# Conjugate-cut loci and injectivity domains on two-spheres of revolution

Bernard Bonnard; Jean-Baptiste Caillau; Gabriel Janin

ESAIM: Control, Optimisation and Calculus of Variations (2013)

- Volume: 19, Issue: 2, page 533-554
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBonnard, Bernard, Caillau, Jean-Baptiste, and Janin, Gabriel. "Conjugate-cut loci and injectivity domains on two-spheres of revolution." ESAIM: Control, Optimisation and Calculus of Variations 19.2 (2013): 533-554. <http://eudml.org/doc/272950>.

@article{Bonnard2013,

abstract = {In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.},

author = {Bonnard, Bernard, Caillau, Jean-Baptiste, Janin, Gabriel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {conjugate and cut loci; injectivity domain; optimal control; optimal transport},

language = {eng},

number = {2},

pages = {533-554},

publisher = {EDP-Sciences},

title = {Conjugate-cut loci and injectivity domains on two-spheres of revolution},

url = {http://eudml.org/doc/272950},

volume = {19},

year = {2013},

}

TY - JOUR

AU - Bonnard, Bernard

AU - Caillau, Jean-Baptiste

AU - Janin, Gabriel

TI - Conjugate-cut loci and injectivity domains on two-spheres of revolution

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2013

PB - EDP-Sciences

VL - 19

IS - 2

SP - 533

EP - 554

AB - In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.

LA - eng

KW - conjugate and cut loci; injectivity domain; optimal control; optimal transport

UR - http://eudml.org/doc/272950

ER -

## References

top- [1] A. Agrachev, U. Boscain and M. Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete Contin. Dyn. Syst.20 (2008) 801–822. Zbl1198.49041MR2379474
- [2] M. Berger, Volume et rayon d’injectivité dans les variétés riemanniennes de dimension 3. Osaka J. Math.14 (1977) 191–200. Zbl0353.53028MR467595
- [3] M. Berger, A panoramic view of Riemannian geometry. Springer-Verlag, Berlin (2003). Zbl1038.53002MR2002701
- [4] G. Besson, Géodésiques des surfaces de révolution. Séminaire de Théorie Spectrale et Géométrie S9 (1991) 33–38.
- [5] V.G. Boltyanskii, Sufficient conditions for optimality and the justification of the dynamic programming method. SIAM J. Control4 (1966) 326–361. Zbl0143.32004MR197205
- [6] B. Bonnard and J.-B. Caillau, Metrics with equatorial singularities on the sphere. HAL preprint No.00319299 (2008) 1–30. Zbl1305.53040MR3262637
- [7] B. Bonnard and J.-B. Caillau, Geodesic flow of the averaged controlled Kepler equation. Forum Math.21 (2009) 797–814. Zbl1171.49030MR2560391
- [8] B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098. Zbl1184.53036MR2542715
- [9] B. Bonnard, J.-B. Caillau and L. Rifford, Convexity of injectivity domains on the ellipsoid of revolution: the oblate case, C. R. Acad. Sci. Paris, Sér. I 348 (2010) 1315–1318. Zbl1225.53006MR2745347
- [10] B. Bonnard, J.-B. Caillau and O. Cots, Energy minimization in two-level dissipative quantum control: the integrable case. Proc. of 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden (2010). Discrete Contin. Dyn. Syst. suppl. (2011) 229–239. Zbl1306.81050MR2987400
- [11] B. Bonnard, G. Charlot, R. Ghezzi and G. Janin, The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry. J. Dyn. Control Syst.17 (2011) 141–161. Zbl1209.53014MR2765542
- [12] B. Bonnard, O. Cots and N. Shcherbakova, Energy minimization problem in two-level dissipative quantum systems. J. Math. Sci. 147 (2012). Zbl1286.81104
- [13] J.-B. Caillau, B. Daoud and J. Gergaud, On some Riemannian aspects of two and three-body controlled problems. Recent Advances in Optimization and its Applications in Engineering. Springer (2010) 205–224. Proc. of the 14th Belgium-Franco-German conference on Optimization, Leuven (2009).
- [14] A. Faridi and E. Schucking, Geodesics and deformed spheres. Proc. Amer. Math. Soc.100 (1987) 522–525. Zbl0622.53025MR891157
- [15] A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex. Amer. J. Math.134 (2012) 109–139. Zbl1241.53031MR2876141
- [16] G.-H. Halphen, Traité des fonctions elliptiques et de leurs applications. Première partie, Gauthier-Villars (1886). JFM22.0447.01
- [17] J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids. Manuscripta Math.114 (2004) 247–264. Zbl1076.53042MR2067796
- [18] G. Janin, Contrôle optimal et applications au transfert d’orbite et à la géométrie presque Riemannienne. Ph.D. thesis, Université de Bourgogne (2010).
- [19] D. Lawden, Elliptic functions and applications. Springer-Verlag (1989). Zbl0689.33001MR1007595
- [20] S.B. Myers, Connections between differential geometry and topology I. Simply connected surfaces II. Duke Math. J. 1 (1935) 376–391; 2 (1936) 95–102. Zbl0012.27502MR1545884JFM61.0787.02
- [21] H. Poincaré, Sur les lignes géodésiques des surfaces convexes. Trans. Amer. Math. Soc.6 (1905) 237–274. Zbl36.0669.01MR1500710JFM36.0669.01
- [22] K. Shiohama, T. Shioya and M. Tanaka, The geometry of total curvature on complete open surfaces. Cambridge University Press (2003). Zbl1086.53056MR2028047
- [23] R. Sinclair and M. Tanaka, The cut locus of a two-sphere of revolution and Toponogov’s comparison theorem. Tohoku Math. J.59 (2007) 379–399. Zbl1158.53033MR2365347
- [24] M. Spivak, A comprehensive introduction to differential geometry II. Publish or Perish (1979). Zbl0439.53001
- [25] C. Villani, Optimal transport, Old and new. Springer-Verlag (2009). Zbl1156.53003MR2459454

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.