Monday, October 27, 2003
2:30 pm Science 2-065

Rachelle Decoste

University of North Carolina, Chapel Hill

Density of closed geodesics in compact nilmanifolds defined by compact semisimple g-modules

Abstract: Nilpotent Lie groups and algebras are interesting for several reasons. Nilpotent Lie groups cannot be studied using the standard comparison methodsof Riemannian geometry because each left invariant metric must have both positive and negative Ricci curvatures. In addition, 2-step nilpotent Lie groups are as close as possible to being abelian, without actually being abelian. One interesting problem associated with nilmanifolds is the distribution of closed geodesics, a study begun by Eberlein, Lee-Park and Mast. We consider nilmanifolds which are constructed from representations of compact, semisimple 2-step nilpotent Lie algebras. Every 2-step nilpotent Lie algebra is isomorphic to one of the following metric examples. Let $W$ be a $p$-dimensional subspace of $\so$, the $q\times q$ real skew symmetric matrices, and let $\n=\Rq\oplus W$. We endow $\n$ with the left invariant metric such that $\mathbb{R}^q$ and $W$ are orthogonal. A case of special interest occurs when $W$ is a semisimple subalgebra of $\so$. After a brief introduction to the history of the problem and some basic definitions,we will discuss this special case. We say that a manifold has the density of closed geodesics property if the vectors tangent to closed, unit speed geodesics are dense in the unit tangent bundle of that manifold. It will be shown that in the case where $W$ is almost any classical Lie algebra, an associated nilmanifold will have this density of closed geodesics property.