Paper classifies 3D conformally flat quasi-Para-Sasakian manifolds.
problem Characterizing 3D conformally flat quasi-Para-Sasakian manifolds.
method Provided necessary and sufficient conditions for conformal flatness and characterized manifolds with η=const.
result Characterization of 3D conformally flat quasi-Para-Sasakian manifolds with η=const.
Constructs metrics with negative curvature on specific manifold types.
problem Creating negatively curved metrics on locally conformally flat manifolds.
method Using Morse functions to construct conformal metrics.
result Successfully constructs conformal metrics with negative sectional curvature.
Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
problem Characterizing projectively flat metrics on Hopf manifolds.
method Partitioning projectively flat metrics into classes based on Chern scalar curvature sign and proving properties of each class.
result Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
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.
New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.
problem Existence of essential conformal transformations in pseudo-Riemannian manifolds.
method Construction of compact locally conformally pseudo-Kähler manifolds with essential conformal transformations.
result Found compact examples of pseudo-Kähler manifolds with essential conformal transformations that are not conformally flat.
The paper examines flatness conditions on normal metric contact pairs and proves properties of Einstein manifolds.
problem The study of flatness conditions on normal metric contact pairs.
method Analysis of conformal, concircular, and quasi-conformal curvature tensors.
result Normal metric contact pair manifolds with flat conformal, concircular, and quasi-conformal curvature tensors are Einstein manifolds with specific scalar and sectional curvatures.
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when k=n/2.
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.
Study classifies half conformally flat GQE manifolds of signature (2,2).
problem Classifying half conformally flat generalized quasi-Einstein manifolds.
method Analysis and examples provided.
result Natural affine quasi-Einstein equation derived.
It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given i…
Paper connects Dirac operators and automorphic forms on conformally flat manifolds.
problem Understanding the relationship between Dirac operators and automorphic forms.
method Analyzes joint work with John Ryan on conformally flat manifolds.
result Summarizes the connection between Dirac operators and automorphic forms.
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.
Nonexistence theorem for product type manifolds, proving no locally conformally flat metrics.
problem Proving nonexistence of locally conformally flat metrics on product manifolds.
method Nonexistence theorem for product type manifolds.
result 4-manifold Σ_g×Σ_h does not admit locally conformally flat metrics for g≥2 and h≥1.
A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conf…
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.
We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.
Extended a formula to higher dimensions with singularities.
problem Generalizing a formula to higher dimensions with singularities.
method Extended a formula to all even dimensions n≥4 for a class of conformally flat manifolds with singular points.
result First formula in dimensions higher than two with isolated conical singularities.
Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.
problem Understanding actions of semisimple Lie groups on pseudo-Riemannian manifolds.
method Analyzing the pseudo-Riemannian Lichnerowicz conjecture in homogeneous settings.
result Compact pseudo-Riemannian manifolds on which a semisimple group acts conformally, essentially and transitively, are conformally flat.
In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n≥5 and with Poincarë exponent less than 2n−4, the set of conformal metrics of positive constant Q and positive …
Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.
problem Proving compactness of locally conformally flat manifolds with positive Ricci curvature.
method Using the Yamabe flow to prove compactness.
result Locally conformally flat manifolds with positive pinched Ricci curvature are compact.
A classification theorem for conformal flat AK2 manifolds is proved.
A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The prob…
A classification theorem for 4-dimensional conformally flat QK3-manifolds is proved.
Study shows mass-capacity inequality for specific geometric manifolds.
problem Establishing mass-capacity inequality for certain geometric manifolds.
method Using conformally flat manifolds with nonnegative scalar curvature.
result Equality implies harmonically conformal to a specific subset of Euclidean space.
Establishes refined singularity estimate for nonnegative n-superharmonic functions in locally conformally flat manifolds.
problem Analyzing volume growth and verifying Cohn-Vossen inequality in locally conformally flat manifolds.
method Refined singularity estimate and characterization of volume growth.
result Analytically characterizes volume growth and verifies Cohn-Vossen inequality.
New geometric variant of factorization homology for conformally flat manifolds.
problem Defining invariants of conformally flat manifolds.
method Introducing a metric-dependent geometric variant of factorization homology.
result Left Kan extensions of conformally flat d-disk algebras define invariants of conformally flat manifolds. New method constructs holonomic immersions from flat submanifolds.
problem Creating holonomic immersions from flat submanifolds.
method Ribaucour transformation and principal coordinate system.
result Holonomic immersions can be constructed using Ribaucour transformation.
In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension n≥3 is locally a warped product with (n−1)-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.
Researchers found M-eigenvalues for higher dimensional conformal flat manifolds.
problem Finding M-eigenvalues for Riemann curvature tensor in higher dimensions.
method Generalized Xiang, Qi and Wei's results to higher dimensions and provided expressions for M-eigenvalues and eigenvectors.
result M-eigenvalues uniquely determine the Riemann curvature tensor and can be complex.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a pp-wave or a warped product.
Proves Lorentzian manifold properties for analytic 3D spaces.
problem Analyzing properties of Lorentzian manifolds.
method Analyzes compact, real-analytic, three-dimensional Lorentzian manifolds.
result Identity conformal group preserves metric or manifold is flat.
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.
An identity of conformal-projective curvature tensor of a statistical manifold is studied in this paper. The relation between the constancy of curvature and conformal-projective flatness of statistical manifolds is also discussed.
New proof confirms noncompact locally conformally flat manifolds are compact.
problem Rigidity of Schouten tensor under conformal transformations.
method Proof of Cheng's theorem using modified Schouten tensor.
result Noncompact locally conformally flat manifolds are compact.
In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points …
Study of f-biharmonic hypersurfaces in conformally flat spaces.
problem Characterize f-biharmonic hypersurfaces in conformally flat spaces. method Analyze f-biharmonicity of totally umbilical hypersurfaces in various contexts. result Properties of f-biharmonic hypersurfaces in nonpositively curved manifolds. Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifo…
Compact complex manifolds with specific group actions are conformally flat.
problem Compact complex manifolds with invariant conformal holomorphic structures.
method Study of manifolds with transitive and essentially acting complex semi-simple Lie groups.
result If a complex semi-simple Lie group acts transitively and essentially, the manifold is conformally flat.
The paper finds universal inequalities for eigenvalues on hyperbolic spaces.
problem Eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds.
method Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
result Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
This paper completes globally hyperbolic conformally flat spacetimes, proving they are topological manifolds.
problem Understanding the structure of spacetimes with specific properties.
method Analyzing globally hyperbolic conformally flat spacetimes, proving their causal completions are topological manifolds.
result Causal completions of globally hyperbolic conformally flat spacetimes are topological manifolds homeomorphic to S x [0, 1].
The study improves Wintgen inequalities for submanifolds in specific geometric spaces.
problem Improving Wintgen inequalities for submanifolds in various geometric spaces.
method Analyzing submanifolds in conformally flat manifolds and deriving inequalities for different types of spaces.
result Derived inequalities for submanifolds in various geometric spaces, including Riemannian manifolds of quasi-constant curvature and warped products.
In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.
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.
Study the fractional Yamabe problem on locally flat conformal infinities of Poincaré-Einstein manifolds.
problem Fractional Yamabe problem on Poincaré-Einstein manifolds with locally flat conformal infinities.
method Algebraic topological argument of Bahri-Coron to bypass positive mass issue.
result Locally flat conformal infinities of Poincaré-Einstein manifolds admit a Riemannian metric of constant fractional scalar curvature.
The study characterizes and classifies specific types of manifolds using conformal and quasi-Einstein properties.
problem Characterizing and classifying manifolds with specific geometric properties.
method Analyzing warped products, contact manifolds, and semi-Riemannian manifolds.
result Characterizations and classifications of weakly conformally flat and quasi-Einstein manifolds.
We show that complete conformally flat manifolds of dimension n>2 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to flat space or to a spherical spaceform. This extends previous works…
Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.
problem Quantifying uniqueness of conformal-harmonic maps on 4-manifolds.
method Proves a quantitative uniqueness result using convexity and second order Hardy inequality.
result Proves a version of second order Hardy inequality on manifolds.
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold …