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.

scholarly journals Generalizations of 3-Sasakian manifolds and skew torsion

2020 ◽  
Vol 20 (3) ◽  
pp. 331-374 ◽  
Author(s):  
Ilka Agricola ◽  
Giulia Dileo
Keyword(s):  
Ricci Curvature ◽  
General Notion ◽  
Heisenberg Groups ◽  
Canonical Connection ◽  
Contact Metric Manifolds ◽  
Special Cases ◽  
Commutator Function ◽  
Killing Function ◽  
Generalized Killing Spinors ◽  
Sasaki Manifolds

AbstractIn the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the new classes should admit ‘good’ metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-(α, δ)-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result of Kashiwada. We study their behaviour under a new class of deformations, called 𝓗-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(α, δ)-Sasaki manifold is Einstein either if α = δ (the 3-α-Sasaki case) or if δ = (2n + 3)α, where dim M = 4n + 3.In the second part we find these adapted connections. We start with a very general notion of φ-compatible connections, where φ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-(α, δ)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.

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]).

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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