Geometric Interpretation of a Vanishing Covariant Derivative

Question

I find many instances where --- allowing a tensor defined on a hypersurface with components given by $T^\alpha$ and a basis given by $\textbf{S}_\alpha$ (where $\textbf{S}_\alpha$ are the orthogonal basis vectors on the hypersurface) --- if the tensor has a vanishing covariant derivative it can simplify equations greatly. Usually, I can tell that these tensors have significance in the geometrical layout of the problem, but I do not understand what "class" of tensors defined on a hypersurface have the following property: $$\nabla_\beta T^\alpha=0$$ Can any of you help me understand the geometrical significance of under what circumstances a Tensor given on a hypersurface has a vanishing Covariant derivative defined on a hypersurface? Thanks Alot