الفهرس Only 14 pages are availabe for public view
 |  | from | 127 |  |  |
| | | from | 127 | | |
Abstract
This thesis deals with a class of smooth manifolds called Grassmann manifolds which are very important manifolds in the differential geometry and physics specially in unified field theory.
For F = ; the real number field, ; the complex number field, ; the algebra of quaternions, let Gp,q(F) denote the real, complex and quaternion Grassmann manifolds of all p-planes in Fp+q. We view Gp,q(F) as a homogeneous Riemannian symmetric space, moreover it is compact and simply connected.
The main purpose of the thesis to study the differential geometry properties of these manifolds. Furthermore, we present a characterization of Gp,q(F) and its non-compact dual G*p,q(F) by means of a particular parallel tensor field T of type (1, 3) satisfying certain algebric conditions and the Weingatren map (or shape operator) on a small geodesic spheres.
The thesis is divided into five chapters and each chapter is divided into subsections.
The first chapter is introductory and is basically intended to make the thesis as self-contained as possible. In this chapter we gave basic definitions and summarized the basic theorems and results which are essential for other chapters.
In second chapter, we study the geometric properties of the real Grassmann manifold Gp,q( ) of p-planes in .
ii
We discuss how this manifold can be seen as a homogeneous Riemannian symmetric space of orthogonal matrices, (p+q) (p) (q). We prove that Gp,q( ) is an Einstein space and has positive curvature. And as a special case G1,q( ) or (Gp,1( ), the real projective space which is defined as the space of all lines, through the origion in (or ) has constant curvature. Then we give characterization of this manifold and its non-compact dual, which was introduced by Blair and Ledger [B2], This characterization, as we mentioned above, is given in terms of a parallel tensor field T of type (1, 3) satisfying certain properties and the Weingatren map on a small geodesic spheres. The main part of this work concerns an algebraic characterization of the tensor T on a vector space V of dimension pq.
The case of Gp,2( ) was studied by A. J. Ledger and B. J. Papantonion [P1] as a Hermitian symmetric space .
In third chapter, similar results are given for the complex Grassmann Manifold Gp,q( ) and its non- compact dual space. As Gp,q( ) is kaehler manifold, then in this case, the algabric characterization of the tensor T, is given on a 2pq- dimensional real vector space V with complex structure J. This case was studied by A. J. Ledger [L1].
iii
The main results which we have achived in this thesis are presented in chapter IV. This chapter is concerned firstly, with some algebric properties of the non-commutative division quaternion algebra , quaternion matrices and their adjoint represetation as real and complex matrices, which can be found in [Z],[K2].
Secondly we study the geometric properties of the quaternion Grassmann manifold Gp,q( ), the set of all p-dimensional subspaces in , see for example [C1],[H]. Then we present a characterization of Gp,q( ) and its non-compact dual G*p,q( ) in terms of a parallel tensor field T of type (1, 3) which has some particular algebraic properties without using the Hermitian structure. The tensor T is characterized in propesition (4.6.2), which is the important part of the proof. Owing to the algebra of quaternion is not commutative, so the conditions imposed here differ from those [L1],[S3], as we identy the tangent space at a point po Gp,q( ) with the vector space of all p×q quaternion matrices but have the advantage of relating closely to those of the real Grassmann [B2].
In fifth chapter, we will present the subjects of fractals, which play an interesting role in mathematics. We explore several different kinds of fractals, explain their mathematical definition, and show how they could be used to generate aesthetically beautiful images.
1