Are there higher-dimensional analogues of sectional curvature? 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)
 A: Sure, there's all kinds of generalizations.  One way to think of sectional curvature is you take a 2-dimensional subspace of the tangent space, exponentiate a small neighbourhood of the origin, and take the Gauss curvature at the point of tangency. 
Given a $k$-dimensional subspace of the tangent space, exponentiate a small neighbourhood of $0$, and take the scalar curvature at the point of tangency.  
Another analogy you could build on is to use the comparison definition of sectional curvature -- measuring the infinitesimal deviation of the length of a circle (exponentiated from a 2-dimensional tangent space) of a given radius from that of a Euclidean circle.  You could do the same for content of spheres of arbitrary dimension in subspaces of the tangent space, etc. I imagine these give very related notions of curvature although I've never worked out the details. 
A: In general the term "sectional curvature" is used in the n-dimensional setting. Basically one computes what is known in theory of surfaces as "Gauss Curvature" for the surface gotten by exponentiating every two plane inside the tangent space of the n-manifold. Then one can show that the following things to build intuition,


*

*If two Riemann curvatures are giving the same sectional curvature for every two plane then the Riemann curvatures are equal as endomorphisms. 

*Ricci curvature along a direction is the sum of the sectional curvatures of all possible two planes spanned by that direction and one more from a basis gotten by extending the given direction. 

*Scalar curvature at a point is the sum of the sectional curvatures of all possible two planes spanned by a chosen basis. 
A theorem of Schur says that sectional curvature if constant on all 2-planes at a point is constant at all point of the manifold. Sectional curvature is the strongest notion of curvature and will made constant under very strong conditions like maximal symmetry or locally geodesic reflecting isometry or transitive action of the isometry group on orthonormal frames. 
