Book Review
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4149884
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Book Information:
Authors:
Andrei Agrachev,
Davide Barilari and
Ugo Boscain
Title:
A comprehensive introduction to sub-Riemannian geometry. From the Hamiltonian viewpoint
Additional book information:
Cambridge Studies in Advanced Mathematics, Vol. 181,
Cambridge University Press,
2020,
xviii+745 pp.,
ISBN 978-1-108-47635-5
H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 379672, DOI 10.1112/plms/s3-25.4.603
A. Bellaïche, The tangent space in sub-Riemannian geometry, J. Math. Sci. (New York) 83 (1997), no. 4, 461–476. Dynamical systems, 3. MR 1442527, DOI 10.1007/BF02589761
A. Belotto da Silva, A. Figalli, P. A., and L. L Rifford, Strong sard conjecture and regularity of singular minimizing geodesics for analytic sub-riemannian structures in dimension 3, arXiv:1810.03347 (2018).
V. G. Boltjanskiĭ, The maximum principle in the theory of optimal processes, Dokl. Akad. Nauk SSSR 119 (1958), 1070–1073 (Russian). MR 0120108
R. W. Brockett, Nonlinear control theory and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1357–1368. MR 804784
Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank $2$ distributions, Invent. Math. 114 (1993), no. 2, 435–461. MR 1240644, DOI 10.1007/BF01232676
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007. MR 2312336, DOI 10.1007/978-3-7643-8133-2
Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105 (German). MR 1880, DOI 10.1007/BF01450011
G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, J. Math. Imaging Vision 24 (2006), no. 3, 307–326. MR 2235475, DOI 10.1007/s10851-005-3630-2
Yves Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A805–A808 (French). MR 313968
Yves Colin de Verdière, Luc Hillairet, and Emmanuel Trélat, Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case, Duke Math. J. 167 (2018), no. 1, 109–174. MR 3743700, DOI 10.1215/00127094-2017-0037
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic geodesics, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 205–209. MR 0423438
G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376. MR 315267, DOI 10.1090/S0002-9904-1973-13171-4
G. B. Folland, Applications of analysis on nilpotent groups to partial differential equations, Bull. Amer. Math. Soc. 83 (1977), no. 5, 912–930. MR 457928, DOI 10.1090/S0002-9904-1977-14326-7
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
A. S. ‘Groupprops’, Formula for dimension of graded component of free lie algebra, https://groupprops.subwiki.org/wiki/Formula_for_dimension_of_graded_component_of_free_Lie_algebra, 2020 (accessed April 20, 2020).
Martin C. Gutzwiller, The quantization of a classically ergodic system, Classical quantum models and arithmetic problems, Lecture Notes in Pure and Appl. Math., vol. 92, Dekker, New York, 1984, pp. 287–351. MR 756248
Eero Hakavuori and Enrico Le Donne, Non-minimality of corners in subriemannian geometry, Invent. Math. 206 (2016), no. 3, 693–704. MR 3573971, DOI 10.1007/s00222-016-0661-9
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
Walker Keener Hughen, The sub-Riemannian geometry of three-manifolds, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–Duke University. MR 2692648
Enrico Le Donne, A metric characterization of Carnot groups, Proc. Amer. Math. Soc. 143 (2015), no. 2, 845–849. MR 3283670, DOI 10.1090/S0002-9939-2014-12244-1
Wensheng Liu and Héctor J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564, x+104. MR 1303093, DOI 10.1090/memo/0564
G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal. 5 (1995), no. 2, 402–433. MR 1334873, DOI 10.1007/BF01895673
John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35–45. MR 806700
R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128 (1990), no. 3, 565–592. MR 1045885
Richard Montgomery, Gauge theory of the falling cat, Dynamics and control of mechanical systems (Waterloo, ON, 1992) Fields Inst. Commun., vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 193–218. MR 1232916
Richard Montgomery, Abnormal minimizers, SIAM J. Control Optim. 32 (1994), no. 6, 1605–1620. MR 1297101, DOI 10.1137/S0363012993244945
Richard Montgomery, Hearing the zero locus of a magnetic field, Comm. Math. Phys. 168 (1995), no. 3, 651–675. MR 1328258
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
E. Nelson, Tensor Analysis, Annals of Mathematics Studies, No. 78, 1967, Princeton University Press, Princeton, N.J.
Pierre Pansu, Une inégalité isopérimétrique sur le groupe de Heisenberg, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 127–130 (French, with English summary). MR 676380
Pierre Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 415–445 (French, with English summary). MR 741395, DOI 10.1017/S0143385700002054
Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
P. Rashevskii, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski ped. inst. Libknexta, 2 (1938), 83–94.
A. I. Šnirel′man, The asymptotic multiplicity of the spectrum of the Laplace operator, Uspehi Mat. Nauk 30 (1975), no. 4 (184), 265–266 (Russian). MR 0413209
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
A. M. Vershik and V. Ya. Gershkovich, The geometry of the nonholonomic sphere for three-dimensional Lie group, Global analysis—studies and applications, III, Lecture Notes in Math., vol. 1334, Springer, Berlin, 1988, pp. 309–331. MR 964707, DOI 10.1007/BFb0080435
Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
L. C. Young, Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. Foreword by Wendell H. Fleming. MR 0259704
References
- H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 379672, DOI 10.1112/plms/s3-25.4.603
- A. Bellaïche, The tangent space in sub-Riemannian geometry, J. Math. Sci. (New York) 83 (1997), no. 4, 461–476. Dynamical systems, 3. MR 1442527, DOI 10.1007/BF02589761
- A. Belotto da Silva, A. Figalli, P. A., and L. L Rifford, Strong sard conjecture and regularity of singular minimizing geodesics for analytic sub-riemannian structures in dimension 3, arXiv:1810.03347 (2018).
- V. G. Boltjanskiĭ, The maximum principle in the theory of optimal processes, Dokl. Akad. Nauk SSSR 119 (1958), 1070–1073 (Russian). MR 0120108
- R. W. Brockett, Nonlinear control theory and differential geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 1357–1368. MR 804784
- Robert L. Bryant and Lucas Hsu, Rigidity of integral curves of rank $2$ distributions, Invent. Math. 114 (1993), no. 2, 435–461. MR 1240644, DOI 10.1007/BF01232676
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- Luca Capogna, Donatella Danielli, Scott D. Pauls, and Jeremy T. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, vol. 259, Birkhäuser Verlag, Basel, 2007. MR 2312336
- Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105 (German). MR 1880, DOI 10.1007/BF01450011
- G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, J. Math. Imaging Vision 24 (2006), no. 3, 307–326. MR 2235475, DOI 10.1007/s10851-005-3630-2
- Yves Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A805–A808 (French). MR 313968
- Yves Colin de Verdière, Luc Hillairet, and Emmanuel Trélat, Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case, Duke Math. J. 167 (2018), no. 1, 109–174. MR 3743700, DOI 10.1215/00127094-2017-0037
- J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic geodesics, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R. I., 1975, pp. 205–209. MR 0423438
- G. B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc. 79 (1973), 373–376. MR 315267, DOI 10.1090/S0002-9904-1973-13171-4
- G. B. Folland, Applications of analysis on nilpotent groups to partial differential equations, Bull. Amer. Math. Soc. 83 (1977), no. 5, 912–930. MR 457928, DOI 10.1090/S0002-9904-1977-14326-7
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
- A. S. ‘Groupprops’, Formula for dimension of graded component of free lie algebra, https://groupprops.subwiki.org/wiki/Formula_for_dimension_of_graded_component_of_free_Lie_algebra, 2020 (accessed April 20, 2020).
- Martin C. Gutzwiller, The quantization of a classically ergodic system, Classical quantum models and arithmetic problems, Lecture Notes in Pure and Appl. Math., vol. 92, Dekker, New York, 1984, pp. 287–351. MR 756248
- Eero Hakavuori and Enrico Le Donne, Non-minimality of corners in subriemannian geometry, Invent. Math. 206 (2016), no. 3, 693–704. MR 3573971, DOI 10.1007/s00222-016-0661-9
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
- Walker Keener Hughen, The sub-Riemannian geometry of three-manifolds, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–Duke University. MR 2692648
- Enrico Le Donne, A metric characterization of Carnot groups, Proc. Amer. Math. Soc. 143 (2015), no. 2, 845–849. MR 3283670, DOI 10.1090/S0002-9939-2014-12244-1
- Wensheng Liu and Héctor J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564, x+104. MR 1303093, DOI 10.1090/memo/0564
- G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal. 5 (1995), no. 2, 402–433. MR 1334873, DOI 10.1007/BF01895673
- John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35–45. MR 806700
- R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128 (1990), no. 3, 565–592. MR 1045885
- Richard Montgomery, Gauge theory of the falling cat, Dynamics and control of mechanical systems (Waterloo, ON, 1992) Fields Inst. Commun., vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 193–218. MR 1232916
- Richard Montgomery, Abnormal minimizers, SIAM J. Control Optim. 32 (1994), no. 6, 1605–1620. MR 1297101, DOI 10.1137/S0363012993244945
- Richard Montgomery, Hearing the zero locus of a magnetic field, Comm. Math. Phys. 168 (1995), no. 3, 651–675. MR 1328258
- Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. Annals of Mathematics Studies, No. 78. MR 0385004
- E. Nelson, Tensor Analysis, Annals of Mathematics Studies, No. 78, 1967, Princeton University Press, Princeton, N.J.
- Pierre Pansu, Une inégalité isopérimétrique sur le groupe de Heisenberg, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 2, 127–130 (French, with English summary). MR 676380
- Pierre Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 415–445 (French, with English summary). MR 741395, DOI 10.1017/S0143385700002054
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
- P. Rashevskii, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski ped. inst. Libknexta, 2 (1938), 83–94.
- A. I. Šnirel′man, The asymptotic multiplicity of the spectrum of the Laplace operator, Uspehi Mat. Nauk 30 (1975), no. 4 (184), 265–266 (Russian). MR 0413209
- A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
- A. M. Vershik and V. Ya. Gershkovich, The geometry of the nonholonomic sphere for three-dimensional Lie group, Global analysis—studies and applications, III, Lecture Notes in Math., vol. 1334, Springer, Berlin, 1988, pp. 309–331. MR 964707, DOI 10.1007/BFb0080435
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
- L. C. Young, Lectures on the calculus of variations and optimal control theory, Foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia-London-Toronto, Ont., 1969. MR 0259704
Review Information:
Reviewer:
Richard Montgomery
Affiliation:
Mathematics Department, University of California, Santa Cruz, Santa Cruz California 95064
Email:
rmont@ucsc.edu
Journal:
Bull. Amer. Math. Soc.
57 (2020), 657-677
DOI:
https://doi.org/10.1090/bull/1701
Published electronically:
June 5, 2020
Review copyright:
© Copyright 2020
American Mathematical Society
