Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$

It is well known that:

The geodesics of $(M,g),$ i.e. the solutions of $\frac{D}{dt}\gamma=0,$ are integral curves for the hamiltonian vector field of $K:u\in TM\to \tfrac{1}{2}g(u,u)\in\mathbb{R}$ w.r.t. the canonical symplectic form.

Question Knowing how to show it using coordinates, I am wondering how to prove it in an intrinsic way.

share|cite|improve this question
I have posted it even on MO, here:… –  Giuseppe Tortorella Feb 16 '12 at 13:09

1 Answer 1

The MathOverflow version of this post has been answered there. There are indeed coordinate free proofs of the fact as the answers there demonstrate.

If someone prefers to have a more detailed answer here, feel free to summarize the MO answers. I just wanted to give this brief CW answer to help people find the answers and to remove this question from the unanswered list.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.