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)