I'd like to cite the part 2 of the book "Foundations of mechanics" of Ralph Abraham, namely "Analytical Dynamics".

I didn't understand how @IanMorris proved this inequality on his post, since I don't understand his "Cayley-Hamilton formula" could you give a reference proving any of these inequalities or explain to me how to prove it, please?

Is there a reference so I can see the proof of this Cayley-Hamilton formula? Does it come from the Cayley-Hamilton theorem? I've never seen it and don't know what is $A^{\wedge k}$.

Do you have ane reference?

@RamiroLafuente Riemannian exponential

I've got it by integration by parts! I wasn't paying attention to the fact that $M$ is closed.

@QuartoBendir either way, i cannot see how is the identity true. Even taking $M=\mathbb{R}^n$ and $\rho\equiv 1$.

@CarloBeenakker I cannot find this definition in this article. I'm used to call $\nabla_i$ the covariant derivative. Can you recomend me a bibliography so I could understand this derivative in a deeper fashion?

@CarloBeenakker I imagine that you get this formula from using musical isomorphisms with $\nabla_i$, correct? But could you please elaborate a little more? I don't know how to get this formula you wrote and also I don't know any book that does that. Another guy said that this could be the Lie derivative of the volume form, which makes some sense.

The action $\Psi: G\times K\to K$ given by $\Psi(g,k)=g\cdot k$ is $C^\infty$

It's still not clear for me.