Volume 18, Issue 4

We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.

We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a closed surface.

We deal with complete generalized linear Weingarten hypersurfaces immersed in hyperbolic spaces. Under appropriate constraints on the image of the Gauss map, we present suitable conditions which guarantee the umbilicity of these hypersurfaces.

We discuss the classification of rank 3 lattices preserved by finite orthogonal groups and derive from it the classification of regular polyhedra in the 3-dimensional torus. This classification is closely related to the classification of regular polyhedra in 3-space.

We exploit an elementary specialization technique to study rational curves on Fano varieties of index one less than their dimension, known as del Pezzo manifolds. First, we study the splitting type of the normal bundles of the rational curves. Second, we prove a simple formula relating the number of rational curves passing through a suitable number of points in the case of threefolds and the analogous invariants for del Pezzo surfaces.

We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on ℂ 3 is irreducible for c ≤ 10.

We prove a stronger version of a criterion of rationality given by Ionescu and Russo. We use this stronger version to define an invariant for rational varieties (we call it rationality degree), and we classify rational varieties for small values of the invariant.

We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.

We consider invariant Einstein metrics on the quaternionic Stiefel manifold V p ℍ n of all orthonormal p -frames in ℍ n . This manifold is diffeomorphic to the homogeneous space Sp( n )/Sp( n − p ) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on V p ℍ n ≅ Sp( n )/Sp( n − p ), where n = k 1 + k 2 + k 3 and p = n − k 3 . We view V p ℍ n as a total space over the generalized Wallach space Sp( n )/(Sp( k 1 )×Sp( k 2 )×Sp( k 3 )) and over the generalized flag manifold Sp( n )/(U( p )×Sp( n − p )).