The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry

By Bernard Bonnard, Gregoire Charlot, Roberta Ghezzi, G. Janin. 2010

48 citations recorded by the source corpus.

We study the tangential case in two-dimensional almost-Riemannian geometry and analyze the connection with the Martinet case in sub-Riemannian geometry. We calculate estimates of the exponential map which allow us to describe the conjugate locus and the cut locus at a tangency point. We prove that this tangency point generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ec9034b9f259ebef194f9740db34b7e9a9aac72f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Gregoire Charlot
2. Roberta Ghezzi
3. Ugo V. Boscain
4. Mario Sigalotti
5. Davide Barilari

In Astronomy and Astrophysics

1. Gregoire Charlot
2. Roberta Ghezzi
3. Ugo V. Boscain
4. Mario Sigalotti
5. Davide Barilari

In Geometry and Topology

1. Gregoire Charlot
2. Roberta Ghezzi
3. Ugo V. Boscain
4. Mario Sigalotti
5. Davide Barilari

Citations in the corpus

• Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces (2010, background): “The structure of the cut and conjugate loci is described in [7] under generic assumptions.”
• Normal forms and invariants for 2-dimensional almost-Riemannian structures (2010, background): “Thus py0 ≡ −(1)2 and one can prove (see [3, 13]) that x0(s) = − √ 2 √ γ cn (K +√γs), y0(s) = − 2 3 √ γ ( √ γs+ 2sn (K +√γs)cn (K +√γs)dn (K +√γs)), where K is the complete elliptic integral of the first kind of modulus 1 √ 2 , and cn , sn and dn denote the classical Jacobi functions of modulus 1 √ 2”