Geometric Hamiltonian matrix on the analogy between geodesic equation and Schr\"{o}dinger equation

By formally comparing the geodesic equation with the Schr"{o}dinger equation on Riemannian manifold, we come up with the geometric Hamiltonian matrix on Riemannian manifold based on the geospin matrix, and we discuss its eigenvalue equation as well. Meanwhile, we get the geometric Hamiltonian function only related to the scalar curvature.