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Deforming metrics of foliations

100%
Rovenski V., Wolak R.
Open Mathematics
|
2013
|
tom 11
|
nr 6
1039-1055
EN
Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.
2
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The BIC of a singular foliation defined by an abelian group of isometries

100%
Saralegi-Aranguren M., Wolak R.
Annales Polonici Mathematici
|
2006
|
tom 89
|
nr 3
203-246
EN
We study the cohomology properties of the singular foliation ℱ determined by an action Φ: G × M → M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $ℍ*_{p̅}(M/ℱ)$ is finite-dimensional and satisfies the Poincaré duality. This duality includes two well known situations: ∙ Poincaré duality for basic cohomology (the action Φ is almost free). ∙ Poincaré duality for intersection cohomology (the group G is compact and connected).
3
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Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations

81%
Royo Prieto J., Saralegi-Aranguren M., Wolak R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom 53
|
nr 4
429-440
EN
It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincaré duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation between these properties. We present an example of a singular isometric flow for which the top-dimensional basic cohomology group is non-trivial, but the basic cohomology does not satisfy the Poincaré Duality. However, we recover the Poincaré Duality in the basic intersection cohomology. It is not accidental that the top-dimensional basic intersection cohomology groups of the example are isomorphic to either 0 or ℝ. We prove that this holds for any singular riemannian foliation of a compact connected manifold. As a corollary, we show that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.
4
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Infinitesimal automorphisms of distributions

60%
Wolak R.
Annales Polonici Mathematici
|
1986-1987
|
tom 47
|
nr 1
1-16
5
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Infinitesimal automorphisms of some G-structures

60%
Wolak R.
Annales Polonici Mathematici
|
1983
|
tom 43
|
nr 3
283-291
6
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Characteristic classes of multifoliations

50%
Wolak R.
Colloquium Mathematicum
|
1987
|
tom 52
|
nr 1
77-91
7
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Preface

41%
Kubarski J., Rybicki T., Wolak R.
Open Mathematics
|
2004
|
tom 2
|
nr 5
i-iii
8
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On G-foliations

40%
Wolak R.
Annales Polonici Mathematici
|
1985
|
tom 46
|
nr 1
371-377
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