2-step Nilmanifolds Research Articles

Correction to: Magnetic Trajectories on 2-Step Nilmanifolds

Correction to: Magnetic Trajectories on 2-Step Nilmanifolds

Symplectic structures on low dimensional 2-step nilmanifolds

Symplectic structures on low dimensional 2-step nilmanifolds

SKT structures on nilmanifolds

The aim of this article is to study the existence of invariant SKT structures on nilmanifolds. More precisely, we give a negative answer to the question of whether there exist a k-step ( $$k>2$$ ) complex nilmanifold admitting an invariant SKT metric. We also provide a construction which serves as a tool to generate examples of invariant SKT structures on 2-step nilmanifolds in arbitrary dimensions.

COMPATIBILITY BETWEEN NON-KÄHLER STRUCTURES ON COMPLEX (NIL)MANIFOLDS

We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced) and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a left-invariant k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics—balanced, pluriclosed and locally conformally Kähler—is a torus. Moreover, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyper-Kähler structure.

Generalized deformation of complex structures on nilmanifolds

On a 2-step nilmanifold with abelian complex structure, there exists an invariant (1,0)-form ρ such that dρ is type-(1,1). It acts on the kernel of ρ by contraction. When this contraction map is non-degenerate, for any given infinitesimal generalized complex deformation Γ1 we construct a solution (Γ,∂→) for the extended Maurer-Cartan equation. It amounts to identifying the obstruction ∂→ for Γ1 to be integrable, and constructing the deformation Γ when the obstruction vanishes. As a consequence of our explicit solutions for (Γ,∂→), we prove that on any real six-dimensional 2-step nilmanifold with abelian complex structure, when the contraction map is non-degenerate, every infinitesimal generalized complex deformation sufficiently close to zero is integrable. We also show that in all dimensions, if the contraction map is skew-Hermitian, then every infinitesimal generalized complex deformation sufficiently close to zero is integrable. Moreover the differential graded algebra controlling the generalized deformation of the underlying abelian complex structure is quasi-isomorphic to the one after deformation.

The anomaly flow on nilmanifolds

We study the Anomaly flow on 2-step nilmanifolds with respect to any Hermitian connection in the Gauduchon line. In the case of flat holomorphic bundle, the general solution to the Anomaly flow is given for any initial invariant Hermitian metric. The solutions depend on two constants \(K_1\) and \(K_2\), and we study the qualitative behaviour of the Anomaly flow in terms of their signs, as well as the convergence in Gromov–Hausdorff topology. The sign of \(K_1\) is related to the conformal invariant introduced by Fu, Wang and Wu. In the non-flat case, we find the general evolution equations of the Anomaly flow under certain initial assumptions. This allows us to detect non-flat solutions to the Hull-Strominger-Ivanov system on a concrete nilmanifold, which appear as stationary points of the Anomaly flow with respect to the Strominger-Bismut connection.

Higher degree Killing forms on 2-step nilmanifolds

We study left-invariant Killing forms of arbitrary degree on simply connected 2-step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.

Some geometric vector fields on 5-dimensional 2-step homogeneous nilmanifolds

In this paper, we examine some geometric vector fields on 2-step nilmanifolds of dimension 5. We show that there is not any invariant concurrent vector field on such spaces. Our results show that in these manifolds each invariant conformal vector field is Killing and every invariant projective field is affine. Also, the space of some other invariant geometric fields such as affine, Killing, and harmonic vector fields on the manifolds are determined.

Killing Forms on 2-Step Nilmanifolds

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $k=2$ or $k=3$, we show that the space of Killing $k$-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

Symplectic harmonicity and generalized coeffective cohomologies

Relations between the symplectically harmonic cohomology and the coeffective cohomology of a symplectic manifold are obtained. This is achieved through a generalization of the latter, which in addition allows us to provide a coeffective version of the filtered cohomologies introduced by Tsai, Tseng and Yau. We construct closed (simply connected) manifolds endowed with a family of symplectic forms $$\omega _t$$ such that the dimensions of these symplectic cohomology groups vary with respect to t. A complete study of these cohomologies is given for 6-dimensional symplectic nilmanifolds, and concrete examples with special cohomological properties are obtained on an 8-dimensional solvmanifold and on 2-step nilmanifolds in higher dimensions.

English

We study a problem of isometric compact 2-step nilmanifolds ${M}/\Gamma $ using some information on their geodesic flows, where $M$ is a simply connected 2-step nilpotent Lie group with a left invariant metric and $\Gamma $ is a cocompact discrete subgroup of isometries of $M$. Among various works concerning this problem, we consider the algebraic aspect of it. In fact, isometry groups of simply connected Riemannian manifolds can be characterized in a purely algebraic way, namely by normalizers. So, suitable factorization of normalizers and expression of a vector bundle as an associated fiber bundle to a principal bundle, lead us to a general framework, namely groupoids. In this way, drawing upon advanced ingredients of Lie groupoids, normal subgroupoid systems and other notions, not only an answer in some sense to our rigidity problem has been given, but also the dependence between normalizers, automorphisms and especially almost inner automorphisms, has been clarified.

A remark on the Bismut–Ricci form on 2-step nilmanifolds

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 an 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-time solution to the pluriclosed flow in 2-step nilmanifolds.

Topological entropy of left-invariant magnetic flows on 2-step nilmanifolds

In this paper, we consider magnetic flows on 2-step nilmanifolds , where the Riemannian metric g and the magnetic field σ are left-invariant. Our first result is that when σ represents a rational cohomology class and its restriction to vanishes on the derived algebra, then the associated magnetic flow has zero topological entropy. In particular, this is the case when σ represents a rational cohomology class and is exact. Our second result is the construction of a magnetic field on a 2-step nilmanifold that has positive topological entropy for arbitrarily high energy levels.

Isometric actions on pseudo-Riemannian nilmanifolds

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-\(H\)-type Lie groups.

Minimal algebras and 2-step nilpotent Lie algebras in dimension 7

We use the methods of Bazzoni and Munoz (Trans Am Math Soc 364:1007–1028, 2012) to give a classification of 7-dimensional minimal algebras, generated in degree 1, over any field \({\mathbf{k}}\) of characteristic \({{\rm char}(\mathbf{k})\neq 2}\) , whose characteristic filtration has length 2. Equivalently, we classify 2-step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of 7-dimensional 2-step nilmanifolds.

A note on canonical Ricci forms on $2$-step nilmanifolds

In this note we prove that any left-invariant almost Hermitian structure on a 2-step nilmanifold is Ricci-flat with respect to the Chern connection and that it is Ricci -flat with respect to another canonical connection if and only if it is cosymplectic (i.e. dω = 0).

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

It is shown that in every dimension n = 3j + 2, j = 1, 2, 3, . . ., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.

Compact symplectic nilmanifolds associated with graphs

In this paper we consider 2-step nilpotent Lie algebras, Lie groups and nilmanifolds associated with graphs. We present a combinatorial construction of the second cohomology group for these Lie algebras. This enables us to characterize those graphs giving rise to symplectic or contact nilmanifolds.

STABILITY OF ABELIAN COMPLEX STRUCTURES

Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

ANOSOV AUTOMORPHISMS ON CERTAIN CLASSES OF NILMANIFOLDS

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