Sectional curvature in GR

1. Jan 26, 2005

hellfire

The terms elliptic, hyperbolic and euclidean geometry are defined according to the sectional curvature, which is a generalization of the Gaussian curvature of a surface. Are there any restrictions on the sectional curvature for spacetimes in general relativity?

The Ricci scalar, being a function of the trace of the energy-momentum tensor $$R = - \kappa T^{\alpha}_{\alpha}$$, must be always positive? Can be the sectional curvature defined as a function of the Ricci scalar?

2. Jan 27, 2005

pervect

Staff Emeritus
This is the first time I've heard of sectional curvature. The thing I find interesting is that it does completely describe the Riemann - Google finds enough hits on the term that I'm sure it does , it's just not particularly obvious how.

For a 2d manifold, there is only one tanget to the surface at any point, so there is only one component of the Riemann tensor. So this case isn't very interesting.

Let's go up to a 3d manifold. Then we can describe a specific tangent 2-plane by two variables (angles, let's say), and the sectional curvature of this 2-plane will be a scalar. So the sectional curvature is a scalar function of two angles for a 3d manifold.

Somehow this scalar function of two angles has to give us the six independent components of the Riemann tensor in 3d. But how?

3. Jan 27, 2005

hellfire

To get the whole information about the curvature you need the Riemann curvature tensor. I guess it is possible to extract sectional curvatures of 2-planes within 3D or 4D spaces from the Riemann curvature tensor (but I do not know how to do this). May be my first question needs some clarification: I was wondering whether it exists a generalization of the sectional curvature which allows to clasify spacetimes in a similar way than elliptical, hyperbolical, etc. spaces are classified and what the physical meaning of such a quantity is (and, further on, whether there is some relation with the Ricci scalar and its sign).