arXiv subjects

Florin Belgun

Publications and source records attributed to Florin Belgun.

13 recordsLinked to original sources

arXiv2025

Projective structures on curves and conformal geometry

Projective structures on curves appear naturally in many areas of mathematics, from extrinsic conformal geometry to analysis, where the main problem is to find qualitative information about the solutions of Hill equations. In this paper, we describe in detail the correspondence between different equivalent definitions of projective structures and their isomorphism classes, correcting long-standing inexactitudes in the literature. As an application, we show that the {\em Yamabe problem for curves} in a conformal/Möbius ambient space has no solutions in general.

math.DG↗

arXiv2025

Weyl structures with special holonomy on compact conformal manifolds

We consider compact conformal manifolds $(M,[g])$ endowed with a closed Weyl structure $\nabla$, i.e. a torsion-free connection preserving the conformal structure, which is locally but not globally the Levi-Civita connection of a metric in $[g]$. Our aim is to classify all such structures when both $\nabla$ and $\nabla^g$, the Levi-Civita connection of $g$, have special holonomy. In such a setting, $(M,[g],\nabla)$ is either flat, or irreducible, or carries a locally conformally product (LCP) structure. Since the flat case is already completely classified, we focus on the last two cases. When $\nabla$ has irreducible holonomy we prove that $(M,g)$ is either Vaisman, or a mapping torus of an isometry of a compact nearly Kähler or nearly parallel $\mathrm{G}_2$ manifold, while in the LCP case we prove that $g$ is neither Kähler nor Einstein, thus reducible by the Berger-Simons Theorem, and we obtain the local classification of such structures in terms of adapted metrics.

math.DG↗

arXiv2018

Left-invariant Einstein metrics on $S^3 \times S^3$

The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \mathrm{SU}(2) \times \mathrm{SU}(2) = S^3 \times S^3$. Einstein metrics are critical points of the total scalar curvature functional for fixed volume. The scalar curvature $S$ of a left-invariant metric $g$ is constant and can be expressed as a rational function in the parameters determining the metric. The critical points of $S$, subject to the volume constraint, are given by the zero locus of a system of polynomials in the parameters. In general, however, the determination of the zero locus is apparently out of reach. Instead, we consider the case where the isotropy group $K$ of $g$ in the group of motions is non-trivial. When $K\not\cong \mathbb{Z}_2$ we prove that the Einstein metrics on $G$ are given by (up to homothety) either the standard metric or the nearly Kähler metric, based on representation-theoretic arguments and computer algebra. For the remaining case $K\cong \mathbb{Z}_2$ we present partial results.

math.DG↗

arXiv2018

Locally conformally symplectic convexity

We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemin-Sternberg. We also prove similar results for the symplectic moment map (defined on the minimal presentation) whose image is then a convex cone. In the special case of a compact toric Vaisman manifold, we obtain a structure theorem.

math.DG↗

arXiv2017

On the irreducibility of locally metric connections

A locally metric connection on a smooth manifold $M$ is a torsion-free connection $D$ on $TM$ with compact restricted holonomy group $\mathrm{Hol}_0(D)$. If the holonomy representation of such a connection is irreducible, then $D$ preserves a conformal structure on $M$. Under some natural geometric assumption on the life-time of incomplete geodesics, we prove that conversely, a locally metric connection $D$ preserving a conformal structure on a compact manifold $M$ has irreducible holonomy representation, unless $\mathrm{Hol}_0(D)=0$ or $D$ is the Levi-Civita connection of a Riemannian metric on $M$. This result generalizes Gallot's theorem on the irreducibility of Riemannian cones to a much wider class of connections. As an application, we give the geometric description of compact conformal manifolds carrying a tame closed Weyl connection with non-generic holonomy.

math.DG↗

arXiv2014

Geodesics and Submanifold Structures in Conformal Geometry

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called Möbius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of Möbius and Laplace structures to the case of surfaces and curves, we develop here the theory of extrinsic conformal geometry for submanifolds, find tensorial invariants of a conformal embedding, and use these invariants to characterize various forms of geodesic submanifolds.

math.DG↗

arXiv2014

On the boundary behaviour of left-invariant Hitchin and hypo flows

We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$, respectively, which are in general geodesically incomplete. Generalizing results of Conti, we prove that for large classes of solvable Lie groups $G$ these manifolds cannot be completed: a complete Riemannian manifold with parallel $SU(3)$-, $G_2$- or $Spin(7)$-structure which is of cohomogeneity one with respect to $G$ is flat, and has no singular orbits. We furthermore classify, on the non-compact Lie group $SL(2,C)$, all half-flat $SU(3)$-structures which are bi-invariant with respect to the maximal compact subgroup $SU(2)$ and solve the Hitchin flow for these initial values. It turns out that often the flow collapses to a smooth manifold in one direction. In this way we recover an incomplete cohomogeneity-one Riemannian metric with holonomy equal to $G_2$ on the twisted product $SL(2,C)\times_{SU(2)} C^2$ described by Bryant and Salamon.

math.DG↗

arXiv2012

On the metric structure of some non-Kähler complex threefolds

We introduce a class of hermitian metrics with {\em Lee potential}, that generalize the notion of l.c.K. metrics with potential introduced in \cite{ov} and show that in the classical examples of Calabi and Eckmann of complex structures on $S^{2p+1}\x S^{2q+1}$, the corresponding hermitian metrics are of this type. These examples satisfy, actually, a stronger differential condition, that we call {\em generalized Calabi-Eckmann}, condition that is satisfied also by the {\em Vaisman} metrics (previously also refered to as {\em generalized Hopf manifolds}). This condition means that, in addition to being with Lee potential, the torsion of the {\em characteristic} (or Bismut) connection is parallel. We give a local geometric characterization of these generalized Calabi-Eckmann metrics, and, in the case of a compact threefold, we give detailed informations about their global structure. More precisely, the cases which can not be reduced to Vaisman structures can be obtained by deformation of locally homogenous hermitian manifolds that can be described explicitly.

math.DG↗

arXiv2010

Essential points of conformal vector fields

For a conformal vector field $ξ$ on a Riemannian manifold, we say that a point is essential if there is no local metric in the conformal class for which $ξ$ is Killing. We show that the only essential points are isolated zeros of $ξ$. As an application, we show that every connected component of the zero set of $ξ$ is totally umbilical.

math.DG↗

arXiv2010

Weyl-parallel forms, conformal products and Einstein-Weyl manifolds

Motivated by the study of Weyl structures on conformal manifolds admitting parallel weightless forms, we define the notion of conformal product of conformal structures and study its basic properties. We obtain a classification of Weyl manifolds carrying parallel forms, and we use it to investigate the holonomy of the adapted Weyl connection on conformal products. As an application we describe a new class of Einstein-Weyl manifolds of dimension 4.

math.DG↗

arXiv2006

A Singularity Theorem for Twistor Spinors

We study spin structures on orbifolds. In particular, we show that if the singular set has codimension greater than 2, an orbifold is spin if and only if its smooth part is. On compact orbifolds, we show that any non-trivial twistor spinor admits at most one zero which is singular unless the orbifold is conformally equivalent to a round sphere. We show the sharpness of our results through examples.

math.DG↗

arXiv2004

Killing Forms on Symmetric Spaces

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew--symmetric. We show that a compact simply connected symmetric space carries a non--parallel Killing $p$--form ($p\ge2$) if and only if it isometric to a Riemannian product $S^k\times N$, where $S^k$ is a round sphere and $k>p$.

math.DG↗

arXiv2002

Symmetries of Contact Metric Manifolds

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo--Riemannian quaternion--Kaehler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that non--regular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds.

math.DG↗