On Differential Geometry and Homogeneous Spaces II

2009 ◽  
pp. 36-39 ◽  
Author(s):  
Bertram Kostant
Keyword(s):  
Differential Geometry ◽  
Homogeneous Spaces

scholarly journals Reduction Theorem for Connections and its Application to the Problem of Isotropy and Holonomy Groups of a Riemannian Manifold

1955 ◽  
Vol 9 ◽  
pp. 57-66 ◽  
Author(s):  
Katsumi Nomizu
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifold ◽  
Symmetric Space ◽  
Homogeneous Spaces ◽  
Holonomy Group ◽  
Complete Riemannian Manifold ◽  
Reduction Theorem ◽  
Riemannian Symmetric Space ◽  
Isotropy Group ◽  
Holonomy Groups

The present paper constitutes, together with [13], a continuation of the study of differential geometry of homogeneous spaces which we started in [11]. Our main result states that if the homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at each point, then the Riemannian manifold is symmetric. The converse is of course one of the well known properties of a Riemannian symmetric space [4]. Besides the results already sketched in [12], we add a few applications of the main theorem.

scholarly journals A classification of Riemannian 3-manifolds with constant principal ricci curvaturesρ1= ρ2≠ ρ3

1993 ◽  
Vol 132 ◽  
pp. 1-36 ◽  
Author(s):  
Oldřich Kowalski
Keyword(s):  
Differential Geometry ◽  
Homogeneous Spaces ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Main Motivation ◽  
Locally Homogeneous ◽  
Riemannian Spaces

This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.

scholarly journals Some Studies on Kaehlerian Homogeneous Spaces

1957 ◽  
Vol 11 ◽  
pp. 77-92 ◽  
Author(s):  
Jun-Ichi Hano ◽  
Yozô Matsushima
Keyword(s):  
Differential Geometry ◽  
Homogeneous Spaces ◽  
Decomposition Theorem ◽  
Canonical Decomposition ◽  
Connected Group ◽  
Group Of Automorphisms ◽  
De Rham ◽  
Real Analytic ◽  
Riemannian Case ◽  
Simply Connected

The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

Applicable Differential Geometry

1987 ◽  
Author(s):  
M. Crampin ◽  
F. A. E. Pirani
Keyword(s):  
Differential Geometry
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