Given a Hilbert manifold $\mathcal H$ (always using the natural Hilbert inner product) and a geodesic $\Gamma(t)$ in this manifold, can one show that the projection of this geodesic onto a submanifold $\mathcal S$ of $\mathcal H$ (corresponding Hilbert subspace) is necessarily also a geodesic in $\mathcal S$? Or show a counterexample? If this is not a valid conclusion, could it be valid under some restriction?
Let me know if I have made any incorrect assumptions or the problem needs further specification.