[Paper Review] Quaternionic Kähler Manifolds of Cohomogeneity One

Andrew Dancer, Andrew Swann|ArXiv.org|Aug 21, 1998

Geometry and complex manifolds7 citations

TL;DR

This paper classifies compact quaternionic Kähler manifolds with a cohomogeneity-one action by a semi-simple group, along with related hyperKähler and 3-Sasakian manifolds under similar symmetry conditions. Using representation theory and geometric analysis of group actions, it establishes that such manifolds are determined by specific symmetric spaces and root system data, providing a complete classification in the cohomogeneity-one setting with explicit examples and structural constraints.

ABSTRACT

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.

Motivation & Objective

• To classify compact quaternionic Kähler manifolds admitting a cohomogeneity-one action by a semi-simple Lie group.
• To extend classification to complete hyperKähler manifolds with cohomogeneity-two actions preserving each complex structure.
• To classify compact 3-Sasakian manifolds that are cohomogeneity one with respect to 3-Sasakian symmetries.
• To analyze non-compact quaternionic Kähler manifolds of cohomogeneity one and their geometric structure.
• To investigate the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras as a related geometric problem.

Proposed method

• Utilizes representation theory of semi-simple Lie groups to analyze orbit structure and isotropy types.
• Applies the theory of symmetric spaces and root systems to classify possible homogeneous models.
• Employs the notion of cohomogeneity-one actions, where the principal orbit has codimension one, to constrain the geometry.
• Analyzes the induced metric and holonomy properties of the manifolds to verify quaternionic Kähler structure.
• Uses the correspondence between 3-Sasakian and hyperKähler geometry to reduce problems to known results on hyperKähler quotients.
• Applies results from the theory of moment maps and Hamiltonian group actions in the hyperKähler setting.

Experimental results

Research questions

• RQ1 Which compact quaternionic Kähler manifolds admit a cohomogeneity-one action by a semi-simple Lie group?
• RQ2 What are the complete hyperKähler manifolds with cohomogeneity-two actions preserving each complex structure?
• RQ3 Which compact 3-Sasakian manifolds are cohomogeneity one with respect to 3-Sasakian symmetries?
• RQ4 What is the structure of non-compact quaternionic Kähler manifolds of cohomogeneity one?
• RQ5 What is the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras under the adjoint action?

Key findings

• Compact quaternionic Kähler manifolds with a cohomogeneity-one action by a semi-simple group are classified and shown to arise from symmetric spaces associated with specific root systems.
• The classification of complete hyperKähler manifolds with cohomogeneity-two actions is completed under the additional hypothesis that each complex structure is preserved by the group action.
• All compact 3-Sasakian manifolds of cohomogeneity one with respect to 3-Sasakian symmetries are classified and shown to correspond to certain symmetric spaces.
• Non-compact quaternionic Kähler manifolds of cohomogeneity one are shown to be modeled on specific symmetric spaces with non-compact duals.
• The cohomogeneity of adjoint orbits in complex semi-simple Lie algebras is determined to be one under the adjoint action of the corresponding group.
• The results establish a strong link between group cohomogeneity and the underlying symmetric space geometry in quaternionic Kähler and related structures.