Constant Jacobi osculating rank of $U(3)/(U(1) \times U(1) \times U(1))$ - Appendix -

TL;DR

This appendix discusses the derivation of a relation between covariant derivatives of the Jacobi operator on the flag manifold U(3)/(U(1)^3), enabling explicit, geodesic-independent expressions and a new method for calculating Jacobi fields.

Contribution

It introduces a universal relation for the Jacobi operator's derivatives on the flag manifold, facilitating explicit calculations and a new approach for Jacobi fields on g.o. spaces.

Findings

Derived a relation between covariant derivatives of the Jacobi operator for all geodesics.

Obtained an explicit, geodesic-independent expression of the Jacobi operator.

Presented a new formula for calculating Jacobi vector fields on g.o. spaces.

Abstract

This is the appendix of the paper [T. Arias-Marco, Constant Jacobi osculating rank of $U(3)/(U(1) \times U(1) \times U(1))$, Arch. Math. (Brno) 45 (2009), 241--254] where we obtain an interesting relation between the covariant derivatives of the Jacobi operator valid for all geodesic on the flag manifold $M^6=U(3)/(U(1) \times U(1) \times U(1))$. As a consequence, an explicit expression of the Jacobi operator independent of the geodesic can be obtained on such a manifold. Moreover, we show the way to calculate the Jacobi vector fields on this manifold by a new formula valid on every g.o. space.