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Luigi Vezzoni

Publications and source records attributed to Luigi Vezzoni.

64 records · Page 4Linked to original sources
arXiv2009

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group $H_{2n + 1}$. Furthermore, we classify Sasakian Lie algebras of dimension 5 and determine which of them carry a Sasakian $α$-Einstein structure. We show that a 5-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either $H_5$ or a semidirect product $\R \ltimes (H_3 \times \R)$. In particular, the compact quotient is an $S^1$-bundle over a 4-dimensional Kähler solvmanifold.

math.DG↗
arXiv2008

Quasi-Kähler manifolds with trivial Chern Holonomy

In this paper we study almost complex manifolds admitting a quasi-Kähler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kähler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.

math.DG↗
arXiv2008

Gray identities, canonical connection and integrability

We characterize quasi Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames introduced by the second author previously. By using such frames we also give a simpler and shorter proof of a Theorem of Goldberg. Furthermore we study almost Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

math.DG↗
arXiv2008

On Symplectc half-flat manifolds

We construct examples of symplectic half-flat manifolds on compact quotients of solvable Lie groups. We prove that the Calabi-Yau structures are not rigid in the class of symplectic half-flat structures. Moreover, we provide an example of a compact 6-dimensional symplectic half-flat manifold whose real part of the complex volume form is d-exact. Finally we discuss the 4-dimensional case.

math.SG↗
arXiv2007

Contact Calabi-Yau manifolds and Special Legendrian submanifolds

We consider a generalization of Calabi-Yau structures in the context of $α$-Sasakian manifolds. We study deformations of a special class of Legendrian submanifolds and classify invariant contact Calabi-Yau structures on 5-dimensional nilmanifolds. Finally we generalize to codimension $r$.

math.DG↗
arXiv2007

Torsion of SU(2)-structures and Ricci curvature in dimension 5

In anlogy with the work of R. Bryant on the Ricci tensor of a G$_2$-structure, we study the intrinsic torsion of an SU$(2)$-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove that the $α$-Einstein condition forces some special SU(2)-structures to be Sasaki-Einstein.

math.DG↗
arXiv2007

On Special Calibrated Almost Complex Structures and Moduli Space

An \emph{$ω$-admissible almost complex structure} on a $2n$-dimensional symplectic manifold $(M,ω)$ is a $ω$-calibrated almost complex structure $J$ admitting a nowhere vanishing $\bar{\partial}_J$-closed $(n,0)$-form $ψ$. After giving some examples we consider the moduli space of admissible almost complex structures and we study infinitesimal deformations. As special case, we write down explicit computations for the complex torus.

math.SG↗
arXiv2006

The Ricci tensor of SU(3)-manifolds

Following the approach of Bryant we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.

math.DG↗