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

Abstract

We 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.

arXiv comment31 pages; matches version published in manuscripta mathematica

References in corpus (9)

• Symmetry structure of special geometries
• String loop corrected hypermultiplet moduli spaces
• HyperKahler and quaternionic Kahler manifolds with S^1-symmetries
• On the hyperkaehler/quaternion Kaehler correspondence
• Quaternionic Kähler metrics associated with special Kähler manifolds
• Twisting Hermitian and hypercomplex geometries
• Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence
• Twist geometry of the c-map
• Curvature of quaternionic Kähler manifolds with $S^1$-symmetry

Cited by in corpus (4)

• Symmetries of quaternionic Kähler manifolds with $S^1$-symmetry
• Curvature of quaternionic Kähler manifolds with $S^1$-symmetry
• A class of locally inhomogeneous complete quaternionic Kähler manifolds
• Symmetries of one-loop deformed q-map spaces