

Proposition (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

.

Proof: 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

Definition (parametrisation by arc-length):

Proposition (characterisation of parametrisation by arc-length):