Bi-Jacobi Fields And Riemannian Cubics For Left-Invariant SO(3)

Lyle Noakes (1) , Tudor S Ratiu (2, 3)

1 UWA - The University of Western Australia
2 EPFL - Ecole Polytechnique Fédérale de Lausanne
3 Mathematical Sciences Program, Skolkovo Institute of Science and Technology

Bi-Jacobi fields are generalized Jacobi fields, and are used to efficiently compute approximations to Riemannian cubic splines in a Riemannian manifold M. Calculating bi-Jacobi fields is straightforward when M is a symmetric space such as bi-invariant SO(3), but not for Lie groups whose Riemannian metric is only left-invariant. Because left-invariant Riemannian metrics occur naturally in applications, there is also a need to calculate bi-Jacobi fields in such cases. The present paper investigates bi-Jacobi fields for left-invariant Riemannian metrics on SO(3), reducing calculations to quadratures of Jacobi fields. Then left-Lie reductions are used to give an easily implemented numerical method for calculating bi-Jacobi fields along geodesics in SO(3), and an example is given of a nearly geodesic approximate Riemannian cubic.

Domaines: Mathématiques [math]

Lyle Noakes, Tudor S Ratiu. Bi-Jacobi Fields And Riemannian Cubics For Left-Invariant SO(3). Communications in Mathematical Sciences, 2016, 14 (1), pp.55-68. ⟨hal-01391496⟩