On Naturally Reductive Homogeneous Spaces Harmonically Embedded into Spheres

1984 ◽  
Vol s2-29 (1) ◽  
pp. 175-180
Author(s):  
Gábor Tóth
Keyword(s):  
Homogeneous Spaces ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive

Classification of five-dimensional naturally reductive spaces

1985 ◽  
Vol 97 (3) ◽  
pp. 445-463 ◽  
Author(s):  
Oldřich Kowalski ◽  
Lieven Vanhecke
Keyword(s):  
General Theory ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Natural Generalization ◽  
Riemannian Symmetric Spaces ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive ◽  
Volume Preserving ◽  
Number Of Authors

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).

scholarly journals Quaternionic Heisenberg groups as naturally reductive homogeneous spaces

2015 ◽  
Vol 12 (08) ◽  
pp. 1560007 ◽  
Author(s):  
Ilka Agricola ◽  
Ana Cristina Ferreira ◽  
Reinier Storm
Keyword(s):  
Homogeneous Spaces ◽  
Heisenberg Groups ◽  
Canonical Connection ◽  
Vertical Distributions ◽  
Metric Connection ◽  
Contact Metric Structure ◽  
Horizontal And Vertical Distributions ◽  
Generalized Killing Spinors ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive

In this paper, we describe the geometry of the quaternionic Heisenberg groups from a Riemannian viewpoint. We show, in all dimensions, that they carry an almost 3-contact metric structure which allows us to define the metric connection that equips these groups with the structure of a naturally reductive homogeneous space. It turns out that this connection, which we shall call the canonical connection because of its analogy to the 3-Sasaki case, preserves the horizontal and vertical distributions and even the quaternionic contact (qc) structure of the quaternionic Heisenberg groups. We focus on the 7-dimensional case and prove that the canonical connection can also be obtained by means of a cocalibrated G2 structure. We then study the spinorial properties of this group and present the noteworthy fact that it is the only known example of a manifold which carries generalized Killing spinors with three different eigenvalues.

POLAR ACTIONS ON BERGER SPHERES

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1019-1023 ◽  
Author(s):  
ANTONIO J. DI SCALA
Keyword(s):  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Type Theorem ◽  
Torus Action ◽  
Polar Action ◽  
Polar Actions ◽  
Berger Spheres ◽  
Berger Sphere ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive

The object of this article is to study a torus action on a so-called Berger sphere. We also make some comments on polar actions on naturally reductive homogeneous spaces. Finally, we prove a rigidity-type theorem for Riemannian manifolds carrying a polar action with a fix point.

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