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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
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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
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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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I think the Lipschitz-Killing curvatures can also be thought of as higher-dimensional generalisations of sectional curvature. The nomenclature of Thorpe explicitly implies this projecteuclid.org/euclid.ijm/1256054993 "Now, between the Riemannian sectional curvature $\gamma$, and the Lipschitz-Killing curvature $K$, there is defined a sequence of intermediate curvatures $\gamma_p$. The function $\gamma_p$, called the $p^{\text{th}}$-sectional curvature, is a smooth function on the bundle of $p$-planes over $X$, and it measures the L-K curvature of geodesic $p$-dimensional submanifolds." –  yasmar Mar 29 at 14:49
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The L-K curvature he refers to is what is used in the generalised Gauss-Bonnet theorem. I don't think he really means geodesic submanifold; just the submanifold that you get by locally exponentiating the $p$-plane (as in sectional curvature). –  yasmar Mar 29 at 14:51

2 Answers 2

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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

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.

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