Some special connections on Weyl manifolds

Weyl Geometry was introduced by H. Weyl to formulate unified field theory in 1918. Although his theory failed for physical reasons, it was interesting not only for physicists but also for mathematicians. A differentiable manifold of dimension $n$ which has a torsion-free connection $D$ and a representative metric $g$ in the conformal class of metrics preserved by $D$ is called a Weyl manifold which satisfies the compatibility condition \begin{equation*} Dg=2\omega\otimes g, \end{equation*} where $\omega$ is a $1$-form. Such a manifold will be denoted by $M_n(g, \omega)$. Under the conformal transformation of the metric ~$\tilde{g}=\lambda^2g$, the $1$-form $\omega$ changes by the law \begin{equation*} \tilde{\omega}= \omega + d \ln \lambda. \end{equation*} The pair $(\tilde{g},\tilde{\omega})$ generates the same Weyl manifold and the process of passing from $(g,\omega)$ to $(\tilde{g},\tilde{\omega})$ is called a gauge transformation. In this work, fist we have primarily examined Weyl manifolds and complex Weyl manifolds with some special non-symmetric connections. Then, we have introduced the notion of quasi-Einstein Weyl manifolds, and we have constructed an example to quasi-Einstein Weyl manifold. Finally, we have studied some applications of Lie derivatives on Weyl manifolds. This work contains six chapters. In chapter I, a brief literature review of Weyl manifolds is given. In chapter II, first, the fundamental definitions and theorems concerning Riemannian manifolds and Weyl manifolds are stated. Then, Riemannian manifolds having semi-symmetric connections, quarter-symmetric connections and Ricci quarter-symmetric connections are reviewed. Moreover, infinitesimal transformations on Riemannian manifolds are given. Finally, some special vector fields on Weyl manifolds are defined. In chapter III, Weyl manifolds with semi-symmetric and Ricci quarter-symmetric connections are studied. First, the known results about Weyl manifolds with semi-symmetric connection are stated. Then, Ricci quarter-symmetric connections are defined on Weyl manifolds and it is proved that on every Weyl manifold, there exists a unique Ricci quarter-symmetric connection associated to every one form. Furthermore, some properties of the curvature tensor according to Ricci quarter-symmetric connection are examined and the necessary and sufficient conditions for the Ricci tensor to be symmetric are obtained. Finally, it is proved that a Weyl manifold with a Ricci quarter-symmetric connection whose sectional curvature is independent of the each plane is locally conformal to an Einstein manifold with a Ricci quarter-symmetric connection. Chapter four consists of five sections. In the first section, a review of Einstein-Weyl manifolds with some known examples and results is given. In the second section, quasi-Einstein Weyl manifolds and nearly quasi-Einstein Weyl manifolds are defined. It is proved that if the associated scalar functions of a quasi-Einstein Weyl manifold having positive scalar curvature are prolonged covariantly constant then the manifold is locally conformal to an Einstein manifold. The necessary condition for the existence of the quasi-Einstein Weyl manifolds is also given. In the third section, an example of three dimensional quasi-Einstein Weyl manifold is constructed. In the last two sections, quasi-Einstein Weyl manifolds with semi-symmetric and Ricci quarter-symmetric connections are examined. Moreover, conformal and generalized concircular transformations of quasi-Einstein Weyl manifolds are studied. The following theorems concerning quasi-Einstein Weyl manifolds are proved: Theorem. A quasi-Einstein Weyl manifold is transformed into a quasi-Einstein Weyl manifold under any generalized concircular mapping. Theorem. Assume that there is a conformal mapping between two Weyl manifolds. If the symmetric parts of the corresponding Ricci tensors of the manifolds are preserved, then the conformal transformation is concircular. In chapter five, complex Weyl manifolds are examined. Particularly, Kahler Weyl and almost contact Weyl manifolds are studied. First, some important pure and hybrid tensors on Kahler Weyl manifolds are reviewed and certain known results are presented. Then, it is proved that a Kahler Weyl manifold with a vanishing conformal curvature tensor or concircular curvature tensor is an Einstein-Weyl manifold. On the other hand, it is showed that the scalar curvature of the Kahler Weyl manifold is zero if the projective curvature tensor of the Kahler Weyl manifold vanishes. After that, Kahler Weyl manifolds with quarter-symmetric connections are introduced and the existence of this connection is proved. Finally, some important tensors according to the quarter-symmetric connections are examined and some related theorems are proved. In the last section of the chapter five, almost contact Weyl manifolds are considered. After giving the basic definitions and equations about almost contact Weyl manifolds, quarter-symmetric connections are defined on them. The existence of the quarter-symmetric connections on almost contact Weyl manifolds is also proved and the properties of the curvature tensor according to quarter-symmetric connection are examined. In chapter six, first, applications of Lie derivative and infinitesimal transformations on Weyl manifolds are studied. Then, the properties of prolonged Lie derivative are obtained. After giving the definition of the generalized motion, the necessary and sufficient condition for a generalized motion to be an affine motion is obtained. Also, the following theorem concerning generalized motions and projective motions is proved: Theorem. An infinitesimal transformation which is both generalized motion and projective motion on a Weyl manifold is an affine motion or the conharmonic curvature tensor of the manifold vanishes. In particular, generalized motions are examined on recurrent Weyl manifolds and some theorems are proved. Finally, by making analogy to solitons on Riemannian manifolds, the notion of solitons on Weyl manifolds is introduced and Weyl manifolds satisfying particular equations are studied by considering some special vector fields.