This question might sound slightly vague, but please bear with me.
If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as well as the implicit (Gauss) curvature and plenty else using the machinery of Riemannian geometry. In particular, notions of curvature are perfectly well defined and understood for such surfaces.
Now suppose the object of my interest is not such a surface per se, but a certain volume (well-behaved subset of $R^3$) that includes this surface as a sort of a "mean". One example would be a real life hollow sphere, whose mean surface is a 2-sphere, but whose wall has non-zero "thickness".
What is the correct notion of curvature (intrinsic as well as extrinsic) for such entities?
From what (little) I know, such an object is a "3-manifold with boundary" so does one need to apply the definition of curvatures that apply to 3 manifolds, making the curvature a tensor? In that case, would the Gauss (scalar) curvature of such an object be 0?
Thanks in advance.