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Andrei Moroianu

Publications and source records attributed to Andrei Moroianu.

At least 37 records · Page 2Linked to original sources
arXiv2025

Quaternion-Kähler manifolds with non-negative quaternionic sectional curvature

Compact Hermitian symmetric spaces are Kähler manifolds with constant scalar curvature and non-negative sectional curvature. A famous result by A. Gray states that, conversely, a compact simply connected Kähler manifold with constant scalar curvature and non-negative sectional curvature is a Hermitian symmetric space. The aim of the present article is to transpose Gray's result to the quaternion-Kähler setting. In order to achieve this, we introduce the quaternionic sectional curvature of quaternion-Kähler manifolds, we show that every Wolf space has non-negative quaternionic sectional curvature, and we prove that, conversely, every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space. The proof makes crucial use of the nearly Kähler twistor spaces of positive quaternion-Kähler manifolds.

math.DG↗
arXiv2024

Conformal vector fields on LCP manifolds

We show that conformal vector fields on compact locally conformally product manifolds are orthogonal to the flat distribution and Killing with respect to the Gauduchon metric.

math.DG↗
arXiv2024

On the ergodicity of the frame flow on even-dimensional manifolds

It is known that the frame flow on a closed $n$-dimensional Riemannian manifold with negative sectional curvature is ergodic if $n$ is odd and $n \neq 7$. In this paper we study its ergodicity in the remaining cases. For $n$ even and $n \neq 8, 134$, we show that: if $n \equiv 2$ mod $4$ or $n=4$, the frame flow is ergodic if the manifold is $\sim 0.3$-pinched, if $n \equiv 0$ mod $4$, it is ergodic if the manifold is $\sim 0.6$-pinched. In the three dimensions $n=7,8,134$, the respective pinching bounds that we need in order to prove ergodicity are $0.4962...$, $0.6212...$, and $0.5788...$. This is a significant improvement over the previously known results and a step forward towards solving a long-standing conjecture of Brin asserting that $0.25$-pinched even-dimensional manifolds have an ergodic frame flow.

math.DS↗
arXiv2024

Submersion constructions for geometries with parallel skew torsion

In the absence of a de Rham decomposition theorem for geometries with torsion, we develop and unify ways to view a geometry with parallel skew torsion as the total space of a locally defined, not necessarily unique Riemannian submersion with totally geodesic fibers. We complete and extend the Cleyton-Swann classification of irreducible such geometries and characterize the cases where the stabilizer of the torsion is larger than the holonomy. As a byproduct, we obtain structure results on Gray manifolds, nearly parallel G$_2$-manifolds and Sasaki manifolds with reducible holonomy.

math.DG↗
arXiv2024

On decomposable LCP structures

We introduce the notion of decomposable locally conformally product (LCP) manifolds and characterize those which are defined on quotients of Riemannian Lie groups by co-compact lattices.

math.DG↗
arXiv2024

The structure of locally conformally product Lie algebras

A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e.~a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the conformal class), with reducible holonomy. A left-invariant LCP structure on a compact quotient $Γ\backslash G$ of a simply connected Riemannian Lie group $(G,g)$ with Lie algebra $\mathfrak{g}$ can be characterized in terms of a closed 1-form $θ\in\mathfrak{g}^*$ and a non-zero subspace $\mathfrak{u}\subset \mathfrak{g}$ satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that $\mathfrak{g}$ is isomorphic to a semidirect product of a non-unimodular Lie algebra acting on an abelian one by a conformal representation. This extends to the general case previous results holding for solvmanifolds. In addition, we construct explicit examples of compact LCP manifolds which are not solvmanifolds.

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arXiv2024

Locally conformally product structures on solvmanifolds

We study left invariant locally conformally product structures on simply connected Lie groups and give their complete description in the solvable unimodular case. Based on previous classification results, we then obtain the complete list of solvable unimodular Lie algebras up to dimension 5 which carry LCP structures, and study the existence of lattices in the corresponding simply connected Lie groups.

math.DG↗
arXiv2024

The Heterotic-Ricci flow and its three-dimensional solitons

We introduce a novel curvature flow, the Heterotic-Ricci flow, as the two-loop renormalization group flow of the Heterotic string common sector and study its three-dimensional compact solitons. The Heterotic-Ricci flow is a coupled curvature evolution flow, depending on a non-negative real parameter $κ$, for a complete Riemannian metric and a three-form $H$ on a manifold $M$. Its most salient feature is that it involves several terms quadratic in the curvature tensor of a metric connection with skew-symmetric torsion $H$. When $κ= 0$ the Heterotic-Ricci flow reduces to the generalized Ricci flow and hence it can be understood as a modification of the latter via the second-order correction prescribed by Heterotic string theory, whereas when $H=0$ and $κ>0$ the Heterotic-Ricci flow reduces to a constrained version of the RG-2 flow and hence it can be understood as a generalization of the latter via the introduction of the three-form $H$. Solutions of Heterotic supergravity with trivial gauge bundle, which we call Heterotic solitons, define a particular class of three-dimensional solitons for the Heterotic-Ricci flow and constitute our main object of study. We prove a number of structural results for three-dimensional Heterotic solitons, obtaining the complete classification of compact three-dimensional strong Heterotic solitons as hyperbolic three-manifolds or quotients of the Heisenberg group equipped with a left-invariant metric. Furthermore, we prove that all Einstein three-dimensional Heterotic solitons have constant dilaton. In this direction, we prove that Einstein Heterotic solitons with constant dilaton are rigid and therefore cannot be deformed into a solution with non-constant dilaton. This is, to the best of our knowledge, the first rigidity result for compact supergravity solutions in the literature.

math.DG↗
arXiv2024

Einstein metrics on conformal products

We show that under some natural geometric assumption, Einstein metrics on conformal products of two compact conformal manifolds are warped product metrics.

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arXiv2024

Adapted metrics on locally conformally product manifolds

We show that the Gauduchon metric $g_0$ of a compact locally conformally product manifold $(M,c,D)$ of dimension greater than $2$ is adapted, in the sense that the Lee form of $D$ with respect to $g_0$ vanishes on the $D$-flat distribution of $M$. We also characterize adapted metrics as critical points of a natural functional defined on the conformal class.

math.DG↗
arXiv2023

Correspondence between Pestov and Weitzenböck identities

The aim of this note is to establish the correspondence between the twisted localized Pestov identity on the unit tangent bundle of a Riemannian manifold and the Weitzenböck identity for twisted symmetric tensors on the manifold.

math.DG↗
arXiv2023

Heterotic solitons on four-manifolds

We investigate four-dimensional Heterotic solitons, defined as a particular class of solutions of the equations of motion of Heterotic supergravity on a four-manifold $M$. Heterotic solitons depend on a parameter $κ$ and consist of a Riemannian metric $g$, a metric connection with skew torsion $H$ on $TM$ and a closed one-form $φ$ on $M$ satisfying a differential system. In the limit $κ\to 0$, Heterotic solitons reduce to a class of generalized Ricci solitons and can be considered as a higher-order curvature modification of the latter. If the torsion $H$ is equal to the Hodge dual of $φ$, Heterotic solitons consist of either flat tori or closed Einstein-Weyl structures on manifolds of type $S^1\times S^3$ as introduced by P. Gauduchon. We prove that the moduli space of such closed Einstein-Weyl structures is isomorphic to the product of $\mathbb{R}$ with a certain finite quotient of the Cartan torus of the isometry group of the typical fiber of a natural fibration $M\to S^1$. We also consider the associated space of essential infinitesimal deformations, which we prove to be obstructed. More generally, we characterize several families of Heterotic solitons as suspensions of certain three-manifolds with prescribed constant principal Ricci curvatures, amongst which we find hyperbolic manifolds, manifolds covered by $\widetilde{\mathrm{Sl}}(2,\mathbb{R})$ and E$(1,1)$ or certain Sasakian three-manifolds. These solutions exhibit a topological dependence in the string slope parameter $κ$ and yield, to the best of our knowledge, the first examples of Heterotic compactification backgrounds not locally isomorphic to supersymmetric compactification backgrounds.

math.DG↗
arXiv2023

On the ergodicity of unitary frame flows on Kähler manifolds

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim_{\mathbb{C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm{U}(m)$-bundle $F_{\mathbb{C}}M$ of unitary frames. We show that if $m \geq 6$ is even, and $m \neq 28$, there exists $λ(m) \in (0, 1)$ such that if $(M, g, J)$ has negative $λ(m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $λ(m)$ satisfy $λ(6) = 0.9330...$, $\lim_{m \to +\infty} λ(m) = \tfrac{11}{12} = 0.9166...$, and $m \mapsto λ(m)$ is decreasing. This extends to the even-dimensional case the results of Brin-Gromov who proved ergodicity of the unitary frame flow on negatively-curved compact Kähler manifolds of odd complex dimension.

math.DS↗
arXiv2022

Weyl-Einstein structures on conformal solvmanifolds

A conformal Lie group is a conformal manifold $(M,c)$ such that $M$ has a Lie group structure and $c$ is the conformal structure defined by a left-invariant metric $g$ on $M$. We study Weyl-Einstein structures on conformal solvable Lie groups and on their compact quotients. In the compact case, we show that every conformal solvmanifold carrying a Weyl-Einstein structure is Einstein. We also show that there are no left-invariant Weyl-Einstein structures on non-abelian nilpotent conformal Lie groups, and classify them on conformal solvable Lie groups in the almost abelian case. Furthermore, we determine the precise list (up to automorphisms) of left-invariant metrics on simply connected solvable Lie groups of dimension 3 carrying left-invariant Weyl-Einstein structures.

math.DG↗
arXiv2021

Conformal vector fields on lcK manifolds

We show that any conformal vector field on a compact lcK manifold is Killing with respect to the Gauduchon metric. Furthermore, we prove that any conformal vector field on a compact lcK manifold whose Kähler cover is neither flat, nor hyperkähler, is holomorphic.

math.DG↗
arXiv2021

Conformal Killing forms on $2$-step nilpotent Riemannian Lie groups

We study left-invariant conformal Killing $2$- or $3$-forms on simply connected $2$-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing $2$- and $3$-forms are: the Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing $2$- and $3$-forms is provided in each case.

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