Restricted Jacobi Fields

Yıl 2021, , 247 - 265, 29.10.2021

Claudiu Remsing Dennis I. Barrett

https://doi.org/10.36890/iejg.945800

Öz

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.

Anahtar Kelimeler

Nonholonomic Riemannian structure, Jacobi field, Schouten curvature tensor, Wagner curvature tensor

Kaynakça

• [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).