arXiv research

Study finds all 4D neutral manifolds.

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

The paper is devoted to the investigation of four-dimensional Kahler manifolds admitting non-affine H-projective mappings. We find all such manifolds which are non-Einstein. In the paper also Kahler manifolds admitting infinitesimal H-projective transformations are determined. It is proved that the class of Kahler mani…

The paper classifies homogeneous hypersurfaces in three 4D Thurston geometries.

In this paper, we investigate geometric properties of some curvature tensors of a four-dimensional Walker manifold. Some characterization theorems are also obtained.

Critical metrics on four-dimensional manifolds are either Einstein or product of two-dimensional manifolds.

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or $\CH_3$ for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, $\CH_2$ manifolds that are not homogeneous.

In this paper, a lot of examples of four-dimensional manifolds with an almost hypercomplex pseudo-Hermitian structure are constructed in several explicit ways. The received 4-manifolds are characterized by their linear invariants in the known aspects.

We study curvature properties of four-dimensional almost Hermitian manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke. We give local structure theorems for such Kaehler manifolds, and find out several examples related to the theorems.

An example of a four-dimensional special complex manifold with Norden metric of constant holomorphic sectional curvature is constructed via a two-parametric family of solvable Lie algebras. The curvature properties of the obtained manifold are studied. Necessary and sufficient conditions for the manifold to be isotropi…

In this short note we prove that any complete four dimensional anti-self-dual (or self-dual) quasi-Einstein manifolds is either Einstein or locally conformally flat. This generalizes a recent result of X. Chen and Y. Wang.

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends…

We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or CH_3 for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, CH_2 manifolds that are not homogeneous. The resulting metrics belong to the class of null electromagnetic radia…

We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. First, we write out form…

A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not admit compact four-dimensional quotients. It also shows that there are solvable g…

Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to the standard 4-sphere.

We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.

The paper studies Einstein-like Walker metrics in Walker manifolds.

Study of Brown--York mass for four-dimensional asymptotically flat manifolds.

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are is…

Survey on four-dimensional Thurston geometries with Riemannian metrics.

Using techniques from the theory of Kirby calculus we give an explicit construction of a four dimensional hyperbolic link complement in a 4-manifold that is diffeomorphic to a standard $S^2 \times S^2$.

We construct invariants of four-dimensional piecewise-linear manifolds, represented as simplicial complexes, with respect to rebuildings that transform a cluster of three 4-simplices having a common two-dimensional face in a different cluster of the same type and having the same boundary. Our construction is based on t…

Proves K-K-W type theorems for specific types of operators.

Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.

The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.

A classification theorem for 4-dimensional conformally flat QK3-manifolds is proved.

In this paper we prove $L^{\infty}$ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…

The four-dimensional sphere is uniquely rigid in terms of scalar curvature.

We establish the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some four dimensional Stein manifolds.

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},\,1].$ In addition, we use the notion of biorthogonal (…

We write out some sequences of linear maps of vector spaces with fixed bases. Each term of a sequence is a linear space of differentials of metric values ascribed to the elements of a simplicial complex - a triangulation of a manifold. If the sequence turns out to be an acyclic complex then one can construct a manifold…

Survey on knot theory's impact on four-dimensional topology.

We classify and investigate locally conformally Kähler structures on four-dimensional solvable Lie algebras up to linear equivalence. As an application we can produce many examples in higher dimension, here including lcK structures on Oeljeklaus-Toma manifolds, and we also give a geometric interpretation of some of the…

In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups…

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.

We obtain an equivariant classification for orientable, closed, four-dimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed four-dimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivari…

We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isom…

In this paper we consider three deeply connected classificational problems on four-dimensional manifolds. First we consider and describe locally regular distributions. Second we give a classification of almost complex structures of general position in terms of distributions. Finally we classify nondegenerate Monge-Ampe…

We extend an $L^2$ energy gap result due independently to Min-Oo and Parker (1982) for Yang-Mills connections on principal $G$-bundles, $P$, over closed, connected, four-dimensional, oriented, smooth manifolds, $X$, from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, includ…

We show that every closed symplectic four-dimensional manifold admits compatible almost Kaehler metrics of negative scalar curvature.

Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection $A$ on a line bundle $L$ and a section $φ$ of another bundle $W^+$ constructed from $L$ and a spinor bundle on a given four-dimensional Riem…

In this paper we study the invariant Walker structures over the conformally flat four-dimensional homogeneous manifolds according to the Seger types of the Ricci operator.

This is the first in a series of papers where we will derive invariants of three-manifolds and framed knots in them from the geometry of a manifold pseudotriangulation put in some way in a four-dimensional Euclidean space. Thus, the elements of the pseudotriangulation acquire Euclidean geometric values such as volumes …

We study curvature properties of four-dimensional Lorentzian manifold with two-symmetry property. We then consider Einstein-like metrics, Ricci solitons and homogeneity over these spaces.

The second Betti number of a smooth, closed, connected and simply connected, four-dimensional spin manifold is greater or equal 11/8 times the abolute value of its signature.