September  2015, 35(9): 4115-4147. doi: 10.3934/dcds.2015.35.4115

Local properties of almost-Riemannian structures in dimension 3

1. 

CNRS, CMAP École Polytechnique, Team GECO, INRIA Saclay - Île-de-France, Route de Saclay, 91128 Palaiseau, France

2. 

Institut Fourier, UMR 5582, Univ. Grenoble Alpes, 100 rue des maths, BP 74, 38402 St Martin d'Hères, France

3. 

CMAP École Polytechnique, Team GECO, INRIA Saclay - Île-de-France, Route de Saclay, 91128 Palaiseau, France

4. 

Laboratoire des Signaux et Systèmes (L2S, UMR 8506), CNRS - CentraleSupelec - Université Paris-Sud, 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France

Received  July 2014 Revised  November 2014 Published  April 2015

A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\mathcal{Z}$ called the singular set. Under the Hörmander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2.
    In this paper we start the study of the 3D case which appears to be richer with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres.
    We finally give some preliminary results about heat diffusion on such manifolds.
Citation: Ugo Boscain, Gregoire Charlot, Moussa Gaye, Paolo Mason. Local properties of almost-Riemannian structures in dimension 3. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4115-4147. doi: 10.3934/dcds.2015.35.4115
References:
[1]

A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 1-12 (2001), Control theory and its applications (Grado, 1998).

[2]

A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian Geometry (Lecture Notes)http://people.sissa.it/agrachev/agrachev_files/notes.html.

[3]

A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var., 2 (1997), 377-448. doi: 10.1051/cocv:1997114.

[4]

A. A. Agrachev, Exponential mappings for contact sub-Riemannian structures, J. Dynam. Control Systems, 2 (1996), 321-358. doi: 10.1007/BF02269423.

[5]

A. A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 793-807. doi: 10.1016/j.anihpc.2009.11.011.

[6]

A. Agrachev, Some open problems, in Geometric Control Theory and Sub-Riemannian Geometry, vol. 5 of Springer INdAM Series, Springer, (2014), 1-13. doi: 10.1007/978-3-319-02132-4_1.

[7]

A. Agrachev, U. Boscain, J.-P. Gauthier and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal., 256 (2009), 2621-2655. doi: 10.1016/j.jfa.2009.01.006.

[8]

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. doi: 10.3934/dcds.2008.20.801.

[9]

A. Agrachev and J.-P. Gauthier, On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 359-382. doi: 10.1016/S0294-1449(00)00064-0.

[10]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, II. doi: 10.1007/978-3-662-06404-7.

[11]

M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.

[12]

D. Barilari, U. Boscain and R. W. Neel, Small-time heat kernel asymptotics at the sub-Riemannian cut locus, J. Differential Geom., 92 (2012), 373-416.

[13]

A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian geometry, vol. 144 of Progr. Math., Birkhäuser, Basel, (1996), 1-78. doi: 10.1007/978-3-0348-9210-0_1.

[14]

G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4), 21 (1988), 307-331.

[15]

G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale. II, Probab. Theory Related Fields, 90 (1991), 377-402. doi: 10.1007/BF01193751.

[16]

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. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081-1098. doi: 10.1016/j.anihpc.2008.03.010.

[17]

B. Bonnard and J.-B. Caillau, Metrics with equatorial singularities on the sphere, Annali di Matematica Pura ed Applicata, 193 (2014), 1353-1382. doi: 10.1007/s10231-013-0333-y.

[18]

U. Boscain, G. Charlot and R. Ghezzi, Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geom. Appl., 31 (2013), 41-62. doi: 10.1016/j.difgeo.2012.10.001.

[19]

U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Lipschitz classification of almost-Riemannian distances on compact oriented surfaces, J. Geom. Anal., 23 (2013), 438-455. doi: 10.1007/s12220-011-9262-4.

[20]

U. Boscain, T. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990. doi: 10.3934/dcdsb.2005.5.957.

[21]

U. Boscain and G. Charlot, Resonance of minimizers for $n$-level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var., 10 (2004), 593-614 (electronic). doi: 10.1051/cocv:2004022.

[22]

U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin and H.-R. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys., 43 (2002), 2107-2132. doi: 10.1063/1.1465516.

[23]

U. Boscain and C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Inst. Fourier (Grenoble), 63 (2013), 1739-1770. doi: 10.5802/aif.2813.

[24]

U. Boscain and M. Sigalotti, High-order angles in almost-Riemannian geometry, in Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 24. Année 2005-2006, vol. 25 of Sémin. Théor. Spectr. Géom., Univ. Grenoble I, (2008), 41-54.

[25]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.

[26]

Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.

[27]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM Journal on Control and Optimization, 47 (2008), 1078-1095. doi: 10.1137/060663003.

[28]

M. Christ, Analytic hypoellipticity breaks down for weakly pseudoconvex Reinhardt domains, Internat. Math. Res. Notices, (1991), 31-40. doi: 10.1155/S1073792891000053.

[29]

M. Christ, Some nonanalytic-hypoelliptic sums of squares of vector fields, Bull. Amer. Math. Soc. (N.S.), 26 (1992), 137-140. doi: 10.1090/S0273-0979-1992-00258-6.

[30]

B. Franchi and E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino, 105-114 (1984), Conference on linear partial and pseudodifferential operators (Torino, 1982).

[31]

B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139 (1977), 95-153. doi: 10.1007/BF02392235.

[32]

V. V. Grušin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.

[33]

M. W. Hirsch, Differential Topology, vol. 33 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994, Corrected reprint of the 1976 original.

[34]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[35]

F. Jean, Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems, 7 (2001), 473-500. doi: 10.1023/A:1013154500463.

[36]

R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, Probab. Theory Related Fields, 74 (1987), 289-294. doi: 10.1007/BF00569994.

[37]

R. Léandre, Minoration en temps petit de la densité d'une diffusion dégénérée, J. Funct. Anal., 74 (1987), 399-414. doi: 10.1016/0022-1236(87)90031-0.

[38]

R. Monti, Regularity results for sub-Riemannian geodesics, Calc. Var. Partial Differential Equations, 49 (2014), 549-582. doi: 10.1007/s00526-012-0592-2.

[39]

L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 4th edition, "Nauka'', Moscow, 1983.

[40]

M. Reed and B. Simon, Methods of Modern Mathematical Physics: Fourier Analysis, Self-adjointness, vol. 2, Academic Press, 1975.

[41]

R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221-263, \urlprefixhttp://projecteuclid.org/getRecord?id=euclid.jdg/1214440436.

[42]

F. Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline\partial$-Neumann problem, Comm. Partial Differential Equations, 3 (1978), 475-642. doi: 10.1080/03605307808820074.

[43]

M. Vendittelli, G. Oriolo, F. Jean and J.-P. Laumond, Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, IEEE Trans. Automat. Control, 49 (2004), 261-266. doi: 10.1109/TAC.2003.822872.

show all references

References:
[1]

A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 1-12 (2001), Control theory and its applications (Grado, 1998).

[2]

A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian Geometry (Lecture Notes)http://people.sissa.it/agrachev/agrachev_files/notes.html.

[3]

A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, Sub-Riemannian sphere in Martinet flat case, ESAIM Control Optim. Calc. Var., 2 (1997), 377-448. doi: 10.1051/cocv:1997114.

[4]

A. A. Agrachev, Exponential mappings for contact sub-Riemannian structures, J. Dynam. Control Systems, 2 (1996), 321-358. doi: 10.1007/BF02269423.

[5]

A. A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 793-807. doi: 10.1016/j.anihpc.2009.11.011.

[6]

A. Agrachev, Some open problems, in Geometric Control Theory and Sub-Riemannian Geometry, vol. 5 of Springer INdAM Series, Springer, (2014), 1-13. doi: 10.1007/978-3-319-02132-4_1.

[7]

A. Agrachev, U. Boscain, J.-P. Gauthier and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal., 256 (2009), 2621-2655. doi: 10.1016/j.jfa.2009.01.006.

[8]

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. doi: 10.3934/dcds.2008.20.801.

[9]

A. Agrachev and J.-P. Gauthier, On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 359-382. doi: 10.1016/S0294-1449(00)00064-0.

[10]

A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, vol. 87 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, II. doi: 10.1007/978-3-662-06404-7.

[11]

M. S. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45-87.

[12]

D. Barilari, U. Boscain and R. W. Neel, Small-time heat kernel asymptotics at the sub-Riemannian cut locus, J. Differential Geom., 92 (2012), 373-416.

[13]

A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian geometry, vol. 144 of Progr. Math., Birkhäuser, Basel, (1996), 1-78. doi: 10.1007/978-3-0348-9210-0_1.

[14]

G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. École Norm. Sup. (4), 21 (1988), 307-331.

[15]

G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale. II, Probab. Theory Related Fields, 90 (1991), 377-402. doi: 10.1007/BF01193751.

[16]

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. H. Poincaré Anal. Non Linéaire, 26 (2009), 1081-1098. doi: 10.1016/j.anihpc.2008.03.010.

[17]

B. Bonnard and J.-B. Caillau, Metrics with equatorial singularities on the sphere, Annali di Matematica Pura ed Applicata, 193 (2014), 1353-1382. doi: 10.1007/s10231-013-0333-y.

[18]

U. Boscain, G. Charlot and R. Ghezzi, Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geom. Appl., 31 (2013), 41-62. doi: 10.1016/j.difgeo.2012.10.001.

[19]

U. Boscain, G. Charlot, R. Ghezzi and M. Sigalotti, Lipschitz classification of almost-Riemannian distances on compact oriented surfaces, J. Geom. Anal., 23 (2013), 438-455. doi: 10.1007/s12220-011-9262-4.

[20]

U. Boscain, T. Chambrion and G. Charlot, Nonisotropic 3-level quantum systems: Complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990. doi: 10.3934/dcdsb.2005.5.957.

[21]

U. Boscain and G. Charlot, Resonance of minimizers for $n$-level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var., 10 (2004), 593-614 (electronic). doi: 10.1051/cocv:2004022.

[22]

U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin and H.-R. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys., 43 (2002), 2107-2132. doi: 10.1063/1.1465516.

[23]

U. Boscain and C. Laurent, The Laplace-Beltrami operator in almost-Riemannian geometry, Ann. Inst. Fourier (Grenoble), 63 (2013), 1739-1770. doi: 10.5802/aif.2813.

[24]

U. Boscain and M. Sigalotti, High-order angles in almost-Riemannian geometry, in Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 24. Année 2005-2006, vol. 25 of Sémin. Théor. Spectr. Géom., Univ. Grenoble I, (2008), 41-54.

[25]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.

[26]

Y. Chitour, F. Jean and E. Trélat, Genericity results for singular curves, J. Differential Geom., 73 (2006), 45-73.

[27]

Y. Chitour, F. Jean and E. Trélat, Singular trajectories of control-affine systems, SIAM Journal on Control and Optimization, 47 (2008), 1078-1095. doi: 10.1137/060663003.

[28]

M. Christ, Analytic hypoellipticity breaks down for weakly pseudoconvex Reinhardt domains, Internat. Math. Res. Notices, (1991), 31-40. doi: 10.1155/S1073792891000053.

[29]

M. Christ, Some nonanalytic-hypoelliptic sums of squares of vector fields, Bull. Amer. Math. Soc. (N.S.), 26 (1992), 137-140. doi: 10.1090/S0273-0979-1992-00258-6.

[30]

B. Franchi and E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino, 105-114 (1984), Conference on linear partial and pseudodifferential operators (Torino, 1982).

[31]

B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents, Acta Math., 139 (1977), 95-153. doi: 10.1007/BF02392235.

[32]

V. V. Grušin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.

[33]

M. W. Hirsch, Differential Topology, vol. 33 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1994, Corrected reprint of the 1976 original.

[34]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[35]

F. Jean, Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems, 7 (2001), 473-500. doi: 10.1023/A:1013154500463.

[36]

R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, Probab. Theory Related Fields, 74 (1987), 289-294. doi: 10.1007/BF00569994.

[37]

R. Léandre, Minoration en temps petit de la densité d'une diffusion dégénérée, J. Funct. Anal., 74 (1987), 399-414. doi: 10.1016/0022-1236(87)90031-0.

[38]

R. Monti, Regularity results for sub-Riemannian geodesics, Calc. Var. Partial Differential Equations, 49 (2014), 549-582. doi: 10.1007/s00526-012-0592-2.

[39]

L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 4th edition, "Nauka'', Moscow, 1983.

[40]

M. Reed and B. Simon, Methods of Modern Mathematical Physics: Fourier Analysis, Self-adjointness, vol. 2, Academic Press, 1975.

[41]

R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221-263, \urlprefixhttp://projecteuclid.org/getRecord?id=euclid.jdg/1214440436.

[42]

F. Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the $\overline\partial$-Neumann problem, Comm. Partial Differential Equations, 3 (1978), 475-642. doi: 10.1080/03605307808820074.

[43]

M. Vendittelli, G. Oriolo, F. Jean and J.-P. Laumond, Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, IEEE Trans. Automat. Control, 49 (2004), 261-266. doi: 10.1109/TAC.2003.822872.

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