Focal Points and the Decrease of Curvature: a Surprising Example

If γ(t) is a unit speed geodesic on a manifold M and J(t) is a non-trivial perpendicular Jacobi field along γ such that J(t0) = 0 and (||J ||2)′(t1) = 0 for some t0 < t1, then the geodesic that is tangent to J(t1) at γ(t1) is said to have a focal point at γ(t0). If this does not happen for any times t0 < t1 and any non-trivial perpendicular Jacobi field J , then we say that there are no focal points along γ. Moreover, if there are no focal points along any geodesic in M , then we say M has no focal points. Although manifolds with no focal points can have sectional curvatures of both signs [4], they have many of the properties of manifolds of nonpositive curvature that are of interest in the study of geodesic flows. (See, for example, [2], [5], and [6].) In the case when M is a surface S with Gaussian curvature K, the existence of a focal point along γ is equivalent to the following: There is a solution u to the scalar Riccati equation u(t) + u′(t) + K(γ(t)) = 0 defined on (t0, t1] such that limt→t+0 u(t) = ∞ and u(t1) = 0. The following comparison lemma (see [1]) is often used in estimating solutions to the Riccati equation.