Ricci Curvature - Direct Geometric Meaning

Direct Geometric Meaning

Near any point p in a Riemannian manifold (M,g), one can define preferred local coordinates, called geodesic normal coordinates. These are adapted to the metric so that geodesics through p corresponds to straight lines through the origin, in such a manner that the geodesic distance from p corresponds to the Euclidean distance from the origin. In these coordinates, the metric tensor is well-approximated by the Euclidean metric, in the precise sense that

In fact, by taking the Taylor expansion of the metric applied to a Jacobi field along a radial geodesic in the normal coordinate system, one has

In these coordinates, the metric volume element then has the following expansion at p:

which follows by expanding the square root of the determinant of the metric.

Thus, if the Ricci curvature Ric(ξ,ξ) is positive in the direction of a vector ξ, the conical region in M swept out by a tightly focused family of short geodesic segments emanating from p with initial velocity inside a small cone around ξ will have smaller volume than the corresponding conical region in Euclidean space, just as the surface of a small spherical wedge has lesser area than a corresponding circular sector. Similarly, if the Ricci curvature is negative in the direction of a given vector ξ, such a conical region in the manifold will instead have larger volume than it would in Euclidean space.

The Ricci curvature is essentially an average of curvatures in the planes including ξ. Thus if a cone emitted with an initially circular (or spherical) cross-section becomes distorted into an ellipse (ellipsoid), it is possible for the volume distortion to vanish if the distortions along the principal axes counteract one another. The Ricci curvature would then vanish along ξ. In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of world-lines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.