Narrow Results

We study hyperbolic Ricci solitons $(M,g,V)$ by imposing certain curvature restrictions on $M$ and conditions on its potential vector field $V$ to ascertain that $M$ reduces to an Einstein manifold with Killing potential vector field.

Necessary and sufficient conditions for a gradient Ricci soliton to be Einstein are given, showing that they can be expressed in terms of upper and lower bounds on the behavior of the Ricci tensor when evaluated on the gradient of the potential function of the soliton.

Let (M, g) be an Asymptotically Locally Hyperbolic (ALH) manifold which is the interior of a conformally compact manifold and (∂M, [γ]) its conformal infinity. Suppose that the Ricci tensor of (M, g) dominates that of the hyperbolic space and that its scalar curvature satisfies a certain decay condition at infinity. If the Yamabe invariant of (∂M, [γ]) is non-negative, we prove that there exists a conformal metric on M with non-negative scalar curvature and whose boundary ∂M has either positive or zero constant inner mean curvature. In the spin case, we make use of a previous estimate obtained by X. Zhang and the authors for the Dirac operator of the induced metric on ∂M. As a consequence, we generalize and simplify the proof of the result by Andersson and Dahl in [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom.16 (1998) 1–27] about the rigidity of the hyperbolic space when the prescribed conformal infinity ∂M is a round sphere. We also provide non-existence results for conformally compact ALH spin metrics when ∂M is conformal to a Riemannian manifold with special holonomy.

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

We study harmonic morphisms by placing them into the context of conformal foliations. Most of the results we obtain hold for fibres of dimension one and codomains of dimension not equal to two. We consider foliations which produce harmonic morphisms on both compact and noncompact Riemannian manifolds.

By using integral formulae, we prove an extension to one-dimensional foliations which produce harmonic morphisms of the well-known result of S. Bochner concerning Killing fields on compact Riemannian manifolds with nonpositive Ricci curvature.

From the noncompact case, we improve a result of R. L. Bryant[9] regarding harmonic morphisms with one-dimensional fibres defined on Riemannian manifolds of dimension at least four with constant sectional curvature. Our method gives an entirely new and geometrical proof of Bryant's result.

The concept of homothetic foliation (or, more generally, homothetic distribution) which we introduce, appears as a useful tool both in proofs and in providing new examples of harmonic morphisms, with fibres of any dimension.

Warped embeddings from a lower dimensional Einstein manifold into a higher dimensional one are analyzed. Explicit solutions for the embedding metrics are obtained for all cases of codimension 1 embeddings and some of the codimension n>1 cases. Some of the interesting features of the embedding metrics are pointed out and potential applications of the embeddings are discussed.

It is well known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equation from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self-duality equations in Yang–Mills gauge theory and so Einstein manifolds correspond to Yang–Mills instantons in SO(4) = SU(2)_L × SU(2)_R gauge theory. Specifically, we prove that any Einstein manifold with or without a cosmological constant always arises as the sum of SU(2)_L instantons and SU(2)_R anti-instantons. This result explains why an Einstein manifold must be stable because two kinds of instantons belong to different gauge groups, instantons in SU(2)_L and anti-instantons in SU(2)_R, and so they cannot decay into a vacuum. We further illuminate the stability of Einstein manifolds by showing that they carry nontrivial topological invariants.

We introduce the concept of generic submanifold in a manifold equipped with a metric mixed 3-structure and investigate the canonical distributions induced on such submanifold. In particular, we obtain necessary and sufficient conditions for the integrability of these distributions and discuss the geometry of leaves. Moreover, some examples are given.

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad(G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2, 1)-bundle over a paraquaternionic Kähler manifold.

This paper characterizes the warping functions for a multiply generalized Robertson–Walker space-time to get an Einstein space $M$ with a quarter-symmetric connection for different dimensions of $M$ (i.e. (1). dim $M=2,$ (2). dim $M\ge 3$) when all the fibers are Ricci flat. Then we have also computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of $M$ (i.e. (1). dim $M=2,$ (2). dim $M=3,$ (3). dim $M\ge 4$) and all the fibers are Ricci flat. In the last section, we have given two examples of multiply generalized Robertson–Walker space-time with respect to quarter-symmetric connection.

Here, the scalar curvature of an Einstein-type manifold or equivalently an almost Yamabe quasi-soliton is explicitly determined in terms of its soliton function and some rigidity theorems are obtained. Among the others its shown; if the soliton function is negative, then every compact conformal Yamabe quasi-soliton is isometric to the standard Euclidean sphere.

In this paper, we characterize the Einstein multiply warped product space with nonpositive scalar curvature. As a result, it is shown that, if $M$ is Einstein multiple-warped product spaces with compact base and nonpositive scalar curvature, then $M$ is simply a Riemannian manifold. Next, we apply our result on Generalized Robertson–Walker space-time and Generalized Friedmann–Robertson–Walker space-time.

The aim of this paper, is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also, some suitable examples of Kenmotsu and ${(\kappa ,\mu )}^{\prime }$-almost Kenmotsu manifolds are constructed to justify our results.

In this paper, we study Einstein standard stationary space-times $(M,g)=(ℝ\times \mathrm{\Sigma },-\beta d{t}^2+dt\otimes g_{\mathrm{\Sigma }}(\xi ,.)+g_{\mathrm{\Sigma }}(\xi ,.)\otimes dt+g_{\mathrm{\Sigma }})$ of dimension $n+1\ge 3$ with $\xi$ is conformal on the space-like hypersurface $(\mathrm{\Sigma },g_{\mathrm{\Sigma }})$. When $\mathrm{\Sigma }$ is assumed to be connected and complete, we give a criterion so that $(M,g)$ is Einstein. In particular, we show that if $(M,g)$ is Einstein and $\mathrm{\Sigma }$ is complete and connected, then the scalar curvature of $M$ and that of $\mathrm{\Sigma }$ are non-positive and $\xi$ is Killing with respect to $g_{\mathrm{\Sigma }}$.

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field $\dot{\alpha }$ to be conformal, where $\alpha$ is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of $\mathrm{\text{ker}}{F}_{\ast }$ then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

The main purpose of this paper is to study and explore some characteristics of static perfect fluid space-time on paracontact metric manifolds. First, we show that if a $K$-paracontact manifold $M^{2n+1}$ is the spatial factor of a static perfect fluid space-time, then $M^{2n+1}$ is of constant scalar curvature $-2n(2n+1)$ and squared norm of the Ricci operator is given by $4n^2(2n+1)$. Next, we prove that if a $(\kappa ,\mu )$-paracontact metric manifold $M^{2n+1}$ with $\kappa >-1$ is a spatial factor of static perfect space-time, then for $n=1$, $M^{2n+1}$ is flat, and for $n>1$, $M^{2n+1}$ is locally isometric to the product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of constant negative curvature $-4$. Further, we prove that if a paracontact metric 3-manifold $M^3$ with $Q\varphi =\varphi Q$ is a spatial factor of static perfect space-time, then $M^3$ is an Einstein manifold. Finally, a suitable example has been constructed to show the existence of static perfect fluid space-time on paracontact metric manifold.

The aim of this paper is to find some important classes of Einstein manifolds using conformal $\eta$-Ricci solitons and conformal $\eta$-Ricci almost solitons. We prove that a Kenmotsu metric as conformal $\eta$-Ricci soliton is Einstein if it is $\eta$-Einstein or the potential vector field $V$ is infinitesimal contact transformation or collinear with the Reeb vector field $\xi$. Next, we prove that a Kenmotsu metric as gradient conformal $\eta$-Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariants. Finally, we construct some examples to illustrate the existence of conformal $\eta$-Ricci soliton, gradient almost conformal $\eta$-Ricci soliton on Kenmotsu manifold.

In this paper, we initiate the study of critical point equation and *-critical point equation within the substructure of Sasakian manifolds. We prove that if a compact Sasakian manifold admits CPE, then either the manifold is Einstein or the potential function is harmonic in an open subset. Later, we demonstrate that if the manifold satisfies *-CPE, it is $\eta$-Einstein and obtain an expression for the *-Ricci tensor. Finally, we construct an example to illustrate the existence of CPE and *-CPE on the Sasakian manifold.