Special quasi Einstein manifolds

In this thesis, we focus on a special class of Riemannian and semi-Riemannian manifolds, which in the mathematical literature, provides several interesting results and relations with many other geometrical structures on manifolds. From historical point of view, the origin of this topic comes from the Einstein manifolds, in which the Ricci tensor and the metric tensor are proportional. The topic of Einstein manifolds and its well-known generalizations, such as quasi-Einstein and $\eta$-Einstein, were intensively studied in literature. In differential geometry, Einstein manifolds represent the metrics of constant Ricci curvature, as well as in physics they have been used to find the exact solutions of Einstein field equations and they are very useful tools to classify the space-times. Einstein field equations, which is the most important equation of the theory of General Relativity, show that the geometry of the manifold can be directly determined by its matter content. For this reason, some generalizations on the Ricci tensor become crucial in order to understand the global character of the manifold. The notion of quasi Einstein manifolds, was first introduced by M. C. Chaki and R. K. Maity in 2000 as a geometric concept that generalizes the Einstein manifolds. On the other hand, they form a model for the solutions of Einstein field equations with perfect fluid matter content. Another natural generalization of the Einstein manifolds is the concept of Ricci soliton, which was introduced by Hamilton. This new concept emerge as a self-similar solution of the Ricci flow equation, arising from the classification problem of manifolds according to their curvatures. The Ricci flow is a parabolic type of partial differential equation that allows the metric (i.e. the shape of manifold) to be transformed into a suitable metric of constant curvature in proportion to the Ricci tensor. Thus, finding more general solutions of the Ricci flow and examining the behavior of these solutions are very important frameworks in the sense of geometry. Therefore, for the analysis of space-times having different curvatures or different matter contents, the Ricci curvature tensor should be more generalized. For this purpose, the notion of generalized quasi Einstein manifolds was introduced by M. C. Chaki in 2001, geometrically. This new concept, in terms of its definition, not only generalizes the quasi Einstein manifolds geometrically, but also forms a model of 4-dimensional space-time that allows heat flux, physically. Due to these reasons, one of the main objectives of this thesis is to investigate some properties of the generalized quasi Einstein manifolds, which are expected to have more interesting properties in both geometric and physical aspects. Later, $m$-Bakry-Emery Ricci tensor, which is closely related to the Ricci solitons and is a natural generalization of Ricci tensor, was introduced and many geometers have described various special manifolds, in which the $m$-Bakry-Emery Ricci tensor is proportional to the metric tensor. One of these special manifolds is the $(m,\rho)$-quasi Einstein manifold, that constitute the other part of this thesis. From historical point of view, these two fields of the study are closely related to the warped product manifolds, which were introduced by Bishop and O'Neill. For example, it is known that Ricci solitons and gradient Ricci solitons under certain conditions, as well as many basic solutions of Einstein field equations, such as Robertson-Walker, Schwarzchild, Reissner-Nordstr\"{o}m-de Sitter space-times have warped product structures. Also, surface of revolutions, sphere and $\mathbb{R}^n-\{0\}$ are locally warped product manifolds. Since both of the above mentioned basic fields are both geometrically formulated and physically linked to some important relativistic models, this thesis study is carried out in two different aspects using the tools of the Riemannian and Lorentzian geometries. In the first chapter, the main topics to be covered in the thesis are introduced and the importance of them are emphasised. A brief review of literature on certain Riemannian and semi-Riemannian manifolds is also given. The second chapter contains the basic notions to be used during the thesis study and the definitions of the various generalizations of the Einstein manifolds, that constitute the first half of the thesis. In the third chapter, some problems related to the characterization of generalized quasi Einstein manifolds under certain symmetry conditions are discussed. First, Ricci-pseudosymmetric and Ricci semisymmetric generalized quasi Einstein manifolds are considered and then such symmetry conditions are extended to the different curvature tensors, such as conformal, concircular, quasi-conformal, $W_2$, projective and pseudo-projective curvature tensors. Then, some classification theorems related to the generalized quasi Einstein manifolds satisfying the above mentioned symmetry conditions have been proved. Later, it is proved that if the covariant derivatives of the Ricci tensor provide certain conditions (e.g. Ricci recurrent, generalized Ricci recurrent), the manifold has the parallel generator vector field and thus the manifold reduces to a quasi Einstein manifold. In the last section of this chapter, the existence of a generalized quasi Einstein manifold is proved by constructing a concrete non-trivial Riemannian metric. In the fourth chapter, first some necessary conditions are investigated so that the generalized quasi Einstein manifold has a warped product structure. In this context, with the help of conformal flatness condition, it is shown that the generator vector field of the $n$-dimensional Ricci semisymmetric generalized quasi Einstein manifold is the proper concircular vector field. As consequence of this, the manifold under consideration is the warped product of a real interval and an $(n-1)$-dimensional Einsteinian fiber. Secondly, in the case that different curvature tensors (e.g. pseudo-projective curvature tensor) are identically zero or conservative (i.e., divergence-free), the relations between the warped product manifolds and the generalized quasi Einstein manifolds have been examined. Also, contrary to the previous case, a generalized quasi Einstein manifold having a warped product structure, has been classified according to its base and fiber with the help of its generators, and then some links with Einstein-like manifolds have been established. The next step in the evolution of the warped products is achieved by Blair, who introduced the notion of $D$-warping. More precisely, on the product manifold $M_1\times M_2$ of a Riemannian manifold $(M_1,g_1)$ with an almost contact metric manifold $(M_2,\phi_2, \xi_2 ,\eta_2, g_2)$, where $\eta_2$ is the dual 1-form of the unit vector field $\xi_2$ with respect to $g_2$, a $D$-warping metric is defined by the metric $g= g_1 +fg_2 +f(f - 1)\eta_2 \otimes \eta_2$, for any positive, smooth function $f$ on $M_1$. In the last part of the fourth chapter, the $D$-warping metric is extended with a slight modification of the metric $g$, by taking into account the 1-form on the second component manifold: For two Riemannian manifolds $(M_i,g_i)$, $i=1,2$, the $D$-general warping metric $g$ is given also by the same metric, but $M_2$ carries a unit vector field only, instead of almost contact metric structure. Then, some geometrical objects, such as the Levi-Civita connection, the Riemannian and the Ricci tensors and also the sectional and the scalar curvature, are calculated on $D$-general warping $(M,g)$, in the context of warped product. A Laplacian formula of $g$ is obtained and the harmonicity of functions and forms on $(M,g)$ is described. Finally, some necessary and sufficient conditions for $D$-general warping $(M,g)$ to be Einstein, quasi-Einstein or $\eta$-Einstein are provided. Chapter five consists of the study of the generalized quasi Einstein space-times. Under the symmetry conditions discussed in the third chapter, the generalized quasi Einstein space-times are characterized in the context of Lorentz metric signature. Then various applications are given, by using some important tools of the theory of General Relativity: It is proved that Ricci symmetric (or recurrent) generalized quasi Einstein space-times can be considered as a model of perfect fluids. Also, such space-time satisfying Einstein's field equations has constant energy density and the isotropic pressure. Moreover, it has the vanishing expansion scalar and the acceleration vector. As a consequence, the possible local cosmological structures of this space-time are of Petrov $I$, $D$ or $O$. Finally, the existence of a generalized quasi Einstein space-times is proved by constructing a concrete non-trivial Lorentzian metric. In the sixth chapter, first a historical overview of the developments on the theory of Ricci solitons and some basic definitions about them are given. Then, the characterizations of an $(m,\rho)-$quasi Einstein manifold admitting closed conformal or parallel vector field are made. Some necessary lemmas related to the formulas of the gradient and the Hessian of the potential function, are proved. By using these lemmas, some rigidity results for this class of manifolds are obtained. It is proved that an $(m,\rho)-$quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form $I\times_{\varphi}M^*$. Then, this result is reduced to the case of an $(m,\rho)-$quasi Einstein manifold admitting a parallel vector field, directly. Therefore, it is shown in this case that the Ricci tensor has the classical quasi Einstein structure and also the similar rigid condition holds. Finally, a non-trivial example of an $(m,\rho)-$quasi Einstein manifold verifying obtained results is constructed.