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Tedi Draghici

Publications and source records attributed to Tedi Draghici.

14 recordsLinked to original sources
arXiv2026

On pluricanonical locally conformally almost Kähler metrics

On an almost complex manifold $(M,J)$, a pluricanonical locally conformally almost Kähler (LCAK) metric $g$ is induced by a locally conformally symplectic structure $(F,θ)$ of the first kind, characterized by the fact that $Dθ$ is $J$-anti-invariant and that the image of the Nijenhuis tensor is $g$-orthogonal to the distribution spanned by $\{θ^\sharp,Jθ^\sharp\}$, where $θ$ is the Lee form and $D$ is the Levi-Civita connection. On a compact complex manifold, pluricanonical locally conformally Kähler (LCK) metrics have parallel Lee form. The same conclusion holds for LCK Chern--Ricci flat Gauduchon metrics. We generalize both results to LCAK metrics. We also observe that on a compact pluricanonical LCAK manifold with a non-trivial Lee form, there is no symplectic form compatible with the same almost complex structure. Moreover, we remark that the pluricanonical LCAK condition implies that the fundamental $2$-form is an eigenform of the Hodge Laplacian, and we give a simple characterization of the pluricanonical LCAK condition on compact manifolds. Finally, we study LCAK metrics with $θ^\sharp$ being real holomorphic, proving in that case $Dθ=0$ when the metric is Gauduchon.

math.DG↗
arXiv2025

On a class of almost Hermitian 4-manifolds

Using an integral identity proved by Sekigawa \cite{Sek87} on compact almost Hermitian 4-manifolds, we naturally obtain a global characterization of the class $\mathcal{AH}_1$ of almost Hermitian 4-manifolds satisfying the first Gray curvature condition from apparently weaker conditions. Then we take steps towards a classification of almost Hermitian 4-manifolds of class $\mathcal{AH}_1$, including proving a uniqueness result on 4-dimensional Lie algebras.

math.DG↗
arXiv2015

A note on tame/compatible almost complex structures on four-dimensional Lie algebras

Four-dimensional, oriented Lie algebras $\mathfrak{g}$ which satisfy the tame-compatible question of Donaldson for all almost complex structures $J$ on $\mathfrak{g}$ are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are tamed but not compatible with symplectic forms.

math.DG↗
arXiv2012

Sasakian metrics with an additional contact structure

The question of whether a Sasakian metric can admit an additional compatible (K-)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some extensions of this result are obtained, mainly in dimensions 3 and 5.

math.DG↗
arXiv2011

A note on exact forms on almost complex manifolds

Reformulations of Donaldson's "tamed to compatible" question are obtained in terms of spaces of exact forms on a compact almost complex manifold $(M^{2n},J)$. In dimension 4, we show that $J$ admits a compatible symplectic form if and only if $J$ admits tamed symplectic forms with arbitrarily given $J$-anti-invariant parts. Some observations about the cohomology of $J$-modified de Rham complexes are also made.

math.SG↗
arXiv2011

Symplectic forms and cohomology decomposition of almost complex 4-manifolds

For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by $J$-invariant, respectively, $J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of $H^2(M, \mathbb{R})$. This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.

math.SG↗
arXiv2011

On the J-anti-invariant cohomology of almost complex 4-manifolds

For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. Computations of the subgroups and their dimensions $h^{\pm}_J$ are obtained for almost complex structures related to integrable ones. We also prove semi-continuity properties for $h^{\pm}_J$.

math.SG↗
arXiv2004

The odd-dimensional Goldberg Conjecture

An odd-dimensional version of the Goldberg conjecture was formulated and proved by Boyer and Galicki, using an orbifold analogue of Sekigawa's formulas, and an approximation argument of K-contact structures with quasi-regular ones. We provide here another proof of this result and give some applications.

math.DG↗
arXiv2003

Symplectic obstructions to the existence of $ω$-compatible Einstein metrics

It is shown that the existence of an $ω$-compatible Einstein metric on a compact symplectic manifold $(M,ω)$ imposes certain restrictions on the symplectic Chern numbers. Examples of symplectic manifolds which do not satisfy these restrictions are given. The results offer partial support to a conjecture of Goldberg.

math.DG↗
arXiv2002

Local models and integrability of certain almost Kahler 4-manifolds

We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure.

math.DG↗
arXiv2001

Local rigidity of certain classes of almost Kahler 4-manifolds

We show that any non-Kahler, almost Kahler 4-manifold for which both the Ricci and the Weyl curvatures have the same algebraic symmetries as they have for a Kahler metric is locally isometric to the (only) proper 3-symmetric 4-dimensional space described by O. Kowalski.

math.DG↗
arXiv2001

A splitting theorem for Kahler manifolds whose Ricci tensors have constant eigenvalues

It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples of any complex dimension n > 1, if one eigenvalue is negative and the other positive or zero, and of any complex dimension n > 2, if the both eigenvalues are negative; there are non-homogeneous examples of complex dimension 2, if one of the eigenvalues is zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct, constant eigenvalues is related to the celebrated (still open) Goldberg conjecture. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete, Einstein, strictly almost Kahler metrics of any even real dimension greater than 4.

math.DG↗