On the Existence of a Codimension 1 Completely Integrable Totally Geodesic Distribution on a Pseudo-Riemannian Heisenberg Group

In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.


Introduction
Given an n-dimensional C ∞ manifold M , a distribution D of codimension k of M is an (n − k)dimensional subbundle of the tangent space T M . A distribution D of a pseudo-Riemannian manifold (M, g) is called totally geodesic if every geodesic tangent to D at some point remains everywhere tangent to D. Among 2-step nilpotent Lie groups with left-invariant metrics, the Heisenberg group is of particular significance. In [8], the second author proved that there are three nonisometric left-invariant Lorentzian metrics on the 3-dimensional Heisenberg group, one of which is flat. This is a strong contrast to the Riemannian case in which there is only one (up to positive homothety) and it is not flat.
Another major difference to the Riemannian case appears. Indeed, T. Hangan proved that the Heisenberg group endowed with a left-invariant Riemannian metric does not admit any codimension 1 completely integrable totally geodesic distribution (see [5] for the three-dimensional case and [6] for the higher dimensional case). Recently, in [9] the second author and N. Rahmani proved that the 3-dimensional Heisenberg group admits a left-invariant Lorentzian metric for which there exists a codimension 1 completely integrable totally geodesic distribution. It is an interesting problem to investigate whether and to what extent results valid in Riemannian Heisenberg group can be extended to the pseudo-Riemannian case. In fact, in a high-dimensional Heisenberg group there are different types of pseudo-Riemannian metrics while there is still essentially only one (up to positive homothety) Riemannian metric. So, the purpose of this paper is to prove, contrary to the Riemannian case, the existence of a codimension 1 completely in-tegrable totally geodesic distribution on a pseudo-Riemannian Heisenberg group of dimension 2p + 1.

Geodesics and totally geodesic distributions on Heisenberg group
Consider R 2p+1 with the elements of the form X = (x 1 , . . . , x p , y 1 , . . . , y p , z). Define the product on R 2p+1 , by Let H 2p+1 = R 2p+1 , g be the (2p + 1)-dimensional Heisenberg group with the left-invariant pseudo-Riemannian metric g defined by (1) With respect to the basis of coordinate vector fields ∂ . . , p for which (1) holds, the nonvanishing metric components are The geodesics equations are obtained in [2] (for the Riemannian case) and in [1] and [4] (for the pseudo-Riemannian case), in more general context of two-step nilpotent Lie groups with a left-invariant metric. Jang and Parker [7] later gave explicit formulas for geodesics on the 3dimensional Heisenberg group endowed with a left-invariant Lorentzian metric. A corresponding description was made in [5] in the Riemannian case.
Here we will find the geodesics equations of H 2p+1 for the left-invariant pseudo-Riemannian metric (1) by integrating their Euler-Lagrange equations. Notice that Hangan obtained in [6] all geodesics of the Riemannian Heisenberg group of dimension 2p + 1.
We now prove the following result Theorem 1. The pseudo-Riemannian metric (1) admits a codimension 1 completely integrable totally geodesic distribution on the Heisenberg group H 2p+1 .