Geometric Structures and Dynamics in Riemannian Manifolds

Riemannian manifolds furnish a rich arena in which geometry and analysis intersect, driven by the study of special holonomy, curvature flows and calibrated submanifolds. Central to this field is the classification of geometric structures—such as G₂, Spin(7), SU(n) and quaternion-Kähler (QK) reductions—by means of differential forms and intrinsic torsion. These structures underpin existence theorems for Einstein metrics and dictate the behaviour of geometric flows, notably the Ricci, Laplacian and harmonic flows. Recent advances have elucidated stability properties of critical points, long-time existence criteria under small-energy conditions and the topology of asymptotically conical spaces. Moreover, explicit cohomogeneity-one metrics and soliton solutions reveal intricate interactions between symmetry, torsion and flow singularities. Applications range from string theory compactifications to the desingularisation of nearly Kähler and nearly G₂ conifolds. As an interconnected body of work, this research not only deepens our understanding of curvature and holonomy but also offers concrete examples of manifolds evolving under geometric flows, highlighting mechanisms of convergence, formation of singularities and the role of symmetry in simplifying nonlinear dynamics.

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Technical terms

Riemannian manifold: A smooth manifold equipped with a positive-definite metric tensor that defines distances, angles and curvature.

Holonomy group: The group formed by parallel transporting tangent vectors around closed loops, encoding the manifold’s curvature and special geometric reductions.

G₂-structure: A reduction of the frame bundle on a seven-manifold determined by a stable 3-form, yielding torsion components that measure deviation from torsion-free (holonomy G₂) metrics.

Quaternion-Kähler structure: A Riemannian structure on 4n-dimensional manifolds whose holonomy lies in Sp(n)·Sp(1), characterised by a parallel 4-form and nonvanishing intrinsic torsion in general.

Geometric flow: An evolution equation for metrics or differential forms driven by curvature, torsion or energy gradients, used to deform structures toward canonical geometries or uncover singularity formation.

Intrinsic torsion: The component of the Levi-Civita connection that measures the failure of a special G-structure (for example G₂ or Sp(n)·Sp(1)) to be torsion-free, decomposing into distinguished modules under the structure group.

References

1. Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds. The Journal of Geometric Analysis (2023).
2. Geometric flows of G2-structures on 3-Sasakian 7-manifolds. Journal of Geometry and Physics (2023).
3. Harmonic Flow of Quaternion-Kähler Structures. The Journal of Geometric Analysis (2024).
4. Cohomogeneity one solitons for the isometric flow of G2-structures. Geometriae Dedicata (2024).