Ricci Solitons on Riemannian Hypersurfaces Generated by Torse-Forming Vector Fields in Riemannian and Lorentzian Manifolds

This paper investigates Ricci solitons on Riemannian hypersurfaces in both Riemannian and Lorentzian manifolds. We provide conditions under which a Riemannian hypersurface, exhibiting specific properties related to a closed conformal vector field of the ambiant manifold, forms a Ricci soliton structure. The characterization involves a delicate balance between geometric quantities and the behavior of the conformal vector field, particularly its tangential component. We extend the analysis to ambient manifolds with constant sectional curvature and establish that, under a simple condition, the hypersurface becomes totally umbilical, implying constant mean curvature and sectional curvature. For compact hypersurfaces, we further characterize the nature of the Ricci soliton.

In this article, we investigate Ricci solitons occurring on spacelike hypersurfaces of Einstein Lorentzian manifolds. We give the necessary and sufficient conditions for a spacelike hypersurface of a Lorentzian manifold, equipped with a closed conformal timelike vector field ξ¯, to be a gradient Ricci soliton having its potential function as the inner product of ξ¯ and the timelike unit normal vector field to the hypersurface. Moreover, when the ambient manifold is Einstein and the hypersurface is compact, we establish that, under certain straightforward conditions, the hypersurface is an extrinsic sphere, that is, a totally umbilical hypersurface with a non-zero constant mean curvature. In particular, if the ambient Lorentzian manifold has a constant sectional curvature, we show that the compact spacelike hypersurface is essentially a round sphere.

The study of Ricci solitons and invariant Ricci solitons with connections of various types has garnered much attention from many mathematicians. Metric connections with vector torsion, or semisymmetric connections, were first studied by E. Cartan on (pseudo) Riemannian manifolds. Later, K. Yano and I. Agricola studied tensor fields and geodesic lines of such connections, while P.N. Klepikov, E.D. Rodionov, and O.P. Khromova considered the Einstein equation of semisymmetric connections on three-dimensional locally homogeneous (pseudo) Riemannian manifolds.
 In the previous paper, the authors studied invariant Ricci solitons with a semisymmetric connection. They are an important subclass of the class of homogeneous Ricci solitons. We obtained the classification of invariant Ricci solitons on three-dimensional Lie groups with a left-invariant Riemannian metric and a semisymmetric connection different from the Levi-Civita connection. Also, the existence of invariant Ricci solitons with a non-conformal Killing vector field was proved for the such case. Moreover, a part of the proofs was obtained using the analytical calculation software packages.
 In this paper, we investigate invariant Ricci solitons on three-dimensional nonunimodular Lie groups with a left-invariant Riemannian metric and a semisymmet-ric connection. Analytical proofs of all theorems completing the classification of such solitons are presented.

In the following paper, we describe Killing fields on 2-symmetric Lorentzian manifolds of dimension four. Killing fields play an important role in the study of Ricci solitons which were introduced by R. Hamilton. Ricci solitons are the generalization of the Einstein metrics on (pseudo)Riemannian manifolds. Ricci soliton equation was studied by many mathematicians on different classes of manifolds. In particular, in the recent author’s papers, solvability of the Ricci soliton equation on 3-symmetric Lorentzian manifolds was proved, and the general solution of the Ricci soliton equation on 2-symmetric Lorentzian manifolds was described. We describe Killing fields using Brinkmann normal coordinates which exist on the class of Lorentzian manifolds, the so-called pp-waves. The system of differential equations that corresponds to the Killing equation can be reduced to a simpler form. This was done by W. Globke and T. Leistner. By applying their result, the general solution of the system was found, the dimension of an algebra of Killing fields was calculated. The results stated in this paper continue author’s research on Ricci solitons on Lorentzian manifolds.

This paper investigates compact Riemannian hypersurfaces immersed in (n+1)-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space Rn+1, Euclidean sphere Sn+1, and de Sitter space S1n+1.

Ricci solitons are an important generalization of Einstein metrics on (pseudo) Riemannian manifolds, and this notion was introduced by R.Hamilton. The problem of solving the Ricci soliton equation is quite difficult, therefore one can assume some restrictions either on a structure of the manifold or on the dimension or on a class of metrics, or on a class of vector fields, which are contained in the Ricci soliton equation. Walker manifolds are one of the most important examples of such restrictions, that is pseudo-Riemannian manifolds admitting a smooth parallel (in sense of Levi-Civita connection) isotropic distribution. The geometry of Walker manifolds and Ricci solitons on them were studied by many mathematicians. In this paper, we investigate the Ricci soliton equation on some Lorentzian manifolds. In particular, we study the Ricci solitons on 2-symmetric Lorentzian manifolds, which are Walker manifolds, as it was proven by D.V. Alekseevsky and A.S. Galaev. K. Onda and B. Batat investigated Ricci solitons on fourdimensional 2-symmetric Lorentzian manifolds, and proved local solvability of the Ricci soliton equation on such manifolds. In this paper we have obtained local solvability of the Ricci soliton equation on fivedimensional 2-symmetric Lorentzian manifolds.

A compact, orientable, Riemannian manifold of dimension n is considered, with the Laplace operator acting on the exterior 2-forms of the manifold. Examining the spectrum, Sp 2 {\text {Sp}^2} , of the Laplace operator acting on 2-forms, the question is raised whether Sp 2 {\text {Sp}^2} exerts an influence on the geometry of the Riemannian manifold. To answer this question, after some preliminaries, two compact, orientable, equispectral, i.e., having the same Sp 2 {\text {Sp}^2} , Riemannian manifolds are considered in §3. (We note, in particular, that equispectral implies that the two manifolds are equidimensional.) Assuming further that the second Riemannian manifold has constant sectional curvature, the paper exhibits all the dimensions, commencing with 2, for which the two Riemannian equispectral manifolds have the same constant sectional curvature. In particular, this implies that for certain dimensions, which are explicitly stated, the Euclidean n-sphere is completely characterized by the spectrum, Sp 2 {\text {Sp}^2} , of the Laplacian on exterior 2-forms. Next, two compact, orientable, equispectral, Einsteinian manifolds are considered. (Again, equispectral implies equidimensional.) Assuming that the second Einsteinian manifold is of constant sectional curvature, the paper exhibits all the dimensions for which the two Einsteinian equispectral manifolds have equal constant sectional curvature. In particular, taking the second manifold to be the standard Euclidean sphere, the paper classifies Einsteinian manifolds, which are equispectral to the sphere, by calculating all the dimensions for which the Einsteinian manifold is isometric to the sphere. In short, if one of the Einsteinian manifolds is the sphere, then for certain dimensions, equispectral implies isometric. In §4, compact, equispectral, Kählerian manifolds are considered, and additional conditions are examined which determine their geometry. Studying two compact, equispectral, Kählerian manifolds, and again assuming that one of the manifolds is of real, constant, holomorphic, sectional curvature, the paper exhibits all the dimensions for which the two manifolds have equal real, constant, holomorphic, sectional curvatures. As a particular case, the paper classifies all the dimensions for which complex projective space, with Fubini-Study metric, is completely characterized by the spectrum, Sp 2 {\text {Sp}^2} , of the Laplacian acting on exterior 2-forms. The calculations were performed by utilizing an electronic computer.

Abstract. A vector field on a Riemannian manifold (M,g) is called con-circular if it satisfies ∇ X v = µX for any vector X tangent to M, where∇is the Levi-Civita connection and µ is a non-trivial function on M. Asmooth vector field ξ on a Riemannian manifold (M,g) is said to definea Ricci soliton if it satisfies the following Ricci soliton equation:12L ξ g + Ric = λg,where L ξ g is the Lie-derivative of the metric tensor g with respect to ξ,Ric is the Ricci tensor of (M,g) and λ is a constant. A Ricci soliton(M,g,ξ,λ) on a Riemannian manifold (M,g) is said to have concircularpotential field if its potential field ξ is a concircular vector field.In the first part of this paper we determine Riemannian manifoldswhich admit a concircular vector field. In the second part we classify Riccisolitons with concircular potential field. In the last part we prove someimportant properties of Ricci solitons on submanifolds of a Riemannianmanifold equipped with a concircular vector field. 1. IntroductionA. Fialkow introduced in [13] the notion of concircular vector fields on aRiemannian manifold M as vector fields which satisfy(1) ∇

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.

The papers of many mathematicians are devoted to the study of conformally Killing vector fields. Being a natural generalization of the concept of Killing vector fields, these fields generate a Lie algebra corresponding to the Lie group of conformal transformations of the manifold. Moreover, they generate the class of locally conformally homogeneous (pseudo) Riemannian manifolds studied by V.V. Slavsky and E.D. Rodionov. Ricci solitons, which R. Hamilton first considered, are another important area of research. Ricci solitons are a generalization of Einstein's metrics on (pseudo) Riemannian manifolds. The Ricci soliton equation has been studied on various classes of manifolds by many mathematicians. In particular, a general solution of the Ricci soliton equation was found on 2-symmetric Lorentzian manifolds of low dimension, and the solvability of this equation in the class of 3-symmetric Lorentzian manifolds was proved. The Killing vector fields make it possible to find the general solution of the Ricci soliton equation in the case of the constancy of the Einstein constant in the Ricci soliton equation. However, the role of the Killing fields is played by conformally Killing vector fields for different values of the Einstein constant. In this paper, we investigate conformal Killing vector fields on 5-dimensional 2-symmetric Lorentzian manifolds. The general solution of the conformal analog of the Killing equation on five-dimensional locally indecomposable 2-symmetric Lorentzian manifolds is described in local coordinates, discovered by A.S. Galaev and D.V. Alekseevsky.

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.

The study of Ricci flows, which describe the deformation of (pseudo) Riemannian metrics on a manifold, and their solutions, Ricci solitons, has garnered much attention from mathematicians. However, previous studies have typically focused on manifolds with Levi-Civita connections. This paper breaks new ground by considering manifolds with semisymmetric connections, which also include the Levi-Civita connection. Metric connections with vector torsion, or semisymmetric connections, were first studied by E. Cartan on (pseudo) Riemannian manifolds. Later, K. Yano and I. Agricola studied tensor fields and geodesic lines of such connections, while P.N. Klepikov,
 E.D. Rodionov, and O.P. Khromova considered the Einstein equation of semisymmetric connections on three-dimensional locally homogeneous (pseudo) Riemannian manifolds. Because the Ricci tensor of a semisymmetric connection is not symmetric in general, we focus on studying the symmetric and skew-symmetric parts of the Ricci tensor. Specifically, we investigate symmetric Ricci flows on three-dimensional Lie groups with J. Milnor's left-invariant (pseudo) Riemannian metric and E. Cartan's semisymmetric connection.

In this article we make a classification of four‐dimensional gradient almost Ricci solitons with harmonic Weyl curvature. We prove first that any four‐dimensional (not necessarily complete) gradient almost Ricci soliton with harmonic Weyl curvature has less than four distinct Ricci‐eigenvalues at each point. If it has three distinct Ricci‐eigenvalues at each point, then is locally a warped product with 2‐dimensional base in explicit form, and if g is complete in addition, the underlying smooth manifold is or . Here is a smooth surface admitting a complete Riemannian metric with constant curvature k. If has less than three distinct Ricci‐eigenvalues at each point, it is either locally conformally flat or locally isometric to the Riemannian product , , where has the Euclidean metric and is a 2‐dimensional Riemannian manifold with constant curvature λ. We also make a complete description of four‐dimensional gradient almost Ricci solitons with harmonic curvature.

As we know from the semi-Riemannian regular interval theorem (Theorem 6.3.1), the existence of an affine solution f of a nonnull eikonal equation on a semi-Riemannian manifold (M, g) yields a splitting of (M, g) into a semi-Riemannian product manifold, provided that ∇ f is a complete vector field on (M, g). We also know from Proposition 6.2.5 that being affine for a solution f of a semi-Riemannian eikonal equation is related to the Ricci curvature of (M,g) and the Hessian tensor of f. Now, by combining these results, one can obtain splitting theorems for certain semi-Riemannian manifolds which admit solutions to nonnull eikonal equations. In fact, in this chapter we obtain such splitting theorems in a more general context for semi-Riemannian, Riemannian and Lorentzian manifolds. We devote sections 7.1, 7.2 and 7.3 to such splitting theorems for semi-Riemannian, Riemannian and Lorentzian manifolds, respectively. Section 7.3 may be considered of separate interest for applications of semi-Riemannian maps in General Relativity.

A hypersurface in a Lorentzian manifold is said to be space-likeif the induced metric on the hypersurface is positive definite. On a space-like hypersurface, the firstfundamental form, the second fundamental form and the mean curvature are defined in the same way as those on a hypersurface in a Riemannian manifold [§1]. It has been proved by Bernstein and others that the only entire minimal hypersurface in a Euclidean space Rn+1 is a linear hyperplane for n^7, but there are other examples for n>7. So, Calabi proposed to study a Lorentzian analoge, called the Bernstein type problem, in Minkowki space i??+1, and this was solved by Cheng and Yau [4] for every n. More precisely, a space-like hypersurface in a Lorentzian manifold is said to be maximal, if the mean curvature is zero. The Bernstein type problem has led to the conclusion that the only entire maximal space-like hypersurface in i? +1 is a linear hyperplane. In order to prove this, Cheng and Yau [4] established the following result: (*) // an entire space-like hypersurface M in R+l has a constant mean curvature H, then the induced Lorentzian metric on M is a complete Riemannian metric and the length of second fundamental form of M is bounded from above by n\H\.