# Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the restriction of a Killing field $X$. The flow associated with $X$ is a 1-parameter group of isometries, $\phi_t$ say. What can be said about the Gaussian curvature of the surface $\phi_t(\sigma)$, the image of $\sigma$ under this flow? Can it be related to the sectional curvatures of $M$?