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Restricted Jacobi Fields

Year 2021, Volume: 14 Issue: 2, 247 - 265, 29.10.2021
https://doi.org/10.36890/iejg.945800

Abstract

We generalize the concept of a Jacobi field to nonholonomic Riemannian geometry, considering both nonholonomic Jacobi fields, and, more generally, restricted Jacobi fields. In the first case, the corresponding Jacobi equation involves the nonholonomic connection and the Schouten curvature tensor. In the second case, the Jacobi equation involves connections and curvature tensors arising in the construction of the Wagner curvature tensor. We also briefly discuss the existence of restricted Jacobi fields.

References

  • [1] Agrachev, A.A., Barilari, D., Boscain, U.: A comprehensive introduction to sub-Riemannian geometry. Cambridge University Press. Cambridge (2020).
  • [2] Agrachev, A.A., Gamkrelidze, R.V.: Feedback-invariant optimal control theory and differential geometry-I. Regular extremals. J. Dyn. Control Sys. 3, 343–389 (1997).
  • [3] Agrachev, A.A., Gamkrelidze, R.V.: Feedback-invariant optimal control theory and differential geometry-II. Jacobi curves for singular extremals. J.Dyn. Control Sys. 4, 583–604 (1998).
  • [4] Barilari, D., Rizzi, L.: On Jacobi fields and a canonical connection in sub-Riemannian geometry. Arch. Math. (Brno) 53, 77–92 (2017).
  • [5] Barrett, D.I., Remsing, C.C.: A note on flat nonholonomic Riemannian structures on three-dimensional Lie groups. Beitr. Algebra Geom. 60, 419–436 (2019).
  • [6] Barrett, D.I., Remsing, C.C.: On geodesic invariance and curvature in nonholonomic Riemannian geometry. Publ. Math. Debrecen 94, 197–213 (2019).
  • [7] Barrett, D.I., Remsing, C.C.: On the Schouten and Wagner curvature tensors. Preprint.
  • [8] Barrett, D.I., Biggs, R., Remsing, C.C., Rossi, O.: Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. J. Geom. Mech. 8, 139–167 (2016).
  • [9] Bellaïche, A., Risler, J-J. (eds): Sub-Riemannian geometry. Birkhäuser. Basel (1996).
  • [10] Bloch, A.M.: Nonholonomic mechanics and control (2nd ed.). Springer. New York (2003).
  • [11] Cantrijn, F., Langerock, B.: Generalised connections over a vector bundle map. Differential Geom. Appl. 18, 295–317 (2003).
  • [12] Cortés Monforte, J.: Geometric, control and numerical aspects of nonholonomic systems. Springer. Berlin (2002).
  • [13] Dragovi´c, V., Gaji´c, B.: The Wagner curvature tensor in nonholonomic mechanics. Regul. Chaotic Dyn. 8, 105–123 (2003).
  • [14] Ehlers,K.: Geometric equivalence on nonholonomic three-manifolds. In: Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington NC, USA. 246–255 (2002).
  • [15] Koiller, J., Rodrigues, P.R., Pitanga, P.: Non-holonomic connections following Élie Cartan. An. Acad. Bras. Cienc. 73, 165–190 (2001).
  • [16] Langerock,B.: Nonholonomic mechanics and connections over a bundle map. J. Phys. A: Math. Gen. 34, L609–L615 (2001).
  • [17] Lewis, A.D.: Affine connections and distributions with applications to nonholonomic mechanics. Rep. Math. Phys. 42, 135–164 (1998).
  • [18] Michor, P.W.: Topics in differential geometry. American Mathematical Society. Providence, RI (2008).
  • [19] Montgomery, R.: A tour of subRiemannian geometries, their geodesics and applications. American Mathematical Society. Providence, RI (2002).
  • [20] Rifford, L.: Sub-Riemannian geometry and optimal transport. Springer. Berlin (2014).
  • [21] Tavares, J.N.: About Cartan geometrization of non-holonomic mechanics. J. Geom. Phys. 45, 1–23 (2003).
  • [22] Vershik, A.M., Gershkovich, V. Ya.: Nonholonomic problems and the theory of distributions. Acta Appl. Math. 12, 181–209 (1988).
  • [23] Vershik, A.M., Gershkovich, V.Ya.: Nonholonomic dynamical systems, geometry of distributions and variational problems. In: Dynamical Systems VII. Springer. Berlin (1994).
Year 2021, Volume: 14 Issue: 2, 247 - 265, 29.10.2021
https://doi.org/10.36890/iejg.945800

Abstract

References

  • [1] Agrachev, A.A., Barilari, D., Boscain, U.: A comprehensive introduction to sub-Riemannian geometry. Cambridge University Press. Cambridge (2020).
  • [2] Agrachev, A.A., Gamkrelidze, R.V.: Feedback-invariant optimal control theory and differential geometry-I. Regular extremals. J. Dyn. Control Sys. 3, 343–389 (1997).
  • [3] Agrachev, A.A., Gamkrelidze, R.V.: Feedback-invariant optimal control theory and differential geometry-II. Jacobi curves for singular extremals. J.Dyn. Control Sys. 4, 583–604 (1998).
  • [4] Barilari, D., Rizzi, L.: On Jacobi fields and a canonical connection in sub-Riemannian geometry. Arch. Math. (Brno) 53, 77–92 (2017).
  • [5] Barrett, D.I., Remsing, C.C.: A note on flat nonholonomic Riemannian structures on three-dimensional Lie groups. Beitr. Algebra Geom. 60, 419–436 (2019).
  • [6] Barrett, D.I., Remsing, C.C.: On geodesic invariance and curvature in nonholonomic Riemannian geometry. Publ. Math. Debrecen 94, 197–213 (2019).
  • [7] Barrett, D.I., Remsing, C.C.: On the Schouten and Wagner curvature tensors. Preprint.
  • [8] Barrett, D.I., Biggs, R., Remsing, C.C., Rossi, O.: Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. J. Geom. Mech. 8, 139–167 (2016).
  • [9] Bellaïche, A., Risler, J-J. (eds): Sub-Riemannian geometry. Birkhäuser. Basel (1996).
  • [10] Bloch, A.M.: Nonholonomic mechanics and control (2nd ed.). Springer. New York (2003).
  • [11] Cantrijn, F., Langerock, B.: Generalised connections over a vector bundle map. Differential Geom. Appl. 18, 295–317 (2003).
  • [12] Cortés Monforte, J.: Geometric, control and numerical aspects of nonholonomic systems. Springer. Berlin (2002).
  • [13] Dragovi´c, V., Gaji´c, B.: The Wagner curvature tensor in nonholonomic mechanics. Regul. Chaotic Dyn. 8, 105–123 (2003).
  • [14] Ehlers,K.: Geometric equivalence on nonholonomic three-manifolds. In: Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington NC, USA. 246–255 (2002).
  • [15] Koiller, J., Rodrigues, P.R., Pitanga, P.: Non-holonomic connections following Élie Cartan. An. Acad. Bras. Cienc. 73, 165–190 (2001).
  • [16] Langerock,B.: Nonholonomic mechanics and connections over a bundle map. J. Phys. A: Math. Gen. 34, L609–L615 (2001).
  • [17] Lewis, A.D.: Affine connections and distributions with applications to nonholonomic mechanics. Rep. Math. Phys. 42, 135–164 (1998).
  • [18] Michor, P.W.: Topics in differential geometry. American Mathematical Society. Providence, RI (2008).
  • [19] Montgomery, R.: A tour of subRiemannian geometries, their geodesics and applications. American Mathematical Society. Providence, RI (2002).
  • [20] Rifford, L.: Sub-Riemannian geometry and optimal transport. Springer. Berlin (2014).
  • [21] Tavares, J.N.: About Cartan geometrization of non-holonomic mechanics. J. Geom. Phys. 45, 1–23 (2003).
  • [22] Vershik, A.M., Gershkovich, V. Ya.: Nonholonomic problems and the theory of distributions. Acta Appl. Math. 12, 181–209 (1988).
  • [23] Vershik, A.M., Gershkovich, V.Ya.: Nonholonomic dynamical systems, geometry of distributions and variational problems. In: Dynamical Systems VII. Springer. Berlin (1994).
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Claudiu Remsing 0000-0002-3972-2591

Dennis I. Barrett This is me

Publication Date October 29, 2021
Acceptance Date July 25, 2021
Published in Issue Year 2021 Volume: 14 Issue: 2

Cite

APA Remsing, C., & Barrett, D. I. (2021). Restricted Jacobi Fields. International Electronic Journal of Geometry, 14(2), 247-265. https://doi.org/10.36890/iejg.945800
AMA Remsing C, Barrett DI. Restricted Jacobi Fields. Int. Electron. J. Geom. October 2021;14(2):247-265. doi:10.36890/iejg.945800
Chicago Remsing, Claudiu, and Dennis I. Barrett. “Restricted Jacobi Fields”. International Electronic Journal of Geometry 14, no. 2 (October 2021): 247-65. https://doi.org/10.36890/iejg.945800.
EndNote Remsing C, Barrett DI (October 1, 2021) Restricted Jacobi Fields. International Electronic Journal of Geometry 14 2 247–265.
IEEE C. Remsing and D. I. Barrett, “Restricted Jacobi Fields”, Int. Electron. J. Geom., vol. 14, no. 2, pp. 247–265, 2021, doi: 10.36890/iejg.945800.
ISNAD Remsing, Claudiu - Barrett, Dennis I. “Restricted Jacobi Fields”. International Electronic Journal of Geometry 14/2 (October 2021), 247-265. https://doi.org/10.36890/iejg.945800.
JAMA Remsing C, Barrett DI. Restricted Jacobi Fields. Int. Electron. J. Geom. 2021;14:247–265.
MLA Remsing, Claudiu and Dennis I. Barrett. “Restricted Jacobi Fields”. International Electronic Journal of Geometry, vol. 14, no. 2, 2021, pp. 247-65, doi:10.36890/iejg.945800.
Vancouver Remsing C, Barrett DI. Restricted Jacobi Fields. Int. Electron. J. Geom. 2021;14(2):247-65.