In the proof of (0x67dbeaae) , we heavily rely on the fact that the curvature tensor of \(\mathbb{S}^d_R\) is covariantly constant. By the Cartan-Ambrose-Hicks theorem , the curvature tensor of every locally symmetric Riemannian manifold is covariantly constant. Therefore, it looks feasible to generalize the Bernstein inequality on those spaces. If we can control the boundary term in the divergence theorem, we can also drop the compactness assumption. In fact, the curvature tensor of the hyperbolic space is the same as the one of the sphere, except for a sign.