,
, ))), that is, 5 i,j=1 v i v j ? i ? j f
, = g = eta g1
, = d2Exp = Simplify [( d2f /. { g1-> Function
, 212 6.2 Basic definitions and statement of the main result, p.214
, 216 6.3.1 Every line field can be realized as a proto-line-field, p.219
,
, 233 6.6.1 What changes if we change the metric g, p.235
, P1 ?? ? (??, ?), F (??) = ?F (?) and G(??) = ?G(?)
, P2 ?? ? [??, 0), F (? + ?) ? F (?) = ? and G(? + ?) ? G(?) = ?
, Let ? > 0 and ? ? [??, ?) be as in Proposition 6.4.6. First consider the case where ? ? 1. The derivative (2F ? G) is then always positive, except possibly at two points in the case ? = 1. Hence 2F ? G is a bijection between [??, ?) and its image. We claim that 2F ? G (mod 2?) is a bijection between [??, ?) and [0, 2?)
, 2F ? G) (?) = 0 and 2F (?) ? G(?) = ? (mod 2?). The case ? ? (0, 1) requires a further study of 2F ? G. From Proposition 6.4.6, we know that (2F ? G) (?) has the same sign as cos(2? + ?) + ?. Let ? 0 ? [0, ?) be such that cos(2? 0 + ?) + ? = 0 and ?2 sin(2? 0 + ?) > 0, and let ? 1 ? [? 0 , ? 0 + ?) be such that cos(2? 1 + ?) + ? = 0 and ?2 sin(2? 1 + ?) < 0. Then cos(2? + ?) + ? > 0 on (? 0 , ? 1 ) and cos(2? + ?) + ? < 0 on (? 1 , ? + ? 0 ). Since (2F ? G) is ?-periodic, the case ? = 1 the set E X is made of a single line through the origin, corresponding to the two values of ? for which
, which correspond to an exceptional set E X made of two lines. Proof of Theorem 6.4.3 when X has a singularity of index ?1. In this case we have that G < 0 and F > 0 on [??, ?). So 2F ?G, as a function from [??, ?) to R is increasing and has total variation 6?. Hence there exists y ? (??, 0) such that (2F ?G)?(2F ?G)(??) is a bijection from (??, y) onto (0, 2?); it exists x ? (y, ?) such that (2F ?G)?(2F ?G)(??) is a bijection from (y, x) onto (2?, 4?); and again (2F ? G) ? (2F ? G)(??) is a bijection from (x, ?) onto (4?, 6?). Hence we have found that there are three solutions to (6.1). Since (2F ? G)(? + ?) = (2F ? G)(?) + ? (mod 2?), 2F ? G)(x), ?, (2F ? G)(?? + x)
, Linearization, blow-up and proof of Theorem 6, vol.2
, The goal of this section is to prove the topological equivalence of a hyper-hyperbolic proto-line-field at a hyperbolic singularity and its linearization. As a direct consequence
, Introduction to Riemannian and Sub-Riemannian geometry, 2016.
, Homotopically invisible singular curves, Calc. Var. Partial Differential Equations, vol.56, issue.4, 2017.
DOI : 10.1007/s00526-017-1203-z
URL : http://arxiv.org/pdf/1603.08937
, Sub-Riemannian curvature in contact geometry, Journal of Geometric Analysis, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01160901
, On sub-Riemannian caustics and wave fronts for contact distributions in the three-space, J. Dynam. Control Systems, vol.6, issue.3, pp.365-395, 2000.
, Exponential representation of flows and a chronological enumeration, Mat. Sb, vol.639, issue.149, p.103, 1978.
, Sub-Riemannian metrics and isoperimetric problems in the contact case, Journal of Mathematical Sciences, vol.103, issue.6, pp.639-663, 2001.
, The classification of critical points, caustics and wave fronts, Translated from the Russian by Ian Porteous and Mark Reynolds, Monographs in Mathematics, vol.I, 1985.
, Exponential mappings for contact sub-Riemannian structures, J. Dynam. Control Systems, vol.2, issue.3, pp.321-358, 1996.
, Catastrophe theory, 1992.
, Filtrations of a Lie algebra of vector fields and the nilpotent approximation of controllable systems, Dokl. Akad. Nauk SSSR, vol.295, issue.4, p.102, 1987.
, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.13, issue.6, pp.635-690, 1996.
DOI : 10.1016/s0294-1449(16)30118-4
URL : https://doi.org/10.1016/s0294-1449(16)30118-4
, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, vol.87, 2004.
, Trace heat kernel asymptotics in 3d contact sub-riemannian geometry, Journal of Mathematical Sciences, vol.195, issue.3, p.139, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00672262
, On the heat diffusion for generic riemannian and sub-riemannian structures, International Mathematics Research Notices, vol.2017, pp.4639-4672, 2016.
URL : https://hal.archives-ouvertes.fr/hal-00879444
, Small-time heat kernel asymptotics at the sub-riemannian cut locus, J. Differential Geom, vol.92, issue.3, p.138
URL : https://hal.archives-ouvertes.fr/hal-00687651
, Lecture notes from the IHP Trimester held at the Institut Henri Poincaré, Paris and from the CIRM Summer School "Sub-Riemannian Manifolds: From Geodesics to Hypoelliptic Diffusion, Geometry, analysis and dynamics on sub-Riemannian manifolds, vol.1, 2016.
, Hypoelliptic diffusion and human vision: a semidiscrete new twist, SIAM J. Imaging Sci, vol.7, issue.2, p.212, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01097156
, Lipschitz classification of almost-Riemannian distances on compact oriented surfaces, J. Geom. Anal, vol.23, issue.1, p.99, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00464414
, Anthropomorphic image reconstruction via hypoelliptic diffusion, SIAM J. Control Optim, vol.50, issue.3, p.212, 2012.
DOI : 10.1137/11082405x
URL : http://arxiv.org/pdf/1006.3735
, The tangent space in sub-Riemannian geometry, SubRiemannian geometry, vol.144, pp.1-78, 1996.
, Caustique mystique. Séminaire Bourbaki, vol.27, pp.19-56, 19841985.
, On binary differential equations and umbilics, Proc. Roy. Soc. Edinburgh Sect. A, vol.111, issue.1-2, p.212, 1989.
DOI : 10.1017/s0308210500025087
, Umbilic points on gaussian random surfaces, Journal of Physics A: Mathematical and General, vol.10, issue.11, p.1809, 1977.
DOI : 10.1088/0305-4470/10/11/009
, Rigidity of integral curves of rank 2 distributions, Invent. Math, vol.114, issue.2, pp.435-461, 1993.
, Stratified Lie groups and potential theory for their sub-Laplacians, vol.125, 2007.
, On sub-Riemannian and Riemannian structures on the heisenberg groups, Journal of Dynamical and Control Systems, vol.22, issue.3, p.138, 2016.
, Intrinsic random walks and sublaplacians in sub-Riemannian geometry, Advances in Mathematics, vol.314, pp.124-184, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01122735
, Graded approximations and controllability along a trajectory, SIAM J. Control Optim, vol.28, issue.4, pp.903-924, 1990.
, On binary differential equations, Nonlinearity, vol.8, issue.2, p.212, 1995.
, Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex, Journal of neuroscience, vol.17, issue.6, pp.2112-2127, 1997.
, On differential equations and umbilici. The London, Edinburgh, and Dublin Philosophical Magazine, Journal of Science, vol.26, issue.176, pp.373-379, 1863.
DOI : 10.1017/cbo9780511703713.032
, Liquid Crystals, 1992.
, Quasi-contact S-R metrics: normal form in R 2n , wave front and caustic in R 4, Acta Appl. Math, vol.74, issue.3, p.136, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00383021
, A cortical based model of perceptual completion in the roto-translation space, J. Math. Imaging Vision, vol.24, issue.3, p.41, 2006.
, Note VII, Sur la forme des lignes de courbure dans le voisinage d'un ombilic, Leçons sur la théorie générale des surfaces. Gauthier-Villars, 1896
, Qualitative theory of planar differential systems, vol.92, 2006.
, Orientational order of motile defects in active nematics, Nat Mater, vol.14, issue.11, p.212, 2015.
, Regions where the exponential map at regular points of sub-Riemannian manifolds is a local diffeomorphism, J. Dyn. Control Syst, vol.15, issue.1, pp.133-156, 2009.
, Small sub-Riemannian balls on R 3, J. Dynam. Control Systems, vol.2, issue.3, pp.359-421, 1996.
, Rectifiability and perimeter in the Heisenberg group, Math. Ann, vol.321, issue.3, pp.479-531, 2001.
, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta mathematica, vol.139, issue.1, pp.95-153, 1977.
DOI : 10.1007/bf02392235
URL : https://doi.org/10.1007/bf02392235
, A note on carnot geodesics in nilpotent lie groups, Journal of Dynamical and Control Systems, vol.1, issue.4, pp.535-549, 1995.
, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), vol.83, issue.125, p.99, 1970.
, Differential topology, vol.33, p.219, 2012.
, The Total Curvature (Curvatura Integra) of a Closed Surface with Riemannian Metric and Poincaré's Theorem on the Singularities of Fields of Line Elements, Differential geometry in the large: seminar lectures New, vol.3, 1956.
, , vol.88, p.216, 2003.
, The sub-Riemannian geometry of three-manifolds
, Thesis (Ph.D.)-Duke University, vol.67, 1995.
, Receptive fields, binocular interaction and functional architecture in the cat's visual cortex, The Journal of physiology, vol.160, issue.1, pp.106-154, 1962.
, Differential geometry from a singularity theory viewpoint, 2016.
, De la ligne géodésique sur un ellipsoïde et des différents usages d'une transformation analytique remarquable, Journal de mathématiques pures et appliquées, vol.6, p.219, 1841.
, Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems, vol.7, issue.4, pp.473-500, 2001.
URL : https://hal.archives-ouvertes.fr/hal-01010757
, Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, SpringerBriefs in Mathematics, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01137580
, Pliability, or the Whitney extension theorem for curves in Carnot groups, Analysis & PDE, vol.37, issue.3, pp.1637-1661, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01285215
, Analyzing oriented patterns. Computer vision, graphics, and image processing, vol.37, pp.362-385, 1987.
DOI : 10.1016/s0734-189x(86)80033-0
, Lusin approximation for horizontal curves in step 2 Carnot groups, Calc. Var. Partial Differential Equations, vol.55, issue.5, 2016.
, How many geodesics join two points on a contact sub-riemannian manifold, Journal of Symplectic Geometry, vol.15, issue.1, pp.247-305, 2017.
DOI : 10.4310/jsg.2017.v15.n1.a7
URL : https://hal.archives-ouvertes.fr/hal-01096718
, Sur les lignes de courbure de l'ellipsoïde, Journal de l'École Polytechnique, IIème cahier, cours de Floréal, vol.3, p.219, 1796.
, A tour of subriemannian geometries, their geodesics and applications. Number 91, 2006.
DOI : 10.1090/surv/091
, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math, vol.129, issue.2, pp.1-60, 1989.
DOI : 10.2307/1971484
, The topology of ridge systems, Annals of human genetics, vol.42, issue.4, pp.435-444, 1979.
, Neurogéométrie de la vision: modèles mathématiques et physiques des architectures fonctionnelles. Editions Ecole Polytechnique, vol.212, 2008.
, Geometric Differentiation, vol.89, 2001.
, The physics of liquid crystals, Number, vol.83, 1995.
, A post-treatment of the homogenization method for shape optimization, SIAM Journal on Control and Optimization, vol.47, issue.3, pp.1380-1398, 2008.
, The multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations, Sovrem. Mat. Fundam. Napravl, vol.19, p.212, 2006.
, Sub-Riemannian geometry and optimal transport, Briefs in Mathematics, 2014.
DOI : 10.1007/978-3-319-04804-8
URL : https://hal.archives-ouvertes.fr/hal-01131787
, Some topics of geometric measure theory in carnot groups, Geometry, analysis and dynamics on sub-Riemannian manifolds, vol.1, p.96, 2016.
, Structurally stable configurations of lines of principal curvature, Asterisque, pp.195-215, 1982.
, Lines of curvature on surfaces, historical comments and recent developments, The São Paulo Journal of Mathematical Sciences, vol.2, issue.1, pp.99-143, 2008.
, Lusin approximation and horizontal curves in Carnot groups, Rev. Mat. Iberoam, vol.32, issue.4, pp.1423-1444, 2016.
DOI : 10.4171/rmi/924
URL : http://arxiv.org/pdf/1412.2531
, Some properties of vector field systems that are not altered by small perturbations, J. Differential Equations, vol.20, issue.2, p.126, 1976.
, Stabilité structurelle et morphogénèse. 1972. Essai d'une théorie générale des modèles, Mathematical Physics Monograph Series
, Some properties of the value function and its level sets for affine control systems with quadratic cost, J. Dynam. Control Systems, vol.6, issue.4, pp.511-541, 2000.
, Contrôle optimal. Mathématiques Concrètes
, Théorie & applications, vol.122, 2005.
, Differentiability of curves in the category of Carnot manifolds, Doklady Mathematics, vol.74, issue.2, pp.686-691, 2006.
, Geometry of Carnot-Carathéodory spaces and differentiability of mappings, The interaction of analysis and geometry, vol.424, pp.247-301, 2007.
, Whitney-type theorems on the extension of functions on Carnot groups, Sibirsk. Mat. Zh, vol.47, issue.4, pp.731-752, 2006.
, Estimating ridge topologies with high curvature for fingerprint authentication systems, IEEE International Conference on Communications, p.213, 2007.
DOI : 10.1109/icc.2007.200
URL : http://researchbank.rmit.edu.au/view/rmit:1573/n2006006559.pdf
, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc, vol.36, issue.1, pp.63-89, 1934.
, A fingerprint orientation model based on 2d fourier expansion (fomfe) and its application to singular-point detection and fingerprint indexing. Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol.29, issue.4, p.212, 2007.
DOI : 10.1109/tpami.2007.1003
, The Whitney extension theorem for C 1 , horizontal curves in the Heisenberg group, The Journal of Geometric Analysis, vol.28, issue.1, pp.61-83, 2018.