On the existence of focal points near closed geodes\-ics on surfaces
Let \(S\) be a compact Riemannian surface. The author introduces the notion of focal points near closed geodesics on \(S\) and gives criteria for the (non)existence of such points. Similarly, analogous definitions for the (non)existence of conjugate points near closed geodesics can be made. The criteria are formulated in terms of the curvature along the closed geodesic and the average values of the partial derivatives of the curvature in the direction perpendicular to the geodesic. These criteria lead to a new family of examples of surfaces with no focal points. The geodesic flows for these examples are ergodic, but not Anosov. Moreover, the author proves that if \(S\) has no focal points and an inequality relating the curvature of \(S\) to the curvature of the horocycles holds, then the horocycles (considered as curves in \(S\)) are uniformly \(C^{2+\text{Lipschitz}}\).
scientific article; zbMATH DE number 1960060