Conjugate Loci of Pseudo-Riemannian 2-Step Nilpotent Lie Groups with Nondegenerate Center

Abstract

We determine the complete conjugate locus along all geodesics parallel or perpendicular to the center (Theorem 2.3). When the center is one-dimensional we obtain formulas in all cases (Theorem 2.5), and when a certain operator is also diagonalizable these formulas become completely explicit (Corollary 2.7). These yield some new information about the smoothness of the pseudoriemannian conjugate locus. We also obtain the multiplicities of all conjugate points. (2000): primary 53C50; secondary 22E25, 53B30, 53C30. Mathematics Subject Classifications

The conjugate locus for the Euler top I. The axisymmetric case

“…Nothwithstanding a great number of studies (see e.g. [17,8,20,13,19,15,5,21] and references therein), there are not yet many known examples of cut and conjugate loci, especially in dimension higher than two.…”

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

“…And J. Kim [11] calculated all conjugate points of H-type groups. These last two works are generalized in a pseudo-Riemannian version by Jang, Parker, and Park [7,8]. J. Lauret [13] introduced the notion of modified H-type by weakening the H-type condition.…”