Let be $M$ a Riemannian manifold with Levi Civita connection. In the equation of geodesics appears the term $$\Gamma^k_{j} \circ \sigma(t) ,$$

where $\sigma :I \rightarrow M$ is a curve and $\Gamma^k_{j}$ are the Christoffel symbols..

If I change the parameter $t$ with $s=s(t)$, is it true that

$$\Gamma^k_{j} \circ \sigma(t)=\Gamma^k_{j} \circ \sigma(t(s))?$$

How can I show it?

EDIT Is this argument correct?

I know that using the Levi Civita connection the Christoffel symbols depend only on the $g_{\alpha,\beta}$, and I show that them are invariant under reparameterization : $$ g_{\alpha,\beta}(\sigma(t))= \langle \frac{\partial}{\partial x^\alpha},\frac{\partial}{\partial x^\beta}\rangle_{\big|\sigma(t)} = \langle \frac{\partial}{\partial x^\alpha},\frac{\partial}{\partial x^\beta}\rangle_{\big|\sigma(t(s))} ,$$ since the point is the same.

  • 1
    $\begingroup$ firstly, term in the Hamilton-Jacobi equations for a geodesic has $\dot \sigma$, not $\sigma$, secondly, the question as it is asked certainly has a negative answer. Surely, Cristoffel symbols only depend on the metric and the chosen local coordinate system, but think of a question to the one you asked: whether $A x(t) = A x(s(t))$ for A some fixed matrix; if your reparametrization is not $s=t$ then you get a different function a priori. $\endgroup$ – Dima Sustretov Oct 11 '15 at 12:10

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