October  2008, 20(4): 801-822. doi: 10.3934/dcds.2008.20.801

A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds


SISSA, via Beirut 2-4, 34014 Trieste, Italy


SISSA, via Beirut 2-4 34014 Trieste


Institut de Mathématiques Élie Cartan de Nancy, UMR 7502 INRIA/Universités de Nancy/CNRS, POB 239, 54506 Vandoeuvre-lès-Nancy, France

Received  December 2006 Revised  August 2007 Published  January 2008

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of \ar s, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular those which concern the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.
Citation: Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801

Eric L. Grinberg, Haizhong Li. The Gauss-Bonnet-Grotemeyer Theorem in space forms. Inverse Problems and Imaging, 2010, 4 (4) : 655-664. doi: 10.3934/ipi.2010.4.655


Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155


Mrinal Kanti Roychowdhury. Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4555-4570. doi: 10.3934/dcds.2018199


M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223


Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Optimal control for an epidemic in populations of varying size. Conference Publications, 2015, 2015 (special) : 549-561. doi: 10.3934/proc.2015.0549


Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2469-2482. doi: 10.3934/jimo.2021076


Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007


Miguel-C. Muñoz-Lecanda. On some aspects of the geometry of non integrable distributions and applications. Journal of Geometric Mechanics, 2018, 10 (4) : 445-465. doi: 10.3934/jgm.2018017


Bing Sun. Optimal control of transverse vibration of a moving string with time-varying lengths. Mathematical Control and Related Fields, 2022, 12 (3) : 733-746. doi: 10.3934/mcrf.2021042


Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014


Minoru Murai, Waichiro Matsumoto, Shoji Yotsutani. Representation formula for the plane closed elastic curves. Conference Publications, 2013, 2013 (special) : 565-585. doi: 10.3934/proc.2013.2013.565


Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung. Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems and Imaging, 2020, 14 (2) : 233-265. doi: 10.3934/ipi.2020011


Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial and Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415


Mohammad Hadi Noori Skandari, Marzieh Habibli, Alireza Nazemi. A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems. Mathematical Control and Related Fields, 2020, 10 (1) : 171-187. doi: 10.3934/mcrf.2019035


Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20.


Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823


Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517


Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007


Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098


Rashad M. Asharabi, Jürgen Prestin. Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4143-4158. doi: 10.3934/cpaa.2020185

2021 Impact Factor: 1.588


  • PDF downloads (100)
  • HTML views (0)
  • Cited by (39)

Other articles
by authors

[Back to Top]