A matrix Harnack inequality for semilinear heat equations

<abstract><p>We derive a matrix version of Li &amp; 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 ^{[<xref ref-type="bibr" rid="b5">5</xref>]} 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>

Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

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.

On static manifolds and related critical spaces with cyclic parallel Ricci tensor

We classify 3-dimensional compact Riemannian manifolds (M 3, g) that admit a non-constant solution to the equation −Δfg +Hess f − f Ric = μ Ric +λg for some special constants (μ, λ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

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 Q m = SOm+2/SO m SO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Q m = SOm+2/SO m SO2 with parallel Ricci tensor.

∗-Ricci tensor on almost Kenmotsu 3-manifolds

In this paper, we obtain the expressions of the ∗-Ricci operator of a three-dimensional almost Kenmotsu manifold [Formula: see text] and find that the ∗-Ricci tensor is not symmetric for [Formula: see text]. We obtain a necessary and sufficient condition for the ∗-Ricci tensor to be symmetric and proved that if the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-[Formula: see text]-manifold [Formula: see text] is symmetric, then [Formula: see text] is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Further, it is shown that the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-manifold [Formula: see text] is parallel if and only if [Formula: see text] is ∗-Ricci flat. In addition, [Formula: see text] satisfying [Formula: see text] is locally isometric to [Formula: see text]. Finally, we discuss about [Formula: see text]-parallel ∗-Ricci tensor on almost Kenmotsu 3-manifolds.

Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators

In this paper, it is proved that the Ricci operator of an almost Kenmotsu 3-h-manifold M is of transversely Killing-type if and only if M is locally isometric to the hyperbolic 3-space {{\mathbb{H}}}^{3}(-1) or a non-unimodular Lie group endowed with a left invariant non-Kenmotsu almost Kenmotsu structure. This result extends those results obtained by Cho [Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J. 45 (2016), no. 3, 435–442] and Wang [Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math. 116 (2016), no. 1, 79–86; Three-dimensional almost Kenmotsu manifolds with \eta -parallel Ricci tensor, J. Korean Math. Soc. 54 (2017), no. 3, 793–805].

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.

Ricci almost solitons and contact geometry

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.

Cyclic-parallel Ricci tensors on a class of almost Kenmotsu 3-manifolds

In this paper, we give a local characterization for the Ricci tensor of an almost Kenmotsu [Formula: see text]-manifold [Formula: see text] to be cyclic-parallel. As an application, we prove that if [Formula: see text] has cyclic-parallel Ricci tensor and satisfies [Formula: see text], (where [Formula: see text] is the Lie derivative of [Formula: see text] along the Reeb flow and both [Formula: see text] and [Formula: see text] are smooth functions such that [Formula: see text] is invariant along the contact distribution), then [Formula: see text] is locally isometric to either the hyperbolic space [Formula: see text] or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.