Homogeneous Spaces - Algebraic Homogeneous Spaces, Topology of Homogeneous Spaces, Sphere, Symmetric Space, Grassmannian, Kostant Polynomial (Paperback)

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 27. Chapters: Algebraic homogeneous spaces, Topology of homogeneous spaces, Sphere, Symmetric space, Grassmannian, Kostant polynomial, Erlangen program, Generalized flag variety, Hyperbolic space, Nilmanifold, Principal homogeneous space, Stiefel manifold, Hermitian symmetric space, Klein geometry, Quaternionic projective space, Quaternion-Kahler symmetric space, Lagrangian Grassmannian, Affine Grassmannian, Parabolic geometry, Weakly symmetric space, Schubert calculus, Iwasawa manifold, Clifford-Klein form. Excerpt: In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point. There are two ways to make this precise, via Riemannian geometry or via Lie theory; the Lie theoretic definition is more general and more algebraic. In Riemannian geometry, the inversions are geodesic symmetries, and these are required to be isometries, leading to the notion of a Riemannian symmetric space. More generally, in Lie theory a symmetric space is a homogeneous space G/H for a Lie group G such that the stabilizer H of a point is an open subgroup of the fixed point set of an involution of G. This definition includes (globally) Riemannian symmetric spaces and pseudo-Riemannian symmetric spaces as special cases. Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. They were first studied extensively and classified by Elie Cartan. More generally, classifications of irreducible and semisimple symmetric spaces have been given by Marcel Berger. They are important in representation theory and harmonic analysis as well as differential geometry. Let M be a connected Riemannian manifold and p a point of M. A map f defined on a neighborhood of p is said to be a geodesic symmetry, if it fix...