arXiv research

Study geometric vector fields on 5D 2-step nilmanifolds.

New proof shows rapid mixing for random walks on nilmanifolds.

Let $G$ be a connected Lie group acting locally simply transitively on a manifold $M$. By connecting curves in $M$ we mean the orbits of one-parameter subgroups of $G$. To block a pair of points $m_1,m_2\in M$ is to find a finite set $B\subset M\setminus{m_1,m_2}$ such that every connecting curve joining $m_1$ and $m_2…

Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is eas…

Study geodesic orbit Lorentz nilmanifolds, proving structural properties.

Researchers extend geodesic orbit properties to pseudo-Riemannian nilmanifolds of specific signature.

J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost Kähler manifolds generalising Kähler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-Kähler structures on several nilmanifolds and on twistor fibrations over hyperbolic space studied…

Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described…

We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…

Geodesic orbit property studied for Lorentz manifolds.

Let $F$ be a left invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left invariant Riemannian metric ${\hat{\textbf{\textit{a}}}}$ and a vector field $X$ which is $I_{\hat{\textbf{\textit{a}}}}(M)$-invariant. If the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci so…

A Hermitian metric on a complex manifold of complex dimension $n$ is called {\em astheno-Kähler} if its fundamental $2$-form $F$ satisfies the condition $\partial \overline \partial F^{n - 2} =0$. If $n =3$, then the metric is {\em strong KT}, i.e. $F$ is $\partial \overline \partial$-closed. By using blow-ups and the …

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the ni…

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded…

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of …

The study of invariant SKT structures on nilmanifolds, focusing on 2-step cases.

Researchers find isospectral but non-diffeomorphic nilmanifolds.

Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.

Analytic torsion matches Ray-Singer for specific nilmanifolds.

We study the geodesic orbit property for nilpotent Lie groups $N$ when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When $N$ acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric…

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left in…

Paper shows spectra can't distinguish naturally reductive manifolds.

The purpose of this paper is to present the first continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, and simultaneously, the first continuous families of Riemannian manifolds with the same marked length spectrum but not the same 1-form spectrum. The examples presented here are Riem…

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

We show that all 6-dimensional nilmanifolds admit generalized complex structures. This includes the five classes of nilmanifold which admit no known complex or symplectic structure. Furthermore, we classify all 6-dimensional nilmanifolds according to which of the four types of left-invariant generalized complex structu…

Study shows infinite real homotopy types for complex nilmanifolds.

Study on holonomy of Obata connection on specific nilmanifolds.

New families of non-singular geodesic orbit nilmanifolds discovered.

Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifold…

We give a necessary and sufficient condition for $k$-step nilmanifolds associated with graphs $(k \geq 3)$ to admit Anosov automorphisms. We also prove nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds.

Study symplectic structures on low dimensional 2-step nilmanifolds.

Study on special metrics on complex nilmanifolds, proving existence and properties.

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety …

Solves a specific Calabi conjecture on special nilmanifolds.

We classify nilmanifolds admitting invariant cocalibrated $G_2$-structures

It is known that the spectrum of the Laplace operator on functions of a closed Riemannian manifold does not determine the integrals of the individual fourth order curvature invariants $\operatorname{scal}^2$, $|\operatorname{ric}|^2$, $|R|^2$, which appear as summands in the second heat invariant $a_2$. We study the an…

Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.

Let (J,g) be a Hermitian structure on a compact nilmanifold M with invariant complex structure J and compatible metric g, which is not required to be invariant. We give classifications of 6-dimensional nilmanifolds M admitting strong Kähler with torsion, balanced or locally conformal Kähler structures (J,g).

Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms…

The Laplace spectrum uniquely identifies five out of eight metrically maximal three-dimensional geometries.

The study classifies complex parallelisable nilmanifolds with unobstructed deformations.

Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.

Study of Bach flow on specific nilmanifolds, converging to a soliton.

Formality of Dolbeault DGAs on complex nilmanifolds restricted to tori.

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.

In this paper, we show that for every non-nilpotent hyperbolic map $f$ on an infra-nilmanifold, the $\textrm{HPer}(f)$ is cofinite in $\mathbb{N}$. This generalizes a similar result for expanding maps. Moreover, we prove that for every nilpotent map $f$ on an infra-nilmanifold, $\textrm{HPer}(f)=\{1\}$.