[Paper Review] Quaternionic contact Einstein manifolds

Stefan Ivanov, Ivan Minchev+1 more|arXiv (Cornell University)|Jun 3, 2013

Geometric Analysis and Curvature Flows12 references11 citations

TL;DR

This paper proves that the qc-scalar curvature of a seven-dimensional quaternionic contact Einstein manifold is constant, resolving a long-standing gap in the theory. Using the qc-conformal curvature tensor and extensions of prior structure equations, the authors characterize qc-Einstein structures via a flat vertical connection and show that regular, Ricci-flat qc-Einstein manifolds fiber over hyper-Kähler manifolds, extending known results from non-zero scalar curvature cases.

ABSTRACT

The main result is that the qc-scalar curvature of a seven dimensional quaternionic contact Einstein manifold is a constant. In addition, we characterize qc-Einstein structures with certain flat vertical connection and develop their local structure equations. Finally, regular qc-Ricci flat structures are shown to fibre over hyper-Kahler manifolds.

Motivation & Objective

• To prove that the qc-scalar curvature of a seven-dimensional quaternionic contact Einstein manifold is constant, resolving an open case left unresolved in prior work.
• To characterize qc-Einstein structures using the flatness of a newly defined connection on the vertical distribution.
• To derive explicit structure equations for qc-Einstein manifolds in terms of defining 1-forms, their exterior derivatives, and the qc-scalar curvature.
• To extend known fibration results from non-vanishing qc-scalar curvature to the case of zero qc-scalar curvature, showing regular qc-Ricci flat structures fiber over hyper-Kähler manifolds.
• To demonstrate that every qc-Einstein manifold with non-zero qc-scalar curvature admits two Einstein metrics, one locally 3-Sasakian and one 'squashed' metric.

Proposed method

• The proof of scalar curvature constancy relies on the qc-conformal curvature tensor and algebraic curvature properties from Kulkarni’s work in four dimensions.
• The authors extend a prior result (IMV, Theorem 1.21) to explicitly describe qc-Einstein structures on open sets of the quaternionic Heisenberg group that are pointwise qc-conformal to the flat structure.
• A new connection on the canonical 3-dimensional vertical distribution is defined and shown to be flat if and only if the manifold is qc-Einstein, providing a characterization.
• Structure equations for qc-Einstein manifolds are derived using the defining 1-forms, their exterior derivatives, and the qc-scalar curvature, generalizing results from IV2 and IV3.
• The fibration result is established by analyzing the Riemannian geometry induced by a one-parameter family of metrics $ h^ u $, derived from the Biquard connection and qc-structure.
• The Ricci and scalar curvatures of the induced metrics $ h^ u $ are computed via the curvature difference formula between the Levi-Civita and Biquard connections, leading to explicit expressions in terms of $ S $ and $ u $.

Experimental results

Research questions

• RQ1 Is the qc-scalar curvature of a seven-dimensional qc-Einstein manifold necessarily constant?
• RQ2 Can qc-Einstein structures be characterized by the flatness of a connection on the vertical distribution?
• RQ3 What are the local structure equations of a qc-Einstein manifold in terms of differential forms and curvature?
• RQ4 Do regular qc-Ricci flat structures fiber over hyper-Kähler manifolds, analogous to the non-zero scalar curvature case?
• RQ5 Do all qc-Einstein manifolds with non-zero qc-scalar curvature admit two distinct Einstein metrics?

Key findings

• The qc-scalar curvature of a seven-dimensional qc-Einstein manifold is constant, resolving Theorem 1.1.
• qc-Einstein structures are characterized by the flatness of a canonical connection on the vertical distribution, providing a new intrinsic criterion.
• The structure equations of a qc-Einstein manifold are explicitly written in terms of the defining 1-forms, their exterior derivatives, and the qc-scalar curvature, extending prior results to the zero scalar curvature case.
• Regular qc-Ricci flat structures are shown to fiber over hyper-Kähler manifolds, generalizing the fibration property from non-vanishing scalar curvature cases.
• For any qc-Einstein manifold with non-zero qc-scalar curvature $ S $, the family of metrics $ h^ u $ yields two Einstein metrics: one locally 3-Sasakian and one 'squashed' metric, both with constant Ricci curvature.
• The Ricci curvature of $ h^ u $ has exactly two distinct constant eigenvalues of multiplicities $ 4n $ and $ 3 $ when $ S = 0 $, confirming the structure in the Ricci-flat case.