  We study the Cauchy problem for Schr\"odinger type stochastic partial
differential equations with uniformly bounded coefficients on a curved space.
We give conditions on the coefficients, on the drift and diffusion terms, on
the Cauchy data, and on the spectral measure associated with the noise, such
that the Cauchy problem admits a unique function-valued mild solution in the
sense of Da Prato and Zabczyc.
