arXiv research

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.

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

Analytic torsion matches Ray-Singer for specific nilmanifolds.

New proof shows rapid mixing for random walks on nilmanifolds.

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

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…

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 on holonomy of Obata connection on specific nilmanifolds.

New families of non-singular geodesic orbit nilmanifolds discovered.

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitra…

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 classify nilmanifolds admitting invariant cocalibrated $G_2$-structures

Solves a specific Calabi conjecture on special nilmanifolds.

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 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, $\textrm{HPer}(f)=\{1\}$.

In this paper, we examine some geometric vector fields on 2-step nilmanifolds of dimension 5.

We consider nilmanifolds with left-invariant complex structure and prove that small deformations of such structures are again left invariant if the Dolbeault-cohomology of the nilmanifold can be calculated using left-invariant forms. By a result of Console and Fino this is generically the case. Our main tool is an anal…

Study on metrics on specific nilmanifolds, finding new examples and properties.

We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermit…

Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.

We prove that if a compact nilmanifold $Γ\backslash G$ is endowed with a Vaisman structure, then $G$ is isomorphic to the Cartesian product of the Heisenberg group with $\mathbb{R}$.

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

Flat hypercomplex nilmanifolds have a specific solvability property.

The paper studies asymptotics and zeta functions on compact nilmanifolds.

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

Paper extends BIP to nilmanifold products and characterizes fixed points.

New families of weakly symmetric nilmanifolds discovered.

Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.

We prove that a compact nilmanifold admits a Sasakian structure if and only if it is a compact quotient of the generalized Heisenberg group of odd dimension by a co-compact discrete subgroup.

Study of symmetry distributions in Lorentzian naturally reductive nilmanifolds.

We find a one-parameter family of non-isomorphic nilpotent Lie algebras $\mathfrak{g}_a$, with $a \in [0,\infty)$, of real dimension eight with (strongly non-nilpotent) complex structures. By restricting $a$ to take rational values, we arrive at the existence of infinitely many real homotopy types of $8$-dimensional ni…

In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long…

Researchers confirm conjecture for complex nilmanifolds in higher dimensions.

Anomaly flow studied on flat and non-flat nilmanifolds.

Study on existence of $p$-Kähler structures on nilmanifolds with nilpotent complex structures.

The paper classifies nilmanifolds with specific SL(3,C) structures.