Leaf closures of Riemannian foliations: A survey on topological and geometric aspects of Killing foliations

It is shown that the structural theory of Molino for Riemannian foliations on compact manifolds and complete Riemannian manifolds may be generalized to a Riemannian foliations with Ehresmann connection. Within this generalization there are no restrictions on the codimension of the foliation and on the dimension of the foliated manifold. For a Riemannian foliation (M,F) with Ehresmann connection it is proved that the closure of any leaf forms a minimal set, the family of all such closures forms a singular Riemannian foliation (M,F¯¯¯¯). It is shown that in M there exists a connected open dense F¯¯¯¯-saturated subset M0 such that the induced foliation (M0,F¯¯¯¯|M0) is formed by fibers of a locally trivial bundle over some smooth Hausdorff manifold. The equivalence of some properties of Riemannian foliations (M,F) with Ehresmann connection is proved. In particular, it is shown that the structural Lie algebra of (M,F) is equal to zero if and only if the leaf space of (M,F) is naturally endowed with a smooth orbifold structure. Constructed examples show that for foliations with transversally linear connection and for conformal foliations the similar statements are not true in general.

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Compact leaves in minimal Riemannian foliations

Indagationes Mathematicae ◽

10.1016/0019-3577(94)90012-4 ◽

1994 ◽

Vol 5 (3) ◽

pp. 375-379

Author(s):

Robert Wolak

Keyword(s):

Riemannian Foliations

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On mean curvature flow of singular Riemannian foliations: Noncompact cases

Differential Geometry and its Applications ◽

10.1016/j.difgeo.2020.101664 ◽

2020 ◽

Vol 72 ◽

pp. 101664

Author(s):

Marcos M. Alexandrino ◽

Leonardo F. Cavenaghi ◽

Icaro Gonçalves

Keyword(s):

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Riemannian Foliations ◽

Singular Riemannian Foliations

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Invariance of basic Hodge numbers under deformations of Sasakian manifolds

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/s10231-020-01044-8 ◽

2020 ◽

Author(s):

Paweł Raźny

Keyword(s):

Elliptic Operators ◽

Holomorphic Foliation ◽

Sasakian Manifolds ◽

Contact Manifolds ◽

Continuity Theorem ◽

Riemannian Foliations ◽

Holomorphic Structure ◽

Sasakian Structure ◽

Hodge Numbers ◽

Small Deformations

Abstract We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the $$\partial {\bar{\partial }}$$ ∂ ∂ ¯ -lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

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