| Chen’s Inequalıtıes And Theır Applıcatıons To Some Space Forms
The main purpose of this thesis is to investigate Chen’s inequalities and their applications to some space forms.
The study consists of four parts. In the first part, a general evaluation of some historical facts about curvatures and further improvements of them called Riemannian invariants which have been defined by B.Y. Chen are presented.
The second part includes five sections. In section 2.1. some definitions and fundamental theorems that will be needed in the content of the thesis are given. In section 2.2. some new types of Riemannian curvature invariants are presented. In section 2.3. Riemannian space forms, Einstein spaces and conformally flat spaces are characterized. In the first part of section 2.4. Chen’s inequalities and the equality cases of them for Riemannian space forms are examined. In this part, firstly sharp inequalities involving and then relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions are investigated. Moreover some special submanifolds which satisfy Chen’s equality are studied. In the second part of this section, a general optimal inequality for arbitrary Riemannian submanifolds is looked over. In section 2.5. a special pointwise inequality in submanifold theory is studied.
In the third part, some characterizations of totally geodesic Riemannian space forms isometrically immersed in a Riemannian space form are obtained and also for each characterization a necessary condition for a Riemannian manifold to be a Riemannian space form and minimal in any Euclidean space regardless of codimension is obtained.
An evaluation of this study is placed in the fourth part.
YILMAZTÜRK Utku ,
Danışman : Prof. Dr. Erhan GÜZEL
Anabilim Dalı : Matematik
Mezuniyet Yılı : 2006
Tez Savunma Jürisi : Prof. Dr. Erhan GÜZEL (Danışman)
Prof. Dr. Bedriye M. ZEREN
Prof. Dr. Yusuf AVCI
Prof. Dr. Musa İLYASOV
Prof. Dr. Hülya ŞENKON
Dostları ilə paylaş: | 
