Pacific Journal of Mathematics

On a constrained variational problem and the spaces of horizontal paths.

Zhong Ge

Article information

Source
Pacific J. Math., Volume 149, Number 1 (1991), 61-94.

Dates
First available in Project Euclid: 8 December 2004

Permanent link to this document
https://projecteuclid.org/euclid.pjm/1102644564

Mathematical Reviews number (MathSciNet)
MR1099784

Zentralblatt MATH identifier
0718.58023

Subjects
Primary: 58E10: Applications to the theory of geodesics (problems in one independent variable)

Citation

Ge, Zhong. On a constrained variational problem and the spaces of horizontal paths. Pacific J. Math. 149 (1991), no. 1, 61--94. https://projecteuclid.org/euclid.pjm/1102644564


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References

  • [I] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Dynamical Systems, III, v3 in the Encyclopedia of Mathematical Sciences Series, 1988, Springer.
  • [2] M. Atiyah and A. N. Pressley, Convexity and loop groups, Progr. Math., 36 (1983), 33-64.
  • [3] L. B. Bergery, Sur certaines fibrations d'espaces homogenes emanniens,Com- positio Math., 30 (1975), 43-61.
  • [4] J. M. Bismut, Large derivations and the Malliavin calculus,Progr. Math., vol.
  • [45] Birkhauser, Basel, 1984.
  • [5] R. Bott,Lectureon Morsetheory,old and new, Bull. Amer. Math. Soc,7 (1982), 331-358.
  • [6] R. Brockett, Control theory and singular Riemannian geometry, in NewDirec- tions in Applied Mathematics, Springer-Verlag, New York (1981).
  • [7] Chan Hong Mo and Tsou Sheung Tsun, Equation of motion for non-abelian monopoles, in Proc. Monopole Meeting, Trieste, Italy, ed. N. S. Craigie, P. Goddard, N. N. Nahm, 261-272.
  • [8] D. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry vol. 2, (1985), Springer.
  • [9] Zhong Ge, Carnot-Caratheodorymetrics and the horizontal paths spaces,toap- pear.
  • [10] M. Gromov, Sur Structures Mtriques pour les Variets Riemanniens, (1981), Cedic-Nathan.
  • [II] U. Hamenstadt, Some regularity result for Carnot-Caratheodory metrics, pre- print.
  • [12] S. Helgason, Differential Geometry and Symmetric Spaces, (1962), Academic Press.
  • [13] R. Hermann, Some differential-geometricaspects of the Lagrangian variational problem, Illinois J. Math., 6 (1962), 634-673.
  • [14] W. Klingenberg, Riemannian Geometry,Walter de Gruyter, 1982.
  • [15] A. Koranyi, Geometric properties of Heisenberg-type group, Adv. Math., 56 (1985), 28-38.
  • [16] J. Milnor, Morse Theory, Ann. Math. Studies 51,Princeton Univ. Press,1963.
  • [17] J. Mitchell, On Carnot-Caratheodory metrics,J. Differential Geom., 21 (1985), 35-45.
  • [18] R. Montgomery, Shortest loops with a fixed holonomy, MSRI preprint (1988).
  • [19] P. Pansu, Croissance des boules et quasiisometries dans les nilvarieties, Ergodic Theory Dynamical Systems, 3 (1983),41-445.
  • [20] R. S. Strichartz, Sub-Riemannangeometry, J. Differential Geom., 24 (1986), 221-263.
  • [21] M. Tamm, subanalytical sets in the calculus of variations, Acta. Math., 146 (1981), 167-199.
  • [22] T. J. S. Taylor, Some aspectsof differential geometry associated with hypoelliptic second order operators, Pacific J. Math., 136 (1989), 355-378.
  • [23] Tsou Sheung Tsuh, Study ofmonopoles using loopspace,in ProceedingMonopole Meeting, Trieste, Italy, ed. N. S. Cragie, P.Goddard, N. N. Nahm.
  • [24] S. Webster, Pseudo-Hermitian structure on a real hyperplane, J. Differential Geom., 13 (1978), 25-40.
  • [25] A. Weinstein, Fat bundle and symplectic manifolds, Adv. in Math., 37 (1980), 239-250.
  • [26] A. Weinstein, Private Communication.
  • [27] S. Smale, Regular curves on Riemannian manifolds, Trans. Amer. Math. Soc, 87 (1958).
  • [28] C. B. Rayner, The exponential map for the Lagrange problem on differentiable manifolds, Phil. Trans. Roy. Soc. London, ser. A, no. 1127, v. 262, pp. 299-344.
  • [29] C. Bar, Geodesies for Camot-Caratheodory metrics, preprint, 1989.