$\eta$-RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON $\delta$- LORENTZIAN TRANS-SASAKIAN MANIFOLDS

The objective of the present research article is to study the $\delta$-Lorentzian trans-Sasakian manifolds conceding the $\eta$-Ricci solitons and gradient Ricci soliton. We shown that a symmetric second order covariant tensor in a $\delta$-Lorentzian trans-Sasakian manifold is a constant multiple of metric tensor. Also, we furnish an example of $\eta$-Ricci soliton on 3-diemsional $\delta$-Lorentzian trans-Sasakian manifold is provide in the region where $\delta$-Lorentzian trans-Sasakian manifold is expanding. Furthermore, we discuss some results based on gradient Ricci solitons on $3$-dimensional $\delta$- Lorentzian trans-Sasakian manifold.

Four-dimensional complete gradient shrinking Ricci solitons

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton ℝ 4 {\mathbb{R}^{4}} , or 𝕊 3 × ℝ {\mathbb{S}^{3}\times\mathbb{R}} , or 𝕊 2 × ℝ 2 . {\mathbb{S}^{2}\times\mathbb{R}^{2}.} In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

η-Ricci Solitons and Gradient Ricci Solitons on f-Kenmotsu Manifolds

The aim of the present research article is to study the f-kenmotsu manifolds admitting the η-Ricci Solitons and gradient Ricci solitons with respect to the semi-symmetric non metric connection.

Positive solutions to Schrödinger equations and geometric applications

A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

INEQUALITIES FOR GRADIENT EINSTEIN AND RICCI SOLITONS

This short note concerns with two inequalities in the geo\-me\-try of gradient Einstein solitons $(g, f, \lambda )$ on a smooth manifold $M$. These inequalities provide some relationships between the curvature of the Riemannian metric $g$ and the behavior of the scalar field $f$ through two quadratic equations satisfied by the scalar $\lambda $. The similarity with gradient Ricci solitons and a slightly generalization involving a $g$-symmetric endomorphism $A$ are provided.

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have polynomial volume growth and at least linear volume growth.

RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON NEARLY KENMOTSU MANIFOLDS

In this paper, we study nearly Kenmotsu manifolds with Ricci soliton and we obtain certain conditions about curvature tensors.