GEODESIC FORMULA OF A CERTAIN CLASS OF PSEUDORIEMANNIAN 2-STEP NILPOTENT GROUPS AND JACOBI OPERATORS ALONG GEODESICS IN PSEUDORIEMANNIAN 2-STEP NILPOTENT GROUPS

Abstract

In this paper, we obtain geodesic formula of a certain class of Pseudoriemmanian 2-step nilpotent groups and show a constancy of represenation matrix of Jacobi oprerators along geodesics in Pseudoriem- manian 2-step nilpotent groups with one dimensional center. These are nonabelian Lie groups that come as close as possible to being abelian, but they admit interesting phenomena that do not arise in abelian groups. During the 1990's many works were done in 2-step nilpotent Lie groups with a left invariant Riemmannian metric or Lorentzian metric. Eberlein (1) and Guederi (2) did general studies on 2-step nilpotent Lie groups with a left invariant Riemman- nian metric and 2-step nilpotent Lie groups with a left invariant Lorentzian metric, respectively. But relatively little was known of the Pseudoriemmanian case. For example , Eberlein and Guederi succeeded in obtaining general ge- odesic formulae of 2-step nilpotent groups with a left invariant Riemmanian metric and with a left invariant Lorentzian metric respectively. But until now a geodesic formula of Pseudoriemmanian 2-step nilpotent groups has not been obtained. In this paper, we obtain a geodesic formula of a special famliy of Pseudoriemmanian 2-step nilpotent groups and show a constancy of represen- ation matrix of Jacobi oprerators along geodesics in Pseudoriemmanian 2-step nilpotent groups with one dimensional center, which is a slight generalization of a result in (6).