# Constant Jacobi osculating rank of $\mathbf{U(3)/(U(1) \times U(1) \times U(1))}$

## Teresa Arias-Marco

Address: Departamento de Matemáticas, Universidad de Extremadura Avenida de Elvas s/n, 06071 Badajoz, Spain

E-mail: ariasmarco@unex.es

Abstract: In this paper 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.

AMSclassification: primary 53C21; secondary 53C25, 53C30, 53C20.

Keywords: naturally reductive space, g.o. space, Jacobi operator, Jacobi osculating rank.