Harmonic Geometry of Nearly Kähler Manifolds

Nearly Kähler manifolds form a distinguished class of almost Hermitian six-dimensional spaces characterised by a non-vanishing intrinsic torsion lying entirely in a single irreducible SU(3)-module. These structures are strictly Einstein and arise naturally in both Riemannian geometry and the study of special holonomy. Harmonic geometry on these backgrounds investigates solutions of natural elliptic differential operators—most notably harmonic forms, spinors and higher-spin fields—as well as pseudoholomorphic curves and moment map constructions that reflect underlying symmetries. Classic examples include the six-sphere S6, the twistor space CP3, the homogeneous spaces S3×S3 and the flag manifold F1,2. The interplay between Levi-Civita and Hermitian connections yields a rich spectrum of harmonic three-forms, Killing spinors and Rarita–Schwinger fields, with direct implications for moduli of metrics and links to supergravity. Meanwhile, the study of J-holomorphic curves in these settings uncovers correspondences with minimal surfaces in S4 and associative submanifolds in G2-cones. Multi-moment map techniques for toric nearly Kähler structures further illuminate cohomological invariants and stability properties. Together, these threads form a coherent programme that bridges pure geometry, analysis of differential operators and mathematical physics, advancing our understanding of both intrinsic curvature phenomena and global topological constraints.

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Harmonic Geometry of Nearly Kähler Manifolds publication trend

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Technical terms

Nearly Kähler manifold: A six-dimensional almost Hermitian manifold whose intrinsic torsion lies in the single SU(3)-representation defining a strict Einstein structure.

Harmonic form: A differential form annihilated by the Laplace operator associated with a chosen connection, representing a cohomology class of minimal energy.

Rarita–Schwinger field: A spin-3/2 field satisfying a first-order Dirac-type equation, generalising Killing spinors and encoding higher-spin harmonic data.

J-holomorphic curve: A map from a Riemann surface into an almost complex manifold that intertwines the complex structures, critical for minimal surface theory.

Moment map: A map from a symplectic or nearly Kähler manifold to the dual of a Lie algebra, encoding conserved quantities under a group action and yielding symplectic quotients.

References

1. Rarita-Schwinger fields on nearly Kähler manifolds. Differential Geometry and its Applications (2023).
2. Transverse J-holomorphic curves in nearly Kähler CP3. Annals of Global Analysis and Geometry (2021).
3. The multi-moment maps of the nearly Kähler S3×S3. Geometriae Dedicata (2018).