# 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)

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(My browser was also having trouble displaying the math. For me, the fix, as suggested on the Meta site, was to hold shift while refreshing math.stackexchange.com. This may help you.) –  Jason DeVito Aug 16 '10 at 1:46
How are you defining $\kappa_i$? –  Ryan Budney Aug 30 '10 at 5:00
@Ryan: I'm following "Riemannian Manifolds" (Lee) and defining the $\kappa_i$'s as the eigenvalues of the shape operator. The shape operator $s$ is the (1,1)-tensor field characterized by $\langle X, sY \rangle = \langle II(X,Y),N \rangle$ for all vector fields $X$, $Y$, where $II$ is the second fundamental form and $N$ is a unit normal vector field. –  Jesse Madnick Aug 30 '10 at 5:25
Okay. So that's not a Riemann manifold idea, it's an idea for a submanifold of a Riemann manifold. $\Pi$ measures how `flat' one manifold is, inside of another, so it's a type of relative curvature. For a $3$-manifold in $\mathbb R^4$ $\kappa_1\kappa_2\kappa_3$ measures the volume distortion of the normal-vector map $\vec n : M \to S^3$. –  Ryan Budney Aug 30 '10 at 7:41
This quantity comes up in Hopf's proof of the Gauss-Bonnet theorem for hypersurfaces in even-dimensional Euclidean spaces, for example. See the integration section in the "Differential Topology" textbook of Guillemin and Pollack for details. –  Ryan Budney Aug 30 '10 at 17:16
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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.

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Very interesting ideas, but are they in common use? Is there a term for what you're describing? In particular, I'd like to know why the scalar curvature of the 2-dimensional subspaces are emphasized more than higher dimensional ones. Are they in any way special? –  Jesse Madnick Aug 16 '10 at 14:47
Yes, they're special. I don't remember whose theorem it is, but the Riemann curvature tensor is equivalent to the sectional curvature. –  Ryan Budney Aug 19 '10 at 7:08