Canonical Quantization of Geometrized Mechanics

A particle in 3D space with certain potential will move in a curved trajectory like a comet in gravitational potential caused by the star. On the other hand, a free particle in curved space also moves according to geometry of that space. In this paper, the connection between potential energy and space metric will be discussed. So the formulation of classical mechanics in geometric terms can be found and the canonical quantization of it can be carried out. At the end of this paper, as an example, we will consider a particle under isotropic harmonic oscillator potential in two-dimensional sphere, carry out the canonical quantization, and then calculate the energies and their states.