Connections between Differential Geometry and Topology

Geodesic flow of the averaged controlled Kepler equation

“…Since the metric is analytic, the cut locus of any point is a finite tree whose extremities are singularities of the conjugate locus [11,15,17]. We now give a complete description of these sets which turn to be completely similar to the cut loci on an oblate ellipsoid of revolution (see §1, Proposition 2).…”

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

“…We study the deformation of the conjugate and cut loci for k ≥ 1 in Figs. [8][9][10][11]. The key point is: when k > 1, θ is not monotonous for all the trajectories.…”

“…Also convexity property of the injectivity domain of the exponential map is related to the continuity property of the Monge transport map T on the surfaces [6]. The structure of the conjugate and cut loci surfaces diffeomorphic to S 2 was investigated in details Bernard Bonnard Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France, e-mail: bernard.bonnard@u-bourgogne.fr Olivier Cots INRIA, 2004 route des lucioles, F-06902 Sophia Antipolis, France, e-mail: olivier.cots@inria.fr Lionel Jassionnesse Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France, e-mail: lionel.jassionnesse@u-bourgogne.fr 1 by Poincaré and Myers [9], [10]. In the analytic case, the cut locus is a finite tree and the extremity of each branch is a cusp point.…”

Deformations of geodesic fields

“…In 1935 Myers [4] showed that on a compact analytic surface, the cut locus can be triangulated as a finite graph. Recently this has been extended to arbitrary dimensions by the work of Buchner, Mather, Hironaka and Kato [2].…”