Chapter 1 Algebraic preliminaries The main goal of this chapter is to introduce some notations and terminologies. We assume that the reader is more or less familiar with the basic concepts of algebraic geometry and linear algebra. Chapter 2 Exceptional groups $G_2(K)$ and $F_4(K)$ This chapter takes up the systematic study of a generalization of the exceptional compact Lie groups $G_2$ and $F_4$ on groups $G_2(K)$ and $F_4(K)$ provided $K$ is Pythagorean formally real field. The main result stated in Theorem 2.48 says that any Hermitian $3 \times 3$-matrix $A \in \mbox{Herm}_3(\mathbb{O}(K))$ can be transformed to a diagonal form by some element of $F_4(K)$. Chapter 3 Stiefel, Grassmann manifolds and generalizations In this chapter we investigate and prove some properties of the classical manifolds of Stiefel, Grassmann and flag manifolds. All along this chapter $\mathcal{A}$ denotes the field of reals, $\mathbb{R}$, the field of complex numbers, $\mathbb{C}$, the skew field of quaternions, $\mathbb{H}$ and, except if otherwise said the octonion division algebra, $\mathbb{O}$. Chapter 4 More classical matrix varieties In this chapter we generalize Stiefel, Grassmann and flag manifolds, defined in Chapter 3, to what we call here i-Stiefel, i-Grassmann and i-flag manifolds.

This "i" comes from idempotent. Those manifolds do not seem to have being enough studied in the literature. In particular, they do not have even a name. As in Chapter 2,$ \mathcal{A}$ denotes the field of reals, $\mathbb{R}$, the field of complex numbers, $\mathbb{C}$, theskew field of quaternions, $\mathbb{H}$ and, occasionally, the octonion division algebra $\mathbb{O}$. Chapter 5 Algebraic generalizations of matrix varieties We use Chapters 1 and 2 to define and extend results of Chapters 3 and 4 to matrix varieties over more general division algebras. That includes extending the classical definitions of Riemannian, Hermitian and symplectic manifolds. All along this chapter $K$ is a formally real Pythagorean field and $\mathcal{A}$ denotes either $K$, the complex $K$-algebra $\mathbb{C}(K)$, the quaternion $K$-algebra $\mathbb{H}(K)$ or the octonion $K$-algebra $\mathbb{O}(K)$. Chapter 6 Curvature, geodesics and distance on matrix varieties In this chapter we study more closely the Riemannian structure of classical matrix manifolds introduced in Chapters 3 and 4.

Here, $\mathcal{A} = \mathbb{R},\, \mathbb{C},\, \mathbb{H}$ and occasionally $\mathbb{O}$. We also extend, whenever it is possible, definitions and results to the general case treated in Chapter 5, where $\mathcal{A} = K,\, \mathbb{C}(K),\, \mathbb{H}(K),\, \mathbb{O}(K)$ for $K$ a Pythagorean formally real field.