We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid.
We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several
classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.
We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyls First Fundamental Theorems, provides a characterization of the recently disc
overed Clifford foliations in terms of basic polynomials. This link also yields new structural results about infinitesimal foliations, such as the existence of non-trivial symmetries.
In this paper we introduce the notion of deformation cohomology for singular foliations and related objects (namely integrable differential forms and Nambu structures), and study it in the local case, i.e., in the neighborhood of a point.
We show that a Riemannian foliation on a topological $n$-sphere has leaf dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.
We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamenta
l properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Mosers deformation arguments for groupoids, we obtain several rigidity and normal form results.
Marcos M. Alexandrino
,Marcelo K. Inagaki
,Mateus de Melo
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(2018)
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"Lie groupoids and semi-local models of Singular Riemannian foliations"
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Marcos Alexandrino
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