CONJUGATE LOCI OF 2-STEP NILPOTENT LIE GROUPS SATISFYING J2z= <Sz, z>A

Abstract

Let n be a 2-step nilpotent Lie algebra which has an inner product <, > and has an orthogonal decomposition <TEX>$n\;=z\;{\oplus}v$</TEX> for its center z and the orthogonal complement v of z. Then Each element z of z defines a skew symmetric linear map <TEX>$J_z\;:\;v\;{\longrightarrow}\;v$</TEX> given by <<TEX>$J_zx$</TEX>, y> = <z, [x, y]> for all x, <TEX>$y\;{\in}\;v$</TEX>. In this paper we characterize Jacobi fields and calculate all conjugate points of a simply connected 2-step nilpotent Lie group N with its Lie algebra n satisfying <TEX>$J^2_z$</TEX> = <Sz, z>A for all <TEX>$z\;{\in}\;z$</TEX>, where S is a positive definite symmetric operator on z and A is a negative definite symmetric operator on v.