Parallel Ricci Tensor Research Articles

η-Ricci--Yamabe and *-η-Ricci--Yamabe solitons in Lorentzian para-Kenmotsu manifolds

Abstract 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 {(\mathrm{LPK})_{n}} ). We study the curvature condition R . S = 0 {R.S=0} and the cyclic parallel Ricci tensor in ( LPK ) n {(\mathrm{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.

Locally conformally flat affine hyperspheres with parallel Ricci tensor

Trying to extend the result of Vrancken et al. (1991) [20] which established the complete classification of locally strongly convex affine hyperspheres with constant sectional curvature, we consider the classification problem of n-dimensional locally strongly convex affine hyperspheres with locally conformally flat affine metrics. As the main result of this paper, we solved the problem under the additional condition that the affine metric has parallel Ricci tensor with respect to its Levi-Civita connection.

Conformal η-Ricci soliton in Lorentzian para Kenmotsu manifolds

The objective of the present paper is to study conformal η-Ricci soliton on Lorentzian Para-Kenmotsu manifolds with some curvature conditions. We study Concircular curvature tensor, Quasi conformal curvature tensor, Codazi type of Ricci tensor and cyclic parallel Ricci tensor in Lorentzian Para-Kenmotsu manifolds. At last we give examples of such manifolds.

Generalized Ricci solitons and Einstein metrics on weak $ K $-contact manifolds

<abstract><p>We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $ K $-contact structures and allow a new look at the classical theory. We characterize weak $ K $-contact manifolds among all weak contact metric manifolds using the property well known for $ K $-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $ K $-contact manifold. We also find sufficient conditions for a weak $ K $-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.</p></abstract>

Certain properties of η-Ricci soliton on η-Einstein para-Kenmotsu manifolds

The objective of present research paper is to be investigate the geometric properties of ?-Ricci solitons on ?-Einstein para-Kenmotsu manifolds. In this manner, we consider ?-Ricci solitons on ?-Einstein para-Kenmotsu manifolds satistfying R.S = 0. Further, we obtain results for ?-Ricci solitons on ?-Einstein para-Kenmotsu manifolds with quasi-conformal flat property. Moreover, we get result for ?-Ricci solitins in ?-Einstein para-Kenmotsu manifolds admitting Codazzi type of Ricci tensor and cyclic parallel Ricci tensor, ?-quasi-conformally semi-symmetric, ?-Ricci symmetric and quasi-conformally Ricci semi-symmetric. At last, we construct an example of a such manifold which justify the existence of proper ?-Ricci solitons

Generalized Killing Ricci tensor for real hypersurfaces in the complex quadric

AbstractFirst we introduce a new notion of generalized Killing Ricci tensor which is equivalent to the notion of cyclic parallel Ricci tensor for real hypersurfaces in the complex quadric . Next, in terms of ‐principal or ‐isotropic unit normal vector fields we give a complete classification of real hypersurfaces in with cyclic parallel Ricci tensor.

On C-totally real minimal submanifolds of the Sasakian space forms with parallel Ricci tensor

On C-totally real minimal submanifolds of the Sasakian space forms with parallel Ricci tensor

On real hypersurfaces of 𝕊²×𝕊²

In this paper, regarding the Riemannian product S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 of two unit 2 2 -spheres as a Kähler surface, we study its real hypersurfaces with typical geometric properties. First, we classify the real hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with isometric Reeb flow and then, by using a Simons’ type inequality, a characterization of these compact real hypersurfaces is provided. Next, we classify Hopf hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with constant product angle function. Finally, we classify Hopf hypersurfaces of S 2 × S 2 \mathbb {S}^2\times \mathbb {S}^2 with parallel Ricci tensor.

A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in <sup>[<xref ref-type="bibr" rid="b5">5</xref>]</sup> for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.</p></abstract>

ETA-RICCI SOLITONS ON LORENTZIAN PARA-KENMOTSU MANIFOLDS

The objective of present research article is to investigate the geometric properties of $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds. In this manner, we consider $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds satisfying $R\cdot S=0$. Further, we obtain results for $\eta$-Ricci solitons on Lorentzian para-Kenmotsu manifolds with quasi-conformally flat property. Moreover, we get results for $\eta$-Ricci solitons in Lorentzian para-Kenmotsu manifolds admitting Codazzi type of Ricci tensor and cyclic parallel Ricci tensor, $\eta$-quasi-conformally semi-symmetric, $\eta$-Ricci symmetric and quasi-conformally Ricci semi-symmetric. At last, we construct an example of a such manifold which justify the existence of proper $\eta$-Ricci solitons.

η-Ricci solitons on N(k)-contact metric manifolds

In this paper, we study ?-Ricci solitons on N(k)-contact metric manifolds. At first we consider ?-Ricci solitons on N(k)-contact metric manifolds with harmonic curvature tensor. Then we study ?-Ricci solitons onN(k)-contact metric manifolds with harmonic Weyl tensor. Moreover, we consider ?-Ricci soliton on N(k)-contact metric manifolds with ?-parallel Ricci tensor. Also ?-Ricci soliton on N(k)-contact metric manifolds satisfying some curvature restrictions under projective curvature tensor have been considered. Finally, the existence of an ?-Ricci soliton on a 3-dimensional N(k)-contact metric manifold is ensured by a proper example.

α-almost Ricci solitons on Kenmotsu manifolds

The current article purports to investigate α-almost Ricci solitons in the framework of Kenmotsu manifolds. Among others, we prove that an α-almost Ricci solitons on a Kenmotsu manifold is expanding. Furthermore, we take into account α-almost Ricci solitons on Kenmotsu manifolds with Codazzi type of Ricci tensor and cyclic parallel Ricci tensor. Also, Kenmotsu manifolds satisfying the curvature condition R.R=Q(S,R) is studied. Ultimately, we consider an example to prove the non-existence of proper α-almost Ricci solitons on Kenmotsu manifolds and verify some results.

A new classification on parallel Ricci tensor for real hypersurfaces in the complex quadric

First we introduce the notion of parallel Ricci tensor ${\nabla }\mathrm {Ric}=0$ for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 with parallel Ricci tensor.

Generalized Quasi-Einstein Manifolds in Contact Geometry

In this study, we investigate generalized quasi-Einstein normal metric contact pair manifolds. Initially, we deal with the elementary properties and existence of generalized quasi-Einstein normal metric contact pair manifolds. Later, we explore the generalized quasi-constant curvature of normal metric contact pair manifolds. It is proved that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic parallel Ricci tensor and the Codazzi type of Ricci tensor are considered, and further prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we characterize normal metric contact pair manifolds satisfying certain curvature conditions related to M-projective, conformal, and concircular curvature tensors. We show that a normal metric contact pair manifold with generalized quasi-constant curvature is locally isometric to the Hopf manifold S2n+1(1)×S1.

η-Ricci solitons on 3-dimensional Kenmotsu manifolds

The purpose of this paper is to study η-Ricci solitons on 3-dimensional Kenmotsu manifolds. First, we prove that an η-Ricci soliton on a 3-dimensional Kenmotsu manifold is an η-Einstein manifold. Besides these, we consider η-Ricci solitons on 3 -dimensional Kenmotsu manifolds with Ricci tensor of Codazzi type and cyclic parallel Ricci tensor. Next, we study conformally at and φ-Ricci symmetric η-Ricci soliton on 3-dimensional Kenmotsu manifolds. Finally, we construct an example to prove the existence of η-Ricci soliton on 3-dimensional Kenmotsu manifold and verify some results.

Ricci almost solitons and contact geometry

Abstract We prove that on a K-contact manifold, a Ricci almost soliton is a Ricci soliton if and only if the potential vector field V is a Jacobi field along the Reeb vector field ξ. Then we study contact metric as a Ricci almost soliton with parallel Ricci tensor. To this end, we consider Ricci almost solitons whose potential vector field is a contact vector field and prove some rigidity results.

On product affine hyperspheres in ℝn+1

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ℝn+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3- and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.

Correction to: Real Hypersurfaces in the Complex Hyperbolic Quadric with Parallel Ricci Tensor

In this corrigendum, we acknowledge that, due to unfortunate circumstances, several passages from Abstract and Introduction of the manuscript “Real hypersurfaces in the complex hyperbolic quadric with parallel Ricci tensor” have been taken from the original paper [8], partly with minor modifications. The correct abstract and introduction are given below.

Real Hypersurfaces in the Complex Hyperbolic Quadric with Parallel Ricci Tensor

We introduce the notion of parallel Ricci tensor for real hypersurfaces in the complex hyperbolic quadric $${Q^m}^* = SO^{o}_{m,2}/SO_mSO_2$$ . According to the $$\mathfrak {A}$$ -principal or the $$\mathfrak {A}$$ -isotropic unit normal vector field N, we give a complete classification of real hypersurfaces in $${Q^m}^* = SO^{o}_{m,2}/SO_mSO_2$$ with Ricci parallelism.

Real Hypersurfaces in the Complex Quadric with Reeb Parallel Ricci Tensor

First we introduce the notion of commuting and parallel Ricci tensor for real hypersurfaces in the complex quadric $$Q^m = SO_{m+2}/SO_2SO_m$$. Then, according to the $$\mathfrak {A}$$-principal or the $$\mathfrak {A}$$-isotropic unit normal vector field N, respectively, we give a classification of real hypersurfaces in $$Q^m = SO_{m+2}/SO_2SO_m$$ with Reeb parallel Ricci tensor.