arXiv subjects

Andrzej Derdzinski

Publications and source records attributed to Andrzej Derdzinski.

At least 19 recordsLinked to original sources

arXiv2026

Almost-product Kähler surfaces

We study the case where the tangent bundle of a Kähler surface splits orthogonally into two integrable complex-line subbundles. This amounts to the presence of a unit-length closed anti-self-dual 2-form, leading in turn to a curvature condition. Therefore, such a decomposition need not exist, even locally, in contrast with general Riemannian four-manifolds which, under the assumption of real-analyticity, were shown by Grant and Vickers [11] to always admit, locally, two mutually orthogonal integrable rank-two distributions. We exhibit local coordinates naturally adapted to a decomposition as above in a Kähler surface, which may be used for simple constructions of examples. We also provide a characterization of the Kähler case within the wider class of almost-Kähler surfaces arising from the coordinates just mentioned. The characterization involves a system of four first-order quasilinear partial differential equations imposed on four unknown functions of four variables and, using Cartan's test, we prove the system's local solvability. Finally, we show that, for such a decomposition in a Kähler surface, one of the summands is holomorphic if and only if the other one has totally geodesic leaves, and describe a general construction of examples with this last property.

math.DG↗

arXiv2026

Scalar-flat Kähler surfaces whose Weyl tensor annihilates the Ricci form

We conjecture that any scalar-flat Kähler surface in which the Weyl tensor acting on 2-forms annihilates the Ricci form must be either Ricci-flat or locally isometric to a Riemannian product of two real surfaces with mutually opposite nonzero constant Gaussian curvatures. This amounts to the nonexistence of proper weakly Einstein anti-self-dual Kähler surfaces. We prove the above conjecture in three special cases: when the manifold is compact, when one of the Ricci eigendistributions is integrable, and when the norms of the Ricci and Weyl tensors are functionally dependent

math.DG↗

arXiv2026

Weakly Einstein curvature tensors

We classify weakly Einstein algebraic curvature tensors in an oriented Euclidean 4-space, defined by requiring that the three-index contraction of the curvature tensor against itself be a multiple of the inner product. This algebraic formulation parallels the geometric notion of weakly Einstein Riemannian four-manifolds, which include conformally flat scalar-flat, and Einstein manifolds. Our main result provides a complete classification of non-Einstein weakly Einstein curvature tensors in dimension four, naturally dividing them into three disjoint five-dimensional families of algebraic types. These types are explicitly constructed using bases that simultaneously diagonalize both the Einstein tensor and the (anti)self-dual Weyl tensors, which consequently proves that such simultaneous diagonalizability follows from the weakly Einstein property. We also point out that our classification has immediate applications, and describe how some known geometric examples that are neither Einstein, nor conformally flat scalar-flat (namely, the EPS space and certain Kähler surfaces) fit within our classification framework.

math.DG↗

arXiv2025

Weakly Einstein conformal products

One says that a Riemannian four-manifold is \emph{weakly Einstein} if the three-index contraction of its curvature tensor against itself equals a function times the metric. Since this includes all four-manifolds that are Einstein, or conformally flat and scalar-flat, the term \emph{proper} may be used for weakly Einstein manifolds (or metrics) not belonging to the latter two classes. We establish two classification-type results about proper weakly Einstein metrics conformal to Riemannian products. This includes constructions of new examples, among them -- some of (local) cohomogeneity two, in contrast with the two previously known narrow classes of examples, having cohomogeneity zero and one. We also exhibit a simple coordinate description of one of the known examples, the EPS space, which shows that it is a conformal product and constitutes a single local-homothety type. Finally, we prove that there exist no proper weakly Einstein manifolds with harmonic curvature.

math.DG↗

arXiv2025

On weakly Einstein Kähler surfaces

Riemannian four-manifolds in which the triple contraction of the curvature tensor against itself yields a functional multiple of the metric are called weakly Einstein. We focus on weakly Einstein Kähler surfaces. We provide several conditions characterizing those Kähler surfaces which are weakly Einstein, classify weakly Einstein Kähler surfaces having some specific additional properties, and construct new examples.

math.DG↗

arXiv2025

Affine vector fields on compact pseudo-Kähler manifolds

It is known that a Killing field on a compact pseudo-Kähler manifold is necessarily (real) holomorphic, as long as the manifold satisfies some relatively mild additional conditions. We provide two further proofs of this fact and discuss the natural open question whether the same conclusion holds for affine -- rather than Killing -- vector fields. The question cannot be settled by invoking the Killing case: Boubel and Mounoud [Trans.Amer. Math. Soc. 368, 2016, 2223--2262] constructed examples of non-Killing affine vector fields on compact pseudo-Riemannian manifolds. We show that an affine vector field v is necessarily symplectic, and establish some algebraic and differential properties of the Lie derivative of the metric along v, such as its being parallel, antilinear and nilpotent as an endomorphism of the tangent bundle. As a consequence, the answer to the above question turns out to be `yes' whenever the underlying manifold admits no nontrivial holomorphic quadratic differentials, which includes the case of compact almost homogeneous complex manifolds with nonzero Euler characteristic.

math.DG↗

arXiv2025

Parallel differential forms of codegree two, and three-forms in dimension six

For a differential form on a manifold, having constant components in suitable local coordinates trivially implies being parallel relative to a torsion-free connection, and the converse implication is known to be true for $p$-forms in dimension $n$ when $p=0,1,2,n-1,n$. We prove the converse for $(n-2)$-forms, and for 3-forms when $n=6$, while pointing out that it fails to hold for Cartan 3-forms on all simple Lie groups of dimensions $n\ge8$ as well as for $(n,p)=(7,3)$ and $(n,p)=(8,4)$, where the 3-forms and 4-forms arise in compact simply connected Riemannian manifolds with exceptional holonomy groups. We also provide geometric characterizations of 3-forms in dimension six and $(n-2)$-forms in dimension $n$ having the constant-components property mentioned above, and describe examples illustrating the fact that various parts of these geometric characterizations are logically independent.

math.DG↗

arXiv2024

Special Ricci-Hessian equations on Kähler manifolds

Special Ricci-Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [Ann. Global Anal. Geom. 34 (2008), 367--380] involve functions $τ$ on $M$ and state that, for some function $α$ of the real variable $τ$, the sum of $α\nabla dτ$ and the Ricci tensor equals a functional multiple of the metric $g$, while $α\nabla dτ$ itself is assumed to be nonzero almost everywhere. Three well-known obvious ``standard'' cases are provided by (non-Einstein) gradient Kähler-Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler-Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $α=2\cotτ$, or $α=2\cothτ$, or $α=2\tanhτ$. We also use the Cartan-Kähler theorem to prove that these three types are actually realized in a ``nonstandard'' way.

math.DG↗

arXiv2024

Nijenhuis geometry of parallel tensors

A tensor -- meaning here a tensor field $Θ$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential $q$-forms, $q=0,1,2,n-1,n$ (in dimension $n$), vectors, bivectors, symmetric $(2,0)$ and $(0,2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type $(1,1)$. In most cases, integrability is equivalent to algebraic constancy of $Θ$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on $Θ$ via a quasilinear first-order differential operator. For $(p,q)=(1,1)$, they include the ordinary Nijenhuis tensor.

math.DG↗

arXiv2024

Compact locally homogeneous manifolds with parallel Weyl tensor

We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak. Previously known examples of compact ECS manifolds, in every dimension greater than 4, were all of rank 1, geodesically complete, and none of them locally homogeneous. By contrast, our new examples -- all of them geodesically incomplete -- realize all odd dimensions starting from 5 and are this time of rank 2, as well as locally homogeneous.

math.DG↗

arXiv2024

Corrections of minor misstatements in several papers on ECS manifolds

In the paper [3] the value of the invariant denoted by d and often called "rank" was misstated for a narrow class of examples of ECS manifolds: they were identified as having d = 1 instead of the correct value d = 2. The error was repeated, by citing [3], in [1], [2] and [4] -- [7]. We briefly describe the class in question, explain why it has d = 2, and list the required corrections of the affected papers.

math.DG↗

arXiv2024

Killing fields on compact pseudo-Kähler manifolds

We show that a Killing field on a compact pseudo-Kähler ddbar manifold is necessarily (real) holomorphic. Our argument works without the ddbar assumption in real dimension four. The claim about holomorphicity of Killing fields on compact pseudo-Kähler manifolds appears in a 2012 paper by Yamada, and in an appendix we provide a detailed explanation of why we believe that Yamada's argument is incomplete.

math.DG↗

arXiv2023

Harmonic curvature in dimension four

We provide a step towards classifying Riemannian four-manifolds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs to one of five otherwise-familiar classes of examples. The main result consists in showing that, if such a manifold (not necessarily compact or even complete) lies outside of the five classes -- a non-vacuous assumption -- then, at all points of a dense open subset, Ric has four distinct eigenvalues, while suitable local coordinates simultaneously diagonalize Ric, the metric and, in a natural sense, also the curvature tensor. Furthermore, in a local orthonormal frame formed by Ricci eigenvectors, the connection form (or, curvature tensor) has just twelve (or, respectively, six) possibly-nonzero components, which together satisfy a specific system, not depending on the point, of homogeneous polynomial equations. A part of the classification problem is thus reduced to a question in real algebraic geometry.

math.DG↗

arXiv2023

Compact flat manifolds and reducibility

Hiss and Szczepański proved in 1991 that the holonomy group of any compact flat Riemannian manifold, of dimension at least two, acts reducibly on the rational span of the Euclidean lattice associated with the manifold via the first Bieberbach theorem. Geometrically, their result states that such a manifold must admit a nonzero proper parallel distribution with compact leaves. We study algebraic and geometric properties of the sublattice-spanned holonomy-invariant rational vector subspaces that exist due to the above theorem, and of the resulting compact-leaf foliations of compact flat manifolds. The class consisting of the former subspaces, in addition to being closed under spans and intersections, also turns out to admit (usually nonorthogonal) complements. As for the latter foliations, we provide descriptions, first -- and foremost -- of the intrinsic geometry of their generic leaves in terms of that of the original flat manifold and, secondly -- as an essentially obvious afterthought -- of the leaf-space orbifold. The general conclusions are then illustrated by examples in the form of generalized Klein bottles.

math.DG↗

arXiv2023

Compact Weyl-parallel manifolds

By ECS manifolds one means pseudo-Riemannian manifolds of dimensions $\,n\ge4\,$ which have parallel Weyl tensor, but not for one of the two obvious reasons: conformal flatness or local symmetry. As shown by Roter [10, 2], they exist for every $\,n\ge4$, and their metrics are always indefinite. The local structure of ECS manifolds has been completely described [3]. Every ECS manifold has an invariant called rank, equal to 1 or 2. Known examples of compact ECS manifolds [4, 6], representing every dimension $\,n\ge5$, are of rank 1. When $\,n\,$ is odd, some further, recently found examples are locally homogeneous [7]. We outline the proof of the author's result, joint with Ivo Terek [5], which states that a compact rank-one ECS manifold, if not locally homogeneous, replaced if necessary by a two-fold isometric covering, must be the total space of a bundle over the circle.

math.DG↗

arXiv2023

The metric structure of compact rank-one ECS manifolds

Pseudo-Riemannian manifolds with nonzero parallel Weyl tensor which are not locally symmetric are known as ECS manifolds. Every ECS manifold carries a distinguished null parallel distribution $\mathcal{D}$, the rank $d \in \{ 1, 2 \}$ of which is referred to as the rank of the manifold itself. Under a natural genericity assumption on the Weyl tensor, we fully describe the universal coverings of compact rank-one ECS manifolds. We then show that any generic compact rank-one ECS manifold must be translational, in the sense that the holonomy group of the natural flat connection induced on $\mathcal{D}$ is either trivial or isomorphic to $\mathbb{Z}_2$. We also prove that all four-dimensional rank-one ECS manifolds are noncompact, this time without assuming genericity, as it is always the case in dimension four.

math.DG↗

arXiv2023

The topology of compact rank-one ECS manifolds

Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as ECS manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the dimension of a certain distinguished null parallel distribution $\,\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a two-fold isometric covering, must be a bundle over the circle with leaves of $\,\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally-homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold the leaves of $\,\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.

math.DG↗

arXiv2023

Rank-one ECS manifolds of dilational type

We study ECS manifolds, that is, pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak, and a rank-one ECS manifold may be called translational or dilational, depending on whether the holonomy group of a natural flat connection in the Olszak distribution is finite or infinite. Some such manifolds are in a natural sense generic, which refers to the algebraic structure of the Weyl tensor. Known examples of compact ECS manifolds, in every dimension greater than 4, are all of rank 1 and translational, some of them generic, none of them locally homogeneous. As we show, generic compact rank-one ECS manifolds must be translational or locally homogeneous, provided that they arise as isometric quotients of a specific class of explicitly constructed "model" manifolds. This result is relevant since the clause starting with "provided that" may be dropped: according to a theorem which we prove in a forthcoming paper, the models just mentioned include the isometry types of the pseudo-Riemannian universal coverings of all generic compact rank-one ECS manifolds.

math.DG↗