Hello,
In general, restricting a function to a smaller set can introduce new critical points. Often times, this is desirable (but maybe not in your case).
Here is a simple example: let f(X) = real(Tr(A*X)) for some non-zero matrix A which has the same size as X, and A* is the conjugate-transpose.
Then, f does not have any critical points (grad f(X) = A \neq 0 for all X).
Yet, f does have a global minimizer and a global maximizer on St(n, p) (because the latter is compact and f is continuous), hence f must have critical points when restricted to St(n, p) (and likely that is what we would want in this scenario).
(I'm assuming your function f is defined on C^{n x p} and you restrict it to St(n, p) = {X : X*X = I} as a subset of C^{n x p}.)
Best,
Nicolas