Natural almost contact structures and their 3D homogeneous models

We consider a natural condition determining a large class of almost contact metric structures. We study their geometry, emphasizing that this class shares several properties with contact metric manifolds. We then give a complete classification of left‐invariant examples on three‐dimensional Lie groups, and show that any simply connected homogeneous Riemannian three‐manifold admits a natural almost contact structure having g as a compatible metric. Moreover, we investigate left‐invariant CR structures corresponding to natural almost contact metric structures.     

Critical point equation on four‐dimensional compact manifolds

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M⁴. In fact, we shall show that for a nontrivial must be isometric to a sphere and f is some height function on      

On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

Rigidity for critical metrics of the volume functional

Geodesic balls in a simply connected space forms , or are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. 7 and Shen 18 to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.     

Critical metrics of the total scalar curvature functional on 4‐manifolds

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980's that every CPE metric must be Einstein. We prove that a 4‐dimensional CPE metric with harmonic tensor must be isometric to a round sphere      

A note on generalized quasi‐Einstein manifolds

In this note, we get a necessary and sufficient condition such that the scalar curvature of generalized m‐quasi‐Einstein manifold with $m=1$ is constant. In particular, we discuss a class of generalized quasi‐Einstein manifolds which are more general than $(m,\rho )$‐quasi‐Einstein manifolds and prove that these manifolds with dimension four are either Einstein or locally conformally flat under some suitable conditions.     

Metrics of Kaluza–Klein type on the anti‐de Sitter space

We introduce and study a new family of pseudo‐Riemannian metrics on the anti‐de Sitter three‐space . These metrics will be called “of Kaluza‐Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle . For any choice of three real parameters , the pseudo‐Riemannian manifold is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures , compatible with , such that is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three‐parameter family of homogeneous metric mixed 3‐structures on the anti‐de Sitter three‐space.     

New gap results on the 4‐dimensional sphere

A result showed by M. Gursky in 4 ensures that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. In this note, we prove that there exists a universal number i ₀ such that any metric g on the 4‐dimensional sphere satisfying and is isometric to the round metric. Moreover, there exists a universal such that any metric g on the 4‐dimensional sphere with nonnegative sectional curvature, and is isometric to the round metric. This last result slightly improves a rigidity theorem also proved in 4 .     

Rigidity of ‐quasi Einstein manifolds

This paper deals with the study on ‐quasi Einstein manifolds. First, we give some characterizations of an ‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an ‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form , where I is a real interval, is an ‐dimensional Riemannian manifold and q is a smooth function on I . Finally, a non‐trivial example of an ‐quasi Einstein manifold verifying our results in terms of the potential function is presented.     

Four‐dimensional contact ‐submanifolds in

The analogue of ‐submanifolds in (almost) Kählerian manifolds is the concept of contact ‐submanifolds in Sasakian manifolds. These are submanifolds for which the structure vector field ξ is tangent to the submanifold and for which the tangent bundle of M can be decomposed as , where is invariant with respect to the endomorphism φ and is antiinvariant with respect to φ. The lowest possible dimension for M in which this decomposition is non trivial is the dimension 4. In this paper we obtain a complete classification of four‐dimensional contact ‐submanifolds in and for which the second fundamental form restricted to and vanishes identically.     

The Riemann extensions with cyclic parallel Ricci tensor

The property of being a D'Atri space (i.e., a Riemannian manifold with volume‐preserving geodesic symmetries) is equivalent, in the real analytic case, to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition L₃ is said to be an L₃‐space. This definition extends easily to the affine case. Here we investigate the torsion‐free affine manifolds and their Riemann extensions as concerns heredity of the condition L₃. We also incorporate a short survey of the previous results in this direction, including also the topic of D'Atri spaces.     

Weakly conformal Finsler geometry

An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz‐Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If X is a weakly conformal complete vector field on a connected Finsler space (M, F) of dimension , then, at least one of the following statements holds: (a) There exists a Finsler metric F₁ weakly conformally equivalent to F such that X is a Killing vector field of the Finsler metric, (b) M is diffeomorphic to the sphere and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on , and (c) M is diffeomorphic to the Euclidean space and the Finsler metric F is weakly conformally equivalent to a Minkowski metric on . The considerations invite further dynamics on Finsler manifolds.     

Pseudo anti‐commuting Ricci tensor and Ricci soliton real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti commuting Ricci tensor for real hypersurfaces in complex hyperbolic two‐plane Grassmannians and prove a complete classification theorem that such a hypersurface must be a tube over a totally real totally geodesic , , a horosphere whose center at the infinity is singular or an exceptional case.     

Some Finsler spaces with homogeneous geodesics

A geodesic in a homogeneous Finsler space is called a homogeneous geodesic if it is an orbit of a one‐parameter subgroup of G . A homogeneous Finsler space is called Finsler g.o. space if its all geodesics are homogeneous. Recently, the author studied Finsler g.o. spaces and generalized some geometric results on Riemannian g.o. spaces to the Finslerian setting. In the present paper, we investigate homogeneous geodesics in homogeneous spaces, and obtain the sufficient and necessary condition for an space to be a g.o. space. As an application, we get a series of new examples of Finsler g.o. spaces.     

Biharmonic hypersurfaces with three distinct principal curvatures in spheres

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmu?‐Montaldo‐Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces are either the hypersphere or the Clifford hypersurface with and . Moreover, we also show that there does not exist proper biharmonic hypersurface with at most three distinct principal curvatures in hyperbolic spaces .     

On the structure of sets with positive reach

We give a complete characterization of compact sets with positive reach (proximally C ¹ sets) in the plane and of one‐dimensional sets with positive reach in . Further, we prove that if is a set of positive reach of topological dimension , then A has its “k‐dimensional regular part” which is a k‐dimensional “uniform” C ^{1, 1} manifold open in A and can be locally covered by finitely many ‐dimensional DC surfaces. We also show that if has positive reach, then can be locally covered by finitely many semiconcave hypersurfaces.     

Symmetry in the geometry of metric contact pairs

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold with decomposable ? is a Calabi‐Eckmann manifold or the Riemannian product of a sphere and . We show that a complete, simply connected, normal metric contact pair manifold with decomposable ?, such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, is the product of globally ?‐symmetric spaces or the product of a globally ?‐symmetric space and . Moreover in the first case the manifold fibers over a locally symmetric space endowed with a symplectic pair.     

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let be a compact gradient shrinking Ricci soliton satisfying with constant. We show that if satisfies , then is Einstein. Here denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in 10 , 14 and 19 . Finally, we prove that if is a complete noncompact gradient steady Ricci soliton satisfying , and if the scalar curvature attains its maximum at some point in the interior of M, then either is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in 3 .     

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let be a connected reductive complex algebraic group with split real form . Consider a strict wonderful ‐variety X equipped with its σ‐equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G‐vector bundle over X . In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E given in terms of the spherical roots of , and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points.