Book Review
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MathSciNet review:
2721044
Full text of review:
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Book Information:
Authors:
Ovidiu Calin and
Der-Chen Chang
Title:
Sub-Riemannian geometry: general theory and examples
Additional book information:
Encyclopedia of Mathematics and Its Applications, vol.~126,
Cambridge University Press,
New York,
2009,
xi+367 pp.,
ISBN 978-0521897303,
US$99.00,
hardcover
M. Born, Natural Philosophy of Cause and Chance, Oxford Press, 1949.
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
Elie Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910), 109–192 (French). MR 1509120
A. Chenciner, Collisions totales, mouvements complètement paraboliques et réduction des homothéties dans le problème des $n$ corps, Regul. Chaotic Dyn. 3 (1998), no. 3, 93–106 (French, with French and Russian summaries). J. Moser at 70 (Russian). MR 1704972, DOI 10.1070/rd1998v003n03ABEH000083
Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105 (German). MR 1880, DOI 10.1007/BF01450011
Mikhael Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79–323. MR 1421823
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
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
P. K. Rashevskii, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknexta, no. 2, 83-94, (1938) [Russian].
A. Shapere and F. Wilczek, Self-propulsion at low Reynolds number, Physical Review Letters, vol. 58, 2051-2054, (1987).
Robert S. Strichartz, Realms of mathematics: elliptic, hyperbolic, parabolic, sub-elliptic, Math. Intelligencer 9 (1987), no. 3, 56–64. MR 895772, DOI 10.1007/BF03023957
Noboru Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), no. 1, 23–84. MR 533089, DOI 10.14492/hokmj/1381758416
L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Chelsea, 1980.
References
- M. Born, Natural Philosophy of Cause and Chance, Oxford Press, 1949.
- L. Capogna, D. Danielli, Donatella, S. Pauls, and J. Tyson, An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progress in Mathematics, 259. Birkhäuser Verlag, Basel, 2007. MR 2312336 (2009a:53053)
- E. Cartan, Les systèmes de Pfaff à cinque variables et lès équations aux dérivées partielles du second ordre, Ann. Sci. Ècole Normale, no. 3, 27, 109-192, (1910). MR 1509120
- A. Chenciner, De l’espace des triangles au probleme des $3$ corps, web page: http://www.imcce.fr/Equipes/ASD/person/chenciner/chen_preprint.php, under the year 2004. MR 1704972 (2000h:70023)
- W.L. Chow, Uber Systeme van Linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., vol. 117, 98-105, (1939). MR 0001880 (1:313d)
- M. Gromov, Carnot-Caratheodory spaces seen from within, pp. 79-323 in Sub-Riemannian Geometry, A. Bellaiche and J-J. Risler, eds., Birhauser, Basel, Switzerland, (1996). MR 1421823 (2000f:53034)
- M. Gromov, Groups of Polynomial Growth and Expanding Maps, Publ. Math. IHES, 53, 53-73, (1981). MR 623534 (83b:53041)
- L. Hörmander, Hypoelliptic second order differential operators, Acta. Math., 119, 147-171, (1967). MR 0222474 (36:5526)
- R. Montgomery, A tour of sub-Riemannian geometries, their geodesics and applications, Math. Surveys Monogr., vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362 (2002m:53045)
- G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Ann. of Math. Stud., vol. 78, Princeton Univ. Press, (1973). MR 0385004 (52:5874)
- P. K. Rashevskii, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. Libknexta, no. 2, 83-94, (1938) [Russian].
- A. Shapere and F. Wilczek, Self-propulsion at low Reynolds number, Physical Review Letters, vol. 58, 2051-2054, (1987).
- R. Strichartz, Realms of mathematics: elliptic, hyperbolic, parabolic, sub-elliptic, Math. Intelligencer, 9, no. 3, 56–64 (1987). MR 895772 (88h:35002)
- N. Tanaka, On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. Journal, v. 8, 23-84, 1979. MR 533089 (80h:53034)
- L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Chelsea, 1980.
Review Information:
Reviewer:
Richard Montgomery
Affiliation:
UC Santa Cruz
Email:
rmont@math.ucsc.edu
Journal:
Bull. Amer. Math. Soc.
47 (2010), 713-722
DOI:
https://doi.org/10.1090/S0273-0979-10-01288-7
Published electronically:
February 8, 2010
Review copyright:
© Copyright 2010
American Mathematical Society