  We provide necessary and sufficient conditions for stochastic invariance of
finite dimensional submanifolds for solutions of stochastic partial
differential equations (SPDEs) in continuously embedded Hilbert spaces with
non-smooth coefficients. Furthermore, we establish a link between invariance of
submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of
submanifolds for finite dimensional SDEs. This provides a new method for
analyzing stochastic invariance of submanifolds for finite dimensional It\^{o}
diffusions, which we will use in order to derive new invariance results for
finite dimensional SDEs.
