Causal symmetric spaces : geometry and harmonic analysis Symmetric Spaces: Basic Structure Theory. Dual Symmetric Spaces. The Module Structure of q. A subspaces. The Hyperboloids. Causal Orientations: Covex Cones and their Automorphisms. Causal Orientations. Semigroups. The Order Compactification. Examples. Symmetric Spaces Related to Tube Domains. Irreducible Causal Symmetric Spaces: Existence of Causal Structures. The Classification of Causal Symmetric Pairs. Clasification of Invariant Cones: SymmetricSL(2 R) Reduction. The Minimal and Maximal Cones. The Linear Convexity Theorem. The Classification. Extension of Cones. The Geometry: The Bounded Realization of H H (K. The Semigroup S(C). The Causal Intervals. Compression Semigroups. The Non Linear Convexity Theorem. The B<+># Order. The Affine Closure of B<+>#. The Order Compactification: Causal Galois Connections. An Alternative Realization of M<+>cpt. NOTE: THE cpt SHOULD APPEAR DIRECTLY OVER A + SIGN. The Stabilizers for M<+>cpt. SEE NOTE. Theo Orbit Structure of M<+>cpt. (NO + SIGN.) The Space SL(3 R) SO(2 1). Holomorphic Representations: Holomorphic Representations of Semigroups. Highest Weight Modules. The Holomorphic Discrete Series. Classical Hardy Spaces. Hardy Spaces. The Cauchy Szegi Kernel. Spherical Functions: The Classical Laplace Transform. Spherical Functions. The Asymptotics. Expansion Formula. The Spherical Laplace Transform. The Abel Transform. Relation to Representation Theory. The Wiener Hopf Algebra. Appendices. Notation. References. Bibliography. Author Index. Subject Index This book is intended to introduce researchers and graduate students to the concepts of causal symmetric spaces. To date results of recent studies considered standard by specialists have not been widely published. This book seeks to bring this information to students and researchers in geometry and analysis on causal symmetric spaces. Includes the newest results in harmonic analysis including Spherical functions on ordered symmetric space and the holmorphic discrete series and Hardy spaces on compactly casual symmetric spaces Deals with the infinitesimal situation coverings of symmetric spaces classification of causal symmetric pairs and invariant cone fields Presents basic geometric properties of semi simple symmetric spaces Includes appendices on Lie algebras and Lie groups Bounded symmetric domains (Cayley transforms) Antiholomorphic Involutions on Bounded Domains and Para Hermitian Symmetric Spaces