A new method of obtaining many-dimensional quasi-exactly-solvable models is suggested. It is based on constructing the generating function with the help of coefficients which obey a finite difference equation. The structure of this equation is selected to obtain the closed second-order differential equation for the generating function. Under some conditions this equation can be thought of as the Schrodinger equation in curved space. For the two-dimensional case the many-parametric class of solution is found explicitly. The spherically-symmetrical case is investigated in detail. It is shown that this case contains spaces of a constant Riemann curvature of both signs.