Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
problem Understanding the rigidity of Ricci-flat manifolds with specific curvature decay.
method Analyzing the gradient of the Green function and using curvature decay conditions.
result Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
Almost-flat manifolds were defined by Gromov as a natural generalisation of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental g…
Study connects G2-structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of G2-structures and flat connections. method Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. result Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. Study describes flat metric moduli spaces on 4D manifolds.
problem Understanding flat metrics on 4D closed manifolds.
method Algebraic and topological description of moduli spaces.
result Algebraic and topological description of moduli spaces of flat metrics.
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by col…
Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.
problem Understanding scalar-flat Kahler 4-manifolds with a Killing field.
method Analysis of manifolds with a Killing field and asymptotic conditions.
result Rigidity results that restrict the behavior of scalar-flat Kahler manifolds at infinity.
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
Study on flat metrics on 3D and 4D manifolds, focusing on topology and algebra.
problem Topology and algebraic structure of flat metrics on manifolds.
method Algebraic and topological descriptions of moduli spaces.
result Algebraic description and topology of moduli spaces for 4D manifolds with a single holonomy generator.
Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.
problem Estimating ∂-operators for flat line bundles. method Uniform L2-estimates for ∂-operators on Kähler manifolds. result Recovers Ueda's lemma for compact Kähler manifolds and generalizes to Ricci-flat manifolds.
We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler met…
Conditions for flat 3-manifolds with diagonal metrics are identified.
problem Characterizing flat 3-manifolds with diagonal metrics.
method Provided necessary and sufficient conditions for flatness.
result Characterized flat manifolds of warped product-type.
The paper explores F-manifolds and metrics, constructing canonical structures.
problem Understanding relationships between F-manifolds and metrics.
method Construction of canonical flat F-manifolds and homogeneous Riemannian F-manifolds.
result Construction of a canonical flat F-manifold associated to an arbitrary Riemannian F-manifold.
Complete classification of Hermitian manifolds with flat Gauduchon connections.
problem Classifying compact Hermitian manifolds with flat Gauduchon connections.
method Analyzing properties of Hermitian manifolds and using Gauduchon connections.
result Established a conjecture about Kähler-like conditions and flatness.
The study finds monotonic properties of harmonic functions on 3-manifolds with a flat end.
problem Understanding harmonic functions on 3-manifolds with specific ends.
method Derives monotonic properties of positive harmonic functions on 3-manifolds with nonnegative scalar curvature and asymptotically flat ends.
result Rigidity characterization of spatial Schwarzschild manifolds with two ends.
Investigates flat bundles over low-dimensional manifolds and their cobordism classes.
problem The cobordism of flat bundles over low-dimensional manifolds.
method Study of flat M-bundles over low-dimensional manifolds, comparing a finite dimensional Lie group G with extDiff0(G) and localizing the holonomy. result Flat M-bundles over low-dimensional manifolds are cobordant to a flat M-bundle. Proves non-existence of certain flat manifolds.
problem Non-existence of asymptotically flat 4-manifolds.
method Analyzes conjecture of Petrunin and Tuschmann.
result Proves conjecture on non-existence of asymptotically flat 4-manifolds.
Study SKT and CYT manifolds with parallel Bismut torsion.
problem Characterize and construct compact complex manifolds with specific geometric properties.
method Characterization via universal cover, construction using mapping torus, investigation of generalized Kaehler structures.
result Existence of non-Bismut flat examples and characterization of universal covers.
The theory of flat Pseudo-Riemannian manifolds and flat affine manifolds is closely connected to the topic of prehomogeneous affine representations of Lie groups. In this article, we exhibit several aspects of this correspondence. At the heart of our presentation is a development of the theory of characteristic classes…
Flat coordinates found for algebraic Frobenius manifolds in low dimensions.
problem Understanding algebraic Frobenius manifolds in small dimensions.
method Using reflection representations of finite Coxeter groups, finding flat coordinates of the Frobenius metric.
result Explicit relations between flat coordinates of the Frobenius metric and intersection form for most known examples up to dimension 4.
The study calculates harmonic functions and 1-forms on specific 4D spaces.
problem Computing harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
method Computed the expansion of harmonic functions and 1-forms.
result Computed the expansion of harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds.
The paper classifies special types of contact metric manifolds with curvature conditions.
problem Classifying N(κ)-contact metric manifolds with specific curvature tensors. method Examining flatness conditions on T-curvature tensor and analyzing specific curvature tensors. result A classification of N(κ)-contact metric manifolds under various curvature conditions. Introduce new boundary mass for asymptotically flat half-manifolds
problem Define boundary mass for asymptotically flat half-manifolds
method Introduce new boundary mass
result Define boundary mass for asymptotically flat half-manifolds
In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional cur…
Statistical manifolds with constant curvature are projectively flat and symmetric.
problem Characterizing statistical manifolds with constant curvature.
method Analyzing the curvature and projective flatness properties of statistical manifolds.
result Statistical manifolds with constant curvature are projectively flat and symmetric.
Compact 3D Cotton-parallel manifolds are always conformally flat.
problem Understanding the properties of compact 3D Cotton-parallel manifolds.
method Analyzing the Cotton tensor and its parallelism condition.
result Compact 3D Cotton-parallel manifolds are conformally flat.
The object of the present paper is to study 3-dimensional conformally flat quasi-Para-Sasakian manifolds. First, the necessary and sufficient conditions are provided for 3-dimensional quasi-Para-Sasakian manifolds to be conformally flat. Next, a characterization of 3-dimensional conformally flat quasi-Para-Sasakian man…
In this paper we study F-manifolds equipped with multiple flat connections (and multiple F-products), that are required to be compatible in a suitable sense. In the semisimple case we show that a necessary condition for the existence of such multiple flat connections can be expressed in terms of the integrability o…
Study on Gauduchon manifolds finds metrics for projectively flat bundles.
problem Existence of Hermitian-Poisson metrics on projectively flat bundles.
method Heat flow techniques and continuity methods.
result Established a correspondence between Hermitian-Poisson metrics and semi-simplicity.
This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.
problem Classifying fibrations of compact flat orbifolds.
method Developed theory for classifying fibrations up to affine equivalence.
result Classified fibrations of compact flat 2-orbifolds.
We present an algorithmic approach to the problem of existence of spin structures on flat manifolds. We apply our method in the cases of flat manifolds of dimensions 5 and 6.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
Classification of specific pseudo-Riemannian manifolds.
problem Classifying conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds.
method Complete classification through mathematical analysis.
result Conformally flat generalized Ricci recurrent pseudo-Riemannian manifolds are either de Sitter or anti-de Sitter spacetimes.
We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact G2 holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive ex…
We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.
Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.
problem None explicitly stated; focuses on description of global sections.
method Complete description of vertex algebra of global sections.
result Complete description of vertex algebra of global sections of chiral de Rham complex.
4 flat 3-manifolds realized in hyperbolic 4-space.
problem Realizing flat 3-manifolds in hyperbolic 4-space.
method Using finite-volume hyperbolic 4-manifolds with cusp sections.
result 6 closed orientable flat 3-manifolds realized as cusp sections.
We study closed non-positively curved Riemannian manifolds M which admit `fat k-flats': that is, the universal cover M~ contains a positive radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection …
Kunneth formula derived for flat affine manifolds and applied to Hessian metrics.
problem Cohomology of flat affine manifolds and Hessian metrics.
method Proved Künneth formula for Dolbeault--Koszul cohomology of flat affine manifolds and applied to Hessian metrics.
result Kunneth formula for Hessian metrics on products and manifolds with hyperbolic manifolds.
The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
problem Proving rigidity for Poincaré-Einstein manifolds with specific conformal infinity.
method Analyzing curvature tensors over level sets of a boundary defining function.
result Rigidity theorem for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
Study gap phenomenon in flat manifolds with Ricci curvature.
problem Understanding curvature decay in flat manifolds.
method Construct solutions to Yamabe flow and analyze curvature decay.
result If curvature decays quickly, manifold must be flat.
New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.
problem Existence of parallel spinors on Ricci-flat manifolds.
method Generalization of existing results to manifolds with non-vanishing Rosenberg index.
result Every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy.
In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat 4-manifolds with curvature decay and controlled holonomy. As a…
Flat space for manifolds with tiny curvature.
problem Understanding manifolds with curvature concentration.
method Analyzing non-compact manifolds with non-negative Ricci curvature and small curvature concentration.
result Manifolds with curvature concentration are flat.