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

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.


Published in:
Communications In Mathematical Sciences, 14, 1, 55-68
Year:
2016
Publisher:
Somerville, Int Press Boston, Inc
ISSN:
1539-6746
Keywords:
Laboratories:




 Record created 2016-02-16, last modified 2018-09-13


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