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I am trying to understand proposition 2.5 in chapter 9 of do Carmo's "Riemannian Geometry". In the proof of the proposition he says:

Let M be a Riemannian manifold and $c:[0,a]\rightarrow M$ a piecewise differentiable curve. We consider a vector field $V$ along $c$ such that $V(0)=V(a)=0$ and if $i\neq0,k$, $V(t_{i})=\frac{dc}{dt}(t_{i}^{+})-\frac{dc}{dt}(t_{i}^{-})$, where $(t_{i},t_{i+1}), i\in\{0,\ldots k-1\}$ are the intervals where $c$ is differentiable.

How do we know this vector field exists? Maybe it has something to do with the parallel transport of $\frac{dc}{dt}(t_{i}^{+})-\frac{dc}{dt}(t_{i}^{-})$ along $c|_{[t_{i},t_{i+1})}$, but I do not know if this contruction makes $V$ piecewise differentiable.

Srinivasa Granujan
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1 Answers1

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Parallel transport will not help, since it will not respect the endpoint values if you move from $t_i$ to $t_{i+1}$. There are several ways to do this. The most straightforward way is to cover $c([a,b])$ with coordinate neighbourhoods $U_k$ (in $M$), choose a finite subcover of $[a,b]$ of connected intervals in the sets $c^{-1}(U_k)$ and define the vector field piecewise locally as constant and globally by using a smooth partition of unity (on $[a,b])$. The desired behavior in the points $t_i$ can be acchieved by choosing an appropriate subset of the coordinate neighbourhoods centered at the $t_i$

Note that, while tedious (as an example: you need to make sure you are working consistently in case of e.g. selfintersections of the curve) this is pretty much technical routine. Since it is a bit tedious I leave the details to you...

Thomas
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  • I have similar question for that. But i encounter this problem from Lee's book Riemannian Geometry which much less same. Only Lee's give us a hint using smooth bump function in coordinate chart. I already made such construction here. https://math.stackexchange.com/questions/2419330/using-bump-function-in-coordinate-chart-to-construct-a-vector-field-v-along-a . – Kelvin Lois Sep 07 '17 at 23:22