Etudiant : AKENTOUR NOUR EDDINE

Filière : Master Mathématiques Pures (MMP)

Encadrant : Pr. KADAOUI ABBASSI MOHAMED TAHAR

Annèe : 2024

Résumé : In this paper, we introduced the theory of spaces of constant curvature by defining many associated concepts and giving some important results which we can resume as follow: • A space of constant curvature is a Riemannian manifold which has a constant sectional curvature. • Model spaces are the complete simply connected Riemannian manifolds of constant curvature. Up to isometry, there is exactly three model spaces; The Euclidean space R n (zero constant curvature), the sphere S n (positive constant curvature) and the hyperbolic spaces Hn (negative constant curvature). • Space forms are the complete connected Riemannian manifolds of constant curvature. A Riemannian manifold is a space form if and only if it is isometric to a quotient of a model space by a group of isometries acting freely and properly discontinuously on that model space. • The Bieberbach theorem from crystallographic geometry caracterise the compact Euclidean space forms by their fundamental group; Two compact Euclidean space forms are affinely equivalent if and only if their fundamental groups are isomorphic, then to classify the compact connected Riemannian manifolds of constant zero curvature we need only to do the job with those groups. Moreover, a fundamental group of a compact Euclidean space form is a crystallographic group, and Charlap gived a complete classification for those groups in [2]. • The spherical space forms of even dimension are exactly, up to isometry, the spheres and the projective spaces. • If M = S n/Γ is a spherical space form, then we can view Γ as the image of a faithful (one to one) orthogonal representation of an abstract finite group.