Proof of the Beltrami theorem

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I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem:

Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is of constant sectional curvature.

the only book I study in this regard is:

"Riemannian geometry" by do Carmo

I'm trying to find more information and/or a proof of it that is fully understandable for me. Is it also satisfied for the Finsler manifolds? Is anyone more knowledgeable of this theorem?

Thanks for any help.