  The Bose-Einstein condensation for an ideal Bose gas and for a dilute weakly
interacting Bose gas in a manifold with nonnegative Ricci curvature is
investigated using the heat kernel and eigenvalue estimates of the Laplace
operator. The main focus is on the nonrelativistic gas. However, special
relativistic ideal gas is also discussed. The thermodynamic limit of the heat
kernel and eigenvalue estimates is taken and the results are used to derive
bounds for the depletion coefficient. In the case of a weakly interacting gas
Bogoliubov approximation is employed. The ground state is analyzed using heat
kernel methods and finite size effects on the ground state energy are proposed.
The justification of the c-number substitution on a manifold is given.
