I recently learned that on Riemannian manifolds, one can define the sectional curvature (http://en.wikipedia.org/wiki/Sectional_curvature) of a (2-dimensional) plane section. I was wondering if a similar concept exists for higher dimensional "space sections."
Here is what got me thinking about this: For 2-dimensional manifolds (surfaces), the sectional curvature is equal to $\kappa_1\kappa_2$, where $\kappa_1$ and $\kappa_2$ are the principal curvatures. Is there a name for the quantity $\kappa_1\kappa_2\kappa_3$ for 3-manifolds, etc., and does it carry similar geometric significance?
(Edit: Typesetting fixed)