Let be a Riemannian manifold. As before, one associates to it the curvature tensor

In the previous post, we saw a quantitative expression of how the curvature is a measure of the deviation from the flatness of . Given , one can try to choose local coordinates around a point which make the metric look like the euclidean metric to order 2 at , i.e. local coordinates such that the coefficients near are given by

However, we saw that the quadratic terms involve precisely the values of the curvature tensor at . Even in the best coordinates, one can’t generally make the coefficients of a metric look euclidean to order 3: the obstruction is precisely the curvature at . Today, I’d like to describe the interpretation of curvature in terms of geodesics. Once again, the material is standard and can be found in introductory textbooks on Riemannian geometry.

**1. Curvature and geodesic deviation**

There’s another way to think of curvature, which also leads to this: curvature measures how nearby geodesics spread. To think about this, suppose we have a one-parameter family of geodesics in , where is the starting point of the variation. One then has a vector field

along the curve , which measures the infinitesimal “spreading” of the one-parameter family . Now, a computation shows that satisfies the equation

in other words that is a **Jacobi field**. Here is covariant differentiation along the curve . (more…)