2011 YÜksek lisans tez özetleri

The Geometry Of Connectıons

In order to examine the geometry of connections, the knowledges of analysis, algebra and topology are intensely needed for review. As a result of the preparations of these knowledges, to see how they are used in differential geometry is the main purpose of this thesis.

This study consists of four main chapters that are differetiable manifolds and some basic concepts defined on differentiable manifolds, linear connections, Riemannian manifolds, tensor bundles and vector bundles.

In the first part, preliminaries used in the next sections are given. Then, by introducing the concept of differentiable manifold, examples are given and tangent space, vector fields, curves, differentiable maps on this structure are examined.

The definition of linear connection on a differentiable manifold is given in the second part. In addition, the condition of being a geodesic of a curve on a differentiable manifold is expressed with respect to a linear connection.

Metric field on a differentiable manifold is defined in the third part. In addition to this, by expressing the the concepts of Riemannian metric, Riemannian manifold and Riemannian connection, it is shown that Riemannian connection is a torsion free and a metric connection. On the other hand, the theorems related to a curve to be a geodesic on a Riemannian manifold are given and some results have been obtained.

In the last part, the notion of tensor bundles are examined and it is proved that this structure is a differentiable manifold. In addition, the concepts of vector bundles, frame bundles, principal fiber bundles are introduced briefly and the relationship between these structures and differentiable manifolds have been revealed.

ÖKTEM Müge

Danışman : Yrd. Doç. Dr. Hakan Mete TAŞTAN

Anabilim Dalı : Matematik

Mezuniyet Yılı : 2011

Tez Savunma Jürisi : Yrd. Doç. Dr. Hakan Mete TAŞTAN

Prof. Dr. Nazım SADIK

Prof. Dr. Leyla ZEREN AKGÜN

Prof. Dr. Mehmet ERDOĞAN

Doç. Dr. Fatma ÖZDEMİR