Schroedinger equation in Differential geometric language I have reading about manifolds and tangents spaces and lie derivatives.
I have been wondering is there is a way to write Schrödinger equation in this formalism?
 A: The simplest curved space (i.e. curved Riemannian manifold) generalization of the Euclidean space forumulation of Schrödinger's equation can be found on page 47 of Geroch's notes on geometric quantum mechanics at the Univ. of Chicago from 1974 which were published in paperback form in 2013 here.
This is (for a classical Hamiltonian at most quadratic in momenta, as not to conflict the famous Groenewold-van Hove no-go theorem)
$$\frac{d}{dt}\psi(x,t) = \frac{1}{i\hbar} \left[\left(\frac{\hbar}{i}\right)^2 g^{ab} \nabla_a \nabla_b \psi(x,t) +\left(\frac{\hbar}{i}\right)\left(A^a (x) \nabla_a \psi (x,t) + \frac{1}{2}\psi (x,t) \nabla_a A^a (x)\right) + V(x,t) \psi (x,t)\right] $$, where $H_{class} = g^{ab} p_a p_b + A^a p_a + V(x)$ and $\nabla_a$ is typically the covariant derivative associated to the Levi-Civita connection. 
A true differential-geometric forumulation of QM is clear in Ashtekar&Schilling's review. This is a true text written by a GRist, with all full machinery of differential geometry. 
