How to compute the riemannian curvature of a compact riemannian manifold?

Question

Let $M$ be a compact Riemannian manifold. How would I compute the Riemannian curvature of $M$ using Gaussian curvature (or otherwise using the exponential map and Jacobi fields)?

Answer 1

If you have the exponential map at $p\in M$, this is easy. Just exponentiate the 2-dimensional direction you want to compute the sectional curvature of. This gives you a surface in $M$ passing through $p$. Its Gaussian curvature at $p$ is precisely the sectional curvature of $M$ in this 2-dimensional direction.