- #1

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Let ##x=(x^1,\ldots,x^m)## be local coordinates in a manifold ##M##; and let ##\{\Gamma^i_{jk}(x)\}## be a connection. Assume that we have a curve ##x=x(t),\quad \dot x\ne 0##. Is this curve geodesic or not?

My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)## satisfies the following system

$$\dot x^k(\ddot x^n+\Gamma^n_{rj}\dot x^r\dot x^j)=\dot x^n(\ddot x^k+\Gamma^k_{rj}\dot x^r\dot x^j).$$

In my taste this system looks strange. Or not?

My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)## satisfies the following system

$$\dot x^k(\ddot x^n+\Gamma^n_{rj}\dot x^r\dot x^j)=\dot x^n(\ddot x^k+\Gamma^k_{rj}\dot x^r\dot x^j).$$

In my taste this system looks strange. Or not?

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