Abstract
Sub-Riemannian geometry is the geometry of a distribution ofk-planes on an-dimensional manifold with a smoothly varying inner product on thek-planes. Singular curves are singularities of the space of paths tangent to the distribution and joining two fixed points. This survey is devoted to the singular curves, which can be length minimizing geodesics, independent of the choice of inner product.
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References
A.A. Agrachev and A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity.Preprint, 1993.
W. Ambrose and I.M. Singer, A theorem on holonomy.Trans. Am. Math. Soc. 75 (1953), 428–453.
A. Bejancu, Geometry of CR-submanifolds.D. Reidel, Dordrecht, Holland, 1986.
J.-M. Bismut, Large deviations and the Malliavin calculus.Birkhauser, 1984.
G.A. Bliss, Lectures on calculus of variations.Univ. of Chicago Press, 1946.
R.W. Brockett, Nonlinear control theory and differential geometry.Proc. of the Int. Congress of Mathematicians, Warszawa, 1983.
R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griffiths, Exterior differential systems, Vol. 18.MSRI Publications, Springer-Verlag, 1991.
R. Bryant and L. Hsu, Rigidity of integral curves of rank two distributions.Invent. Math. 114 (1993), 435–461.
E. Cartan, Lès systemes de Pfaff a cinque variables et lès equations aux derivees partielles du second ordre.Ann. Sci. Ècole Normale 27 (1910), No. 3, 109–192.
L. Chaves A finiteness theorem for fat bundles.Topology. (1994), Aug., 493–499.
W.L. Chow, Uber Systeme van Linearen partiellen Differentialgleichungen erster Ordnung.Math. Ann. 117 (1939), 98–105.
M. Christ, Analytic hypoellipticity breaks down for weakly pseudoconvex Reinhardt domains.Preprint, UCLA, 1991.
K.J. Falconer, The geometry of fractal sets.Cambridge Univ. Press, 1985.
Zhong Ge, Horizontal path space and Carnot-Caratheodory metrics.Pac. J. Math. 161 (1993), 255–286.
—, Caustics in optimal control: an example when the symmetry is broken.Lect. Appl. Math. 29 (1993), 203–212.
Zhong Ge, On the cut points and conjugate points ina constrained variational problem.Fields Inst. Commun. 1 (1993).
—, On a constrained variational problem and the space of horizontal paths.Pac. J. Math. 149 (1991), 61–94.
V. Gershkovich, Engel structures on four-dimensional manifolds.Preprint, Univ. of Melbourne, 1993.
M. Gromov, J. Lafontaine, and P. Pansu, Structures metriques pour les varietes Riemanniennes.Paris, Cedic, 1981.
M. Gromov, Carnot-Caratheodory spaces seen from within.IHES preprint M/94/6, 1994, 221 pp.
A. Guichardet, On rotation and vibration motions of molecules.Ann. Inst. H. Poincare, Phys. Theor. 40 (1984), No. 3, 329–342.
V. Guillemin and A. Uribe, The trace formula for vector bundles.Bull. Am. Math. Soc. 14 (1986), No. 2, 222–224.
V. Guillemin, Some spectral properties of periodic potentials. inLect. Notes Math. 1256 (1987),Springer, 192–214.
—, The Laplace operator on thenth tensor power of a line bundle: eigenvalues which are uniformly bounded inn.Asympt. Anal. 1 (1988), 105–113.
U. Hamenstädt, Some regularity theorems for Carnot-Caratheodory metricsJ. Differ. Geom. 32 (1990), 819–850.
—, Zur Theorie von Carnot-Caratheodory Metriken und ihren Anwendungen.Bonner Math. Schriften, Univ. Bonn, (1987), No. 180, 1–64.
R. Hermannn, Some differential geometric aspects of the Lagrange variational problem.Indiana Math. J. (1962), 634–673.
L. Hsu, Calculus of variations via the Griffiths formalism.J. Differ. Geom. 36 (1991), No. 3, 551–591.
Th. Kaluza, Zum unitatsproblem der physik,Berlin Berichte (1921), 966 pp.
R. Kerner, Generalization of the Kaluza-Klein theory for an arbitrary non-Abelian group.Ann. Inst. Henri Poincare 9 (1968), 143–152.
I. Kupka, Abnormal extremals.Preprint, 1992.
W-S. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank two distributions.Preprint, Trans. Am. Math. Soc., 1994 (to appear).
J. Martinet, Sur les singularites des formes differentialles.Ana. Insp. (Grenoble) 20 (1970), No. 1, 95–198.
J. Mitchell, On Carnot-Caratheodory metrics.J. Differ. Geom. 21 (1985), 34–45.
R. Montgomery, Abnormal minimizers.SIAM J. Control Optim. (1994) (to appear).
—, The isoholonomic problem and some applications.Commun. Math. Phys.,128 (1990), 565–592.
R. Montgomery, Singular extremals on Lie groups,MCSS (1995) (to appear).
—, Gauge theory and control theory. In Nonholonomic motion planning. J. Canny and Z. Li, eds.,Kluwer Acad., Norwell, MA, USA, 1993, 343–378.
—, Gauge theory and the falling cat. In Dynamics and control of mechanical systems. M. Enos, ed., Fields Inst. Communications Series, Vol.1,AMS Pub., Providence, RI, 1993, 193–218.
—, Generic distributions and finite dimensional Lie algebras.J. Differ. Equ. 103 (1993), 387–393.
R. Montgomery, Hearing the zero locus of a magnetic field.Preprint, 1994.
M. Morse and S. Myers, The problems of Lagrange and Mayer with variable endpoints.Proc. Am. Acad. 66 (1931), No. 6, 236–253.
R. Murray, Nilpotent bases for a class of non-integrable distributions with applications to trajectory generation for nonholonomic systems. CDS Technical Memorandum, No. CIT-CDS 92-002,Calif. Inst. of Tech., 1993.
A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields.Acta Math. 155 (1985), 103–147.
P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symetriques de rang un.Ann. Math. 129 (1989), 1–60.
L. S. Pontrjagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes.Wiley Interscience, 1962.
C.B. Rayner, The exponential map for the Lagrange problem on differentiable manifolds.Philos. Trans. R. Soc. London, Ser. A, Math. and Phys. Sci. 262 (1967), No. 1127, 299–344.
L. P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups.Acta Math. 137 (1976), 247–320.
A. Shapere and F. Wilczek, Self-propulsion at low Reynolds number.Phys. Rev. Lett. 58 (1987), 2051–2054.
—, Geometric phases in physics.World Scientific, Singapore, 1989.
E.D. Sontag, Mathematical control theory.Springer-Verlag, New York, 1990.
S. Sternberg, On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field.Proc. Natl. Acad. Sci. USA 74 (1977), 5253–5254.
R. Strichartz, Sub-Riemannian geometry.J. Differ. Geom. 24 (1983), 221–263.
—, Corrections to “Sub-Riemanniand geometry”.J. Differ. Geom. 30 (1989), No. 2, 595–596.
A.M. Vershik and V. Ya Gerhskovich, An estimate of the functional dimension of the space of orbits of germs of generic distributions.Math. USSR, Zametki 44:45 (1988), 596–603.
—, Nonholonomic dynamical systems, geometry of distributions and variational problems. In Dynamical Systems VII. Ed. V.I. Arnol'd and S.P. Novikov, Vol. 16 of the Encyclopaedia of Mathematical Sciences series,Springer-Verlag, NY, 1994. (Russian original 1987.)
A. Weinstein, Fat bundles and symplectic manifolds.Adv. Math. 37 (1980), 239–250.
—, A universal phase space for a particle in a Yang-Mills field.Lett. Math. Phys. 2 (1978), 417–420.
S.K. Wong, Field and particle equations for the classical Yang-Mills field and particles with isotopic spin.Nuovo Cimento 65a (1970), 689–693.
L.C. Young, Lectures on the calculus of variations and optimal control theory.Chelsea, 1980.
M. Ya. Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian equations. Transl. of Math. Monographs series, Vol. 113,A.M.S., Providence, RI, 1992.
—, Singularities and normal forms of odd-dimensional Pfaff equations. (Russian)Funkts. Anal. Ego Prilozhen. 23 (1989), No. 1, 70–71.
—, Normal forms of germs of 2-dimensional distributions in ℝ4. (Russian)Funkts. Anal. Ego Prilozhen. 24 (1990), No. 2, 81–82, English translation:Funct. Anal. Appl. 24, (1990).
J.W. Zwanziger, M. Koenig, and A. Pines, Berry's phase. InAnnual Review of Chemistry, 1990.