Minimal volume invariants,topological sphere theorems and biorthogonal curvature on 4‐manifolds

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4‐manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.     

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.     

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      

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      

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

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.     

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.     

Classification of surfaces in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map

In this article, we study submanifolds in a pseudo‐sphere with 2‐type pseudo‐spherical Gauss map. We give a characterization theorem for Lorentzian surfaces in the pseudo‐sphere with zero mean curvature vector in and 2‐type pseudo‐spherical Gauss map. We also prove that non‐totally umbilical proper pseudo‐Riemannian hypersurfaces in a pseudo‐sphere with non‐zero constant mean curvature has 2‐type pseudo‐spherical Gauss map if and only if it has constant scalar curvature. Then, for we obtain the classification of surfaces in with 2‐type pseudo‐spherical Gauss map. Finally, we give an example of surface with null 2‐type pseudo‐spherical Gauss map which does not appear in Riemannian case, and we give a characterization theorem for Lorentzian surfaces in with null 2‐type pseudo‐spherical Gauss map.     

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k‐Einstein (in the sense that their 2k‐Ricci tensor is constant) or have constant 2k‐Gauss‐Bonnet curvature. The results hold for a family of manifolds containing all non‐flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.     

Ricci η‐recurrent real hypersurfaces in 2‐dimensional nonflat complex space forms

Let M be a Hopf hypersurface in a nonflat complex space form $M^2(c)$, $c\ne 0$, of complex dimension two. In this paper, we prove that M has η‐recurrent Ricci operator if and only if it is locally congruent to a homogeneous real hypersurface of type (A) or (B) or a non‐homogeneous real hypersurface with vanishing Hopf principal curvature. This is an extension of main results in [17, 21] for real hypersurfaces of dimension three. By means of this result, we give some new characterizations of Hopf hypersurfaces of type (A) and (B) which generalize those in [14, 18, 26].     

Helicoidal flat surfaces in the 3‐sphere

In this paper, helicoidal flat surfaces in the 3‐dimensional sphere are considered. A complete classification of such surfaces, that generalizes a classification of rotational flat surfaces, is given in terms of the first and second fundamental forms for asymptotic parameters. The result consists in a relation between helicoidal flat surfaces and linear solutions of the corresponding homogeneous wave equation for the angle function.     

4-dimensional minimal CR submanifolds of the sphere S satisfying Chen's equality

In this paper, we classify 4-dimensional minimal CR submanifolds M of the nearly Kähler 6-sphere S⁶(1) which satisfy Chen's equality, i.e. , where δM(p)=τ(p)−infK(p) for every p∈M.     

Asymptotic results on the existence of 4‐RGDDs and uniform 5‐GDDs

In this paper, we continue to investigate the existence of 4‐RGDDs and uniform 5‐GDDs. It is proved that the necessary conditions for the existence of such designs are also sufficient with a finite number of possible exceptions. As an application, the known results on the existence of uniform 4‐frames are also improved. © 2004 Wiley Periodicals, Inc.     

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 .     

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

In this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.     

On the geometry of complex ‐spaces

It is known that applying an ‐homothetic deformation to a complex contact manifold whose vertical space is annihilated by the curvature yields a condition which is invariant under ‐homothetic deformations. A complex contact manifold satisfying this condition is said to be a complex ‐space. In this paper, we deal with the questions of Bochner, conformal and conharmonic flatness of complex ‐spaces when , and prove that such kind of spaces cannot be Bochner flat, conformally flat or conharmonically flat.