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We prove dynamical stability and instability theorems for Poincaré–Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and Bérard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either (X - α) (X - α ¯) (type I), where α ∈ C \ R , or X 2 (type II). Firstly, we obtain a complete description of Ricci-parallel metrics of type I. In particular, such a Ricci-parallel metric is uniquely determined by an Einstein metric and an invariant symmetric parallel complex structure up to isometry and scaling. Then we study Ricci-parallel metric Lie algebras of type II by using double extension process. Surprisingly, we find that every double extension of a metric Abelian Lie algebra is Ricci-parallel and the converse holds for Lorentz Ricci-parallel metric nilpotent Lie algebras of type II. Moreover, we construct infinitely many new explicit examples of Ricci-parallel metric Lie algebras which are not Einstein. [ABSTRACT FROM AUTHOR]

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial L 1 space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages. [ABSTRACT FROM AUTHOR]

We establish two key results regarding pseudo symmetric and pseudo Ricci symmetric space-times. Firstly, we demonstrate that in pseudo symmetric generalized Robertson-Walker space-times either the scalar curvature remains constant or the associated vector field B i is irrotational. Secondly, in pseudo Ricci symmetric generalized Robertson-Walker space-times, we establish that either the scalar curvature is zero or the associated vector field A i is irrotational. We identify the conditions to ensure both B i and A i of these manifolds are acceleration-free and vorticity-free. We provide evidence that a pseudo symmetric and pseudo Ricci symmetric GRW space-time can be described as a perfect fluid. In a pseudo symmetric space-time, the state equation is given by p = 4 - n 2 n - 2 μ , whereas in a pseudo Ricci symmetric space-time, the state equation takes the form p = 3 - n n - 1 μ , where p and μ are the isotropic pressure and the energy density. It is noteworthy that if n = 4 , a pseudo symmetric space-time corresponds to the dust matter era, while a pseudo Ricci symmetric space-time corresponds to the phantom era. [ABSTRACT FROM AUTHOR]

In this paper, we show that for an Sp(k + 1)-invariant metric ĝ on S 4 k + 3 (k ⩾ 1) close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with (S 4 k + 3 , [ g ^ ]) as its conformai infinity is unique up to isometry. Moreover, by the result in Li et al. (2017), g is the Graham-Lee metric (see Graham and Lee (1991)) on the unit ball B1 ⊂ ℝ4k+4. We also give an a priori estimate of the Einstein metric g. As a by-product of the a priori estimates, based on the estimate and Graham-Lee and Lee's seminal perturbation results (see Graham and Lee (1991) and Lee (2006)), we directly use the continuity method to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity (S 4 k + 3 , [ g ^ ]) when the metric ĝ is Sp(k + 1)-invariant. We also generalize the results to the case of conformal infinity (S 15 , [ g ^ ]) with ĝ a Spin(9)-invariant metric in the appendix. [ABSTRACT FROM AUTHOR]

In this article, we investigate the Riemannian and semi-Riemannian metrics on the base space of the Boothby-Wang fibration of a closed regular non-Sasakian (κ , μ) -manifold. To this end, we study a natural class of deviations of the projection map from being (semi-)Riemannian submersions. We consider deviations that preserve the canonical bilegendrian structure on the given (κ , μ) -manifold. This approach gives a unified framework to analyze rigidity results in both categories. As a consequence, in the Riemannian category, we obtain uniqueness of Sasakian structure on the given (κ , μ) -manifold which orthogonalizes the canonical bilegendrian structure. In the semi-Riemannian category, we obtain an explicit description of the finitely many para-Sasakian structures which orthogonalize the canonical bilegendrian structure. [ABSTRACT FROM AUTHOR]

We describe all Lorentzian semi-direct extensions of the Heisenberg group which are conformally Einstein. As a by side result, Bach-flat left-invariant Lorentzian metrics on semi-direct extensions of the Heisenberg group are classified, thus providing new background solutions in conformal gravity. [ABSTRACT FROM AUTHOR]

We introduce another distinguished closed vector field on a Riemannian manifold which we shall call torse-forming-like and we relate it to other kind of vector fields such as torse-forming, self-torse-forming and semi-torse-forming vector fields. For a warped product manifold, we obtain different types of vector fields on the factor manifolds when the warped product is endowed with a torse-forming-like vector field. We study the nature of the tangential component of such a vector field for some special cases of submanifolds, namely, totally umbilical (in particular, totally geodesic) or having Codazzi (in particular, parallel) shape operator, and we show that the tangential component on a hypersurface of a torse-forming-like vector field is a Killing vector field if and only if the hypersurface is quasi-umbilical. In particular, if the hypersurface is isometrically immersed in a real space-form, then it is a space of quasi-constant curvature, hence, it is a quasi-Einstein manifold. Finally, we characterize the metallic shaped hypersurfaces with the normal direction generated by the normal component of a torse-forming-like vector field having Killing tangential component to the hypersurface. [ABSTRACT FROM AUTHOR]

Given a non-Kähler Calabi–Yau compact orbifold with isolated singularities endowed with a Chern–Ricci flat balanced metric, we study, via a gluing construction, the existence of Chern–Ricci flat balanced metrics on its crepant resolutions, and discuss applications to the search of solutions for the Hull–Strominger system. We also describe the scenario of singular threefolds with ordinary double points, and see that similarly is possible to obtain balanced approximately Chern–Ricci flat metrics. [ABSTRACT FROM AUTHOR]

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of almost contact metric manifolds. In this paper we study the new structure of this type, called the weak nearly Sasakian structure. We find conditions that are satisfied by almost contact manifolds and under which the contact distribution is curvature invariant and weak nearly Sasakian manifolds admit two types of totally geodesic foliations. Our main result generalizes the theorem by Cappelletti-Montano and Dileo (Ann Matem Pura Appl 195:897-922, 2016) to the context of weak almost contact geometry. [ABSTRACT FROM AUTHOR]

In this paper, we derive the second order estimate to the 2nd Hessian type equation on a compact almost Hermitian manifold. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study η-Ricci–Yamabe solitons (η-RYS) and * -η-Ricci–Yamabe solitons (* -η-RYS) in Lorentzian para-Kenmotsu n-manifolds (briefly, (LPK) n ). We study the curvature condition R. S = 0 and the cyclic parallel Ricci tensor in (LPK) n admitting η-RYS. Furthermore, we study M-projectively flat and quasi-M-projectively flat Lorentzian para-Kenmotsu manifolds admitting * -η-RYS. Finally, we give two examples of Lorentzian para-Kenmotsu manifolds admitting η-RYS and * -η-RYS to verify some of our results. [ABSTRACT FROM AUTHOR]

We show that shrinking Kähler-Ricci solitons over a compact Kähler manifold are gradient shrinking Kähler-Ricci solitons. The proof relies on a remarkable identity on the kernels of a real and a complex elliptic operator proved in our solution of the variational stability problem for gradient shrinking Kähler-Ricci solitons in Pali (Complex Manifolds 3(1):41–144, 2016). [ABSTRACT FROM AUTHOR]

We prove that there exists a gradient expanding Ricci soliton asymptotic to any given cone over the product of a round sphere and a Ricci flat manifold. In particular we obtain asymptotically conical expanding Ricci solitons with positive scalar curvature on R 3 × S 1. More generally we construct continuous families of gradient expanding Ricci solitons on trivial vector bundles over products of Einstein manifolds with arbitrary Einstein constants. [ABSTRACT FROM AUTHOR]

The present article deals with static perfect fluid spacetimes on f-Kenmotsu 3-manifolds. At first, we demonstrate if a 3-dimensional f-Kenmotsu manifold with constant scalar curvature as the spatial factor of a static perfect fluid spacetime, then either it is a space of constant sectional curvature or g r a d ψ is pointwise collinear with ξ and the warping function of the static perfect fluid spacetime is given by ψ = k 1 t + k 2 , k 1 ≠ 0 . As a result, we establish that if a cosymplectic manifold of dimension three with constant scalar curvature is the spatial factor of a static perfect fluid spacetime, then either it is flat or, the manifold becomes a space of constant sectional curvature. Next, we show that under certain restrictions if a 3-dimensional f-Kenmotsu manifold is the spatial factor of a static perfect fluid spacetime, then either the manifold is a space of constant sectional curvature or, the manifold is locally isometric to either the flat Euclidean space R 3 or the Riemannian product R × M 2 (c) , where M 2 (c) represents a Kahler surface with constant curvature c ≠ 0 , provided ξ ψ = 0 and ξ f ~ = 0 . Lastly, we have cited an example of an f-Kenmotsu manifold to validate our result. [ABSTRACT FROM AUTHOR]

A Riemannian metric on a closed manifold is said to be geometrically formal if the wedge product of any two harmonic forms is harmonic; equivalently, the interior product of any two harmonic forms is harmonic. Given a Riemannian foliation on a closed manifold, we say that a bundle-like metric is transversely geometrically formal if the interior product of any two basic harmonic forms is basic harmonic. In this paper, we examine the geometric and topological consequences of this condition. [ABSTRACT FROM AUTHOR]

This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product Qεm×R{{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}}. In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and δ\delta -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.

Inspired by K. Fujita's algebro-geometric result that complex projective space has maximal degree among all K-semistable complex Fano varieties, we conjecture that the height of a K-semistable metrized arithmetic Fano variety $\mathcal {X}$ of relative dimension $n$ is maximal when $\mathcal {X}$ is the projective space over the integers, endowed with the Fubini–Study metric. Our main result establishes the conjecture for the canonical integral model of a toric Fano variety when $n\leq 6$ (the extension to higher dimensions is conditioned on a conjectural 'gap hypothesis' for the degree). Translated into toric Kähler geometry, this result yields a sharp lower bound on a toric invariant introduced by Donaldson, defined as the minimum of the toric Mabuchi functional. Furthermore, we reformulate our conjecture as an optimal lower bound on Odaka's modular height. In any dimension $n$ it is shown how to control the height of the canonical toric model $\mathcal {X},$ with respect to the Kähler–Einstein metric, by the degree of $\mathcal {X}$. In a sequel to this paper our height conjecture is established for any projective diagonal Fano hypersurface, by exploiting a more general logarithmic setup. [ABSTRACT FROM AUTHOR]

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups. [ABSTRACT FROM AUTHOR]

Recently, Liu-Deng studied projectively flat homogeneous (α , β) -metrics and showed that if these metrics are not Riemannian nor locally Minkowskian, then the Finsler metrics are left invariant Randers metrics on the hyperbolic space H n as a solvable Lie group (Liu and Deng in Forum Math 27:3149–3165, 2015). In this paper, we study homogeneous projectively flat (or projective) general Finsler metrics. First, we prove that homogeneous projectively flat Finsler metrics have vanishing E ¯ -curvature if and only if they have almost isotropic S-curvature if and only if they have relatively isotropic L-curvature. In any cases, the Finsler metric reduces to a locally Minkowskian metric or a Riemannian metric of constant sectional curvature. This yields a classification of homogeneous projective Finsler metrics with the above mentioned non-Riemannian curvatures properties. Finally, we show that Liu-Deng’s Randers metrics are Douglas metrics which have not isotropic S-curvature nor relatively isotropic L-curvature. [ABSTRACT FROM AUTHOR]

This short note deals with compact and complete and non-compact gradient Yamabe solitons (M, g, f) such that it has metric of constant scalar curvature. Firstly, we give a new proof of triviality for gradient compact Yamabe solitons. Also, under some integral conditions, we are able to improve a result due to Ma and Miquel (Ann Global Anal Geom 42:195–205, 2012). Finally, we obtain that the Yamabe metric becomes rotationally symmetric. Results for k-Yamabe solitons are also obtained here. [ABSTRACT FROM AUTHOR]

We consider the existence of cohomogeneity one solitons for the isometric flow of G 2 -structures on the following classes of torsion-free G 2 -manifolds: the Euclidean R 7 with its standard G 2 -structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G 2 -manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R 7 is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} equation on a Riemannian CRVF-manifold M and show that either M is Einstein or the potential function is pointwise collinear with ζ on an open set U of M. Next, we show that if a Riemannian CRVF-manifold M is the spatial factor of a 핊 ⁢ ℙ ⁢ 픽 {\mathbb{SPF}} with a Batch tensor then it is a Batch flat space-time manifold. [ABSTRACT FROM AUTHOR]

In this article, we prove that there are at least two new non-naturally reductive Ad(SO(l) × SO(k) × SO(k) × SO(k))-invariant Einstein metrics on compact simple Lie group SO(l + 3k) (k < l). It implies that every compact simple Lie group SO(n) (n > 12) admits at least 2 ([ n − 1 4 ] − 2) non-naturally reductive left invariant Einstein metrics. Moreover, we obtain that there are at least two families of invariant Einstein–Randers metrics on the compact Lie group SO(n) (n > 12). Besides, we construct non-naturally reductive left invariant (α, β) metrics on the Lie group SO(n) (n > 12). Finally, we examine the isometric problem for those Riemannian Einstein metrics. [ABSTRACT FROM AUTHOR]

In this paper, we show that complete Bach-flat Schouten solitons with n ≥ 4 are rigid. When n = 3 we are able to conclude rigidity under a more general condition, namely when the Bach tensor is divergence-free. These results imply rigidity of locally conformally flat Schouten solitons for n ≥ 3 . [ABSTRACT FROM AUTHOR]

We prove that every compact, homogeneous η -Einstein almost cosymplectic manifold is a cosymplectic manifold. [ABSTRACT FROM AUTHOR]

In this paper, we study conformal bi-slant Riemannian submersions from locally product manifolds onto Riemannian manifold as a generalization of invariant, anti-invariant, semi-invariant, slant, semi-slant and hemi-slant Riemannian submersions. We investigate the geometry of foliations determined by vertical and horizontal distributions, and obtain the geometry of leaves of these distributions. In the end, we give a non-trivial example. [ABSTRACT FROM AUTHOR]

We show that if the universal cover of a closed smooth manifold admitting a metric with non-negative Ricci curvature is formal, then the manifold itself is formal. We reprove a result of Fiorenza–Kawai–Lê–Schwachhöfer, that closed orientable manifolds with a non-negative Ricci curvature metric and sufficiently large first Betti number are formal. Our method allows us to remove the orientability hypothesis; we further address some cases of non-closed manifolds. [ABSTRACT FROM AUTHOR]

We show that, for any given q ≥ 0 {q\geq 0} , any Sasakian structure on a closed manifold M is approximated in the C q {C^{{q}}} -norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C 0 {C^{0}} -norm to a C q {C^{{q}}} -approximation. [ABSTRACT FROM AUTHOR]

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold: first, we obtain various curvature inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Kenmotsu space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion. The equality cases of all the inequalities are studied. Moreover, these curvature inequalities are studied under two different cases: the structure vector field ξ being vertical or horizontal. [ABSTRACT FROM AUTHOR]

In this paper we have found a number of commutator formulas between the weighted divergence, the weighted Laplacian, the weighted Lichnerowicz Laplacian and the Lie derivtive on closed orientable gradient Ricci shrinkers, then using them we have generalized a Theorem of Cao and Zhu about the necessary condition for linear stability of a gradient Ricci shrinker. Recentlly they have also extended our results. [ABSTRACT FROM AUTHOR]

This paper explores the properties of Tanno tensor on quasi contact metric manifolds and provides a characterization of these manifolds using Tanno tensor. Furthermore, it presents formulas for skew-symmetric and symmetric parts of Tanno tensor. [ABSTRACT FROM AUTHOR]

In this note, we show that the potential vector field of a Cotton soliton (M, g, V) is an infinitesimal harmonic transformation, and we use it to give another proof of the triviality of compact Cotton solitons. Moreover, we extend this triviality result to the complete case by imposing certain regularity conditions on the potential vector field V. [ABSTRACT FROM AUTHOR]

First, we derive various curvature inequalities involving Ricci and scalar curvatures of horizontal and vertical distributions of a quasi bi-slant Riemannian submersion from complex space forms onto a Riemannian manifold. Then we establish our main results which are the Chen–Ricci inequalities for such quasi bi-slant Riemannian submersions. Finally, we determine the equality cases of these inequalities. [ABSTRACT FROM AUTHOR]

We provide conditions for a compact gradient hyperbolic Ricci and a compact gradient hyperbolic Yamabe soliton to be trivial, hence, the manifold to be an Einstein manifold in the first case, and a manifold of constant scalar curvature, in the second case. In particular, we prove that for a compact gradient hyperbolic Yamabe soliton of dimension > 2 , if the second Lie derivative of the metric in the direction of the potential vector field is trace-free and divergence-free, then the above conclusion is reached. [ABSTRACT FROM AUTHOR]

We provide a direct proof of the result “A closed Yamabe soliton of dimension ≥ 3 has constant scalar curvature” through the derivation of a divergence formula. Next, we show that, if the potential vector field of a Yamabe soliton of dimension ≥ 3 leaves the Ricci tensor invariant, then the scalar curvature is constant. Finally, we provide a classification of a complete non-trivial gradient Kähler Yamabe soliton, showing that it is isometric to either (i) a product of a complex Euclidean space and a Kähler manifold of non-zero constant scalar curvature, or (ii) a complex Euclidean space. [ABSTRACT FROM AUTHOR]

This article investigates generalized Ricci almost solitons, also known as GRA solitons, on contact metric manifolds, including the gradient case. At first, we establish that a complete K-contact or Sasakian manifold endowed with a closed GRA soliton satisfying 4 c 1 c 2 ≠ 1 is compact Einstein with scalar curvature 2 n (2 n + 1) . As for the gradient case, it exhibits an isometry to the unit sphere S 2 n + 1 . Subsequently, we identify a few adequate conditions under which a non-trivial complete K-contact manifold with a GRA soliton is trivial (η -Einstein). Following that, we establish certain results on H-contact and complete contact manifolds. We also demonstrate that a non-Sasakian (k , μ) -contact manifold with a closed GRA soliton is flat for dimension 3, and for higher dimensions, it is locally isometric to the trivial bundle R n + 1 × S n (4) , provided 4 c 1 c 2 (1 - 2 n) ≠ 1 and c 2 ≠ 0 . Finally, we discuss a few applications of GRA solitons in general relativity. These include characterizing PF spacetimes with a concircular velocity vector field and determining a sufficient condition for a GRW spacetime to be a PF spacetime. [ABSTRACT FROM AUTHOR]