الفهرس 
Basic Definitions and TheoremsOnly 14 pages are availabe for public view
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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 = R; the real number field, C; the complex number field, Q; the quaternion
number field, let () denote the real, complex and quaternion Grassmann manifolds of all p-planes in +. We view (Q) as a homogeneous Riemannian symmetric space, moreover it is compact and simply connected.
Apart from being a vital and exciting field in mathematics with interesting results, projective spaces have various applications in design theory, coding theory, physics, combinatorics, number theory and extremal combinatorial problems. We consider real, complex and quaternion projective spaces. The geometric characteristic that we focus on are the sectional curvatures. These depict how curved the space is in some 2-dimensional subspace of the tangent space at a given point. The compact rank 1 symmetric spaces are the sphere, and the projective spaces over the reals, complex numbers, and quaternions. We are interested in the homogeneous spaces that have strictly positive sectional curvatures.
The new period of progress in dynamics in a complex domain started mostly because of the attractive computer graphics of Mandelbrot, connected to the pretty pictures of Julia and Mandelbrot sets, and also to fractals. The quaternion Mandelbrot set is one of the most important sets in mathematics. We give some properties of the quaternion algebra.
Then, we introduce the quaternion dynamical system. We are concerned with analytical investigation of the quaternion dynamical system.