scholarly journals THE IDENTIFICATION OF CONFORMAL HYPERCOMPLEX AND QUATERNIONIC MANIFOLDS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 913-932 ◽  
Author(s):  
ERIC BERGSHOEFF ◽  
STEFAN VANDOREN ◽  
ANTOINE VAN PROEYEN
Keyword(s):  
Killing Vector ◽  
Kähler Manifolds ◽  
One Dimension ◽  
Complex Structures ◽  
Kahler Manifolds ◽  
Type Formula ◽  
New Type ◽  
Quaternionic Manifolds ◽  
Explicit Relation ◽  
Quaternionic Kähler

We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. "conformal hypercomplex" manifolds) and quaternionic manifolds of one dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between the structure of these manifolds is presented, including curvatures and symmetries. An important role is played by "ξ transformations," relating connections on quaternionic manifolds, and a new type "[Formula: see text] transformations" relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds.

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

scholarly journals Hyperkähler cones and instantons on quaternionic Kähler manifolds

2020 ◽  
Vol 58 (3) ◽  
pp. 291-323
Author(s):  
Chandrashekar Devchand ◽  
Massimiliano Pontecorvo ◽  
Andrea Spiro
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Quaternionic Kähler

scholarly journals QUATERNIONIC KAHLER MANIFOLDS OF COHOMOGENEITY ONE

1999 ◽  
Vol 10 (05) ◽  
pp. 541-570 ◽  
Author(s):  
ANDREW DANCER ◽  
ANDREW SWANN
Keyword(s):  
Simple Group ◽  
Lie Algebras ◽  
Complex Structure ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action ◽  
Hyperkähler Manifolds ◽  
Adjoint Orbits ◽  
Quaternionic Kähler

Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic Kähler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.

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