Volume 18, Issue 3

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫 G from a simple directed graph G in 2005. There is a natural inner product on 𝔫 G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫 g . We classify singularity properties of the Lie algebra 𝔫 g in terms of the graph G . A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N , a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

Let X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set X τ . We show that X is diffeomorphic to the normal bundle of X τ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ * g = g .

We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds ( M , g ) which are traceless cyclic with respect to some quotient expression M = G / K and reductive decomposition 𝔤 = 𝔨 ⊕ 𝔪. Using transversally symmetric fibrations of noncompact type, we give a list of them.

In earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X , which is a generating function for Gromov–Witten invariants of X , coincides with the classical period of its mirror partner f . In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with 13$\begin{array}{} \frac{1}{3} \end{array} $(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

In this paper, we consider a special class of solvable Lie groups such that for any x , y in their Lie algebras, [ x , y ] is a linear combination of x and y . We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

The classical Loomis–Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer’s dual Loomis–Whitney inequality.

A multi-fan (respectively multi-polytope), introduced first by Hattori and Masuda, is a purely combinatorial object generalizing an ordinary fan (respectively polytope) in algebraic geometry. It is well known that an ordinary fan or polytope is associated with a toric variety. On the other hand, we can geometrically realize multi-fans in terms of torus manifolds. However, it is unfortunate that two different torus manifolds may correspond to the same multi-fan. The goal of this paper is to give some criteria for a multi-polytope to be an ordinary polytope in terms of the Duistermaat–Heckman functions and winding numbers. Moreover, we also prove a generalized Pick formula and its consequences for simple lattice multi-polytopes by studying their Ehrhart polynomials.

We investigate a class of complex Finsler metrics on a domain D ⊂ ℂ n . Necessary and sufficient conditions for these metrics to be strongly pseudoconvex complex Finsler metrics, or complex Berwald metrics, are given. The complex Berwald metrics constructed in this paper are neither trivial Hermitian metrics nor conformal changes of complex Minkowski metrics. We give a characterization of complex Berwald metrics which are of isotropic holomorphic curvatures, and also give characterizations of complex Finsler metrics of this class to be Kähler Finsler or weakly Kähler Finsler metrics. Moreover, in the strongly convex case, we give characterizations of complex Finsler metrics of this class to be projectively flat Finsler metrics or dually flat Finsler metrics.

We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincaré similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form.