arXiv research

Extended a formula to higher dimensions with singularities.

The study proves constraints on the structure of compact half-conformally flat manifolds.

To each flat conformal structure (FCS) of hyperbolic type in the sense of Kulkarni-Pinkall, we associate, for all $θ\in[(n-1)π/2,nπ/2[$ and for all $r>\opTan(θ/n)$ a unique immersed hypersurface $Σ_{r,θ}=(M,i_{r,θ})$ in $\Bbb{H}^{n+1}$ of constant $θ$-special Lagrangian curvature equal to $r$. We show that these hypers…

In this paper, we focus our study on the ends of a locally conformally flat complete manifold with finite total $Q$-curvature. We prove that for such a manifold, the integral of the $Q$-curvature equals an integral multiple of a dimensional constant $c_n$, where $c_n$ is the integral of the $Q$-curvature on the unit $n…

In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In th…

The paper classifies and proves properties of ALE manifolds and orbifolds.

Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.

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 …

We discuss several kinds of Willmore surfaces of flat normal bundle in this paper. First we show that every S-Willmore surface with flat normal bundle in $S^n$ must locate in some $S^3\subset S^n$, from which we characterize Clifford torus as the only non-equatorial homogeneous minimal surface in $S^n$ with flat normal…

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the $σ_k$ curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, nea…

Study relationships between submanifolds and ambient Kahler 4-manifolds' fundamental groups.

Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.

New examples of special geometric solitons found.

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat - that is, everywh…

Defines conditions for unique conformal metrics on manifolds with specific curvature properties.

Paper classifies 3D conformally flat quasi-Para-Sasakian manifolds.

The article classifies G2-structures with conformally flat metrics.

New method constructs holonomic immersions from flat submanifolds.

The paper extends the study of conformally flat spaces to four dimensions.

The paper proves inequalities for scalar curvature on various manifolds.

Constructs metrics with negative curvature on specific manifold types.

Direct proof for all conformally flat isoparametric submanifolds in Euclidean space.

Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.

A spacetime can be embedded in an enveloping space with all its extensions.

A space has the maximal number of CKVs if and only if it is conformally flat.

New geometric variant of factorization homology for conformally flat manifolds.

In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound (…

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 $k \neq n/2$.

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 biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.

The paper classifies and studies conformally flat hypersurfaces in 4D space.

Compact 3D Cotton-parallel manifolds are always conformally flat.

The paper classifies and proves rigidity for compact conformally flat hypersurfaces in S^(n+1).

In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.

Classification of specific pseudo-Riemannian manifolds.

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…

Study classifies half conformally flat GQE manifolds of signature (2,2).

There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …

Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…

The paper examines flatness conditions on normal metric contact pairs and proves properties of Einstein manifolds.

Study how past eon's matter affects present eon in Penrose's cyclic cosmology.

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 $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive …

Semisimple Lie groups act transitively on pseudo-Riemannian manifolds, making them flat.

Paper connects Dirac operators and automorphic forms on conformally flat manifolds.

Paper connects hypersurfaces in 4D to surfaces in 3-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.