Sectional Curvature Pinching of Two-Step Nilmanifolds

Nilmanifolds are homogeneous Riemannian manifolds admitting a transitive nilpotent Lie group of isometries. By classical results (Wolf, Milnor), nilmanifoldsare always of mixed curvature. Two-step nilmanifolds are particularly important, as they play a crucial role in the classification of quarter-pinched homogeneous manifolds of negative curvature by Eberlein and Heber. Given a two-step nilmanifold, we study its pinching constant, which is the ratio of the minimum and maximum of sectional curvature. 

A prototype of a two-step nilmanifold is the 3-dimensional Heisenberg group (so-called Nil). In this case, it is well known that the pinching constant is -3. In this talk, we show that for any two-step nilmanifold, the pinching constant lies in the compact interval [-3, -3/2]. We give examples of two-step nilmanifolds that achieve the bounds -3 and -3/2, respectively. Moreover, we discuss why the bounds -3 and -3/2 are special in terms of rigidity.