Riemannian Foliations and Geometric Structures

Riemannian foliations partition a smooth manifold into a collection of disjoint submanifolds, called leaves, that locally resemble parallel layers. Each leaf inherits a Riemannian metric from the ambient space, and the transverse geometry—capturing how leaves sit side by side—carries additional structure. Such foliations link differential geometry, topology and dynamical systems, providing a unified language to describe objects as diverse as geodesic flows, torus actions and symplectic decompositions. The theory addresses both regular foliations, where all leaves have the same dimension, and singular foliations, admitting leaves of varying dimensions. Central themes include the study of holonomy groups, which measure how nearby leaves twist around one another, and the classification of foliations via transverse connections and curvature conditions. Applications span gauge theory, where foliated Hilbert-space techniques control the evolution of invariant hypersurfaces; symplectic geometry, in which taut foliations define characteristic submanifolds of symplectic pairs; and global topology, where the fundamental groups of leaves reflect the large-scale structure of the underlying manifold. Recent advances continue to deepen our understanding of leaf topology, the rigidity of transverse structures and the interplay between local curvature constraints and global invariants.

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Riemannian Foliations and Geometric Structures publication trend

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Technical terms

Riemannian foliation: A decomposition of a Riemannian manifold into leaves so that geodesics perpendicular to one leaf remain perpendicular to all encountered leaves.

Leaf: A connected submanifold in a foliation, locally resembling a slice or layer of the ambient manifold.

Holonomy: The transformation obtained by parallel transporting around a closed path within a leaf, measuring the twisting of nearby leaves.

Transverse structure: Geometric data defined on directions orthogonal to the leaves, often given by a metric or connection.

Symplectic pair: A pair of closed two-forms on a manifold whose kernels define complementary, transverse foliations and jointly satisfy nondegeneracy.

Equifocality: A property of regular leaves in a singular foliation where the end-point map of a normal foliated vector field has constant rank, enabling global reconstruction of the foliation.

References

1. Symplectic Pairs and Intrinsically Harmonic Forms †. Mathematics (2023).
2. Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory. Calculus of Variations and Partial Differential Equations (2024).
3. Equifocality of a singular Riemannian foliation. Proceedings of the American Mathematical Society (2008).
4. On the topology of leaves of singular Riemannian foliations. Revista Matemática Iberoamericana (2024).