scholarly journals Riemannian foliations with Ehresmann connection

2018 ◽  
Vol 20 (4) ◽  
pp. 395-407
Author(s):  
N. I. Zhukova
Keyword(s):  
Linear Connection ◽  
Riemannian Foliation ◽  
Minimal Set ◽  
Foliated Manifold ◽  
Trivial Bundle ◽  
Riemannian Foliations ◽  
Ehresmann Connection ◽  
The Family ◽  
Leaf Space ◽  
Complete Riemannian Manifolds

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.

scholarly journals Equivariant basic cohomology of Riemannian foliations

2018 ◽  
Vol 2018 (745) ◽  
pp. 1-40 ◽  
Author(s):  
Oliver Goertsches ◽  
Dirk Töben
Keyword(s):  
Fixed Point ◽  
Lie Group ◽  
Betti Number ◽  
Group Actions ◽  
Fixed Point Set ◽  
Foliated Manifold ◽  
Riemannian Foliations ◽  
Basic Cohomology ◽  
Lie Group Actions ◽  
Point Set

Abstract The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.

scholarly journals Riemannian foliations and the kernel of the basic Dirac operator

2012 ◽  
Vol 20 (2) ◽  
pp. 145-158
Author(s):  
Vladimir Slesar
Keyword(s):  
Dirac Operator ◽  
Mean Curvature ◽  
Riemannian Foliation ◽  
Curvature Form ◽  
Harmonic Mean ◽  
Riemannian Foliations ◽  
Bochner Technique ◽  
The Mean ◽  
Traditional Setting ◽  
Rela Tions

Abstract In this paper, in the special setting of a Riemannian foliation en- dowed with a bundle-like metric, we obtain conditions that force the vanishing of the kernel of the basic Dirac operator associated to the metric; this way we extend the traditional setting of Riemannian foli- ations with basic-harmonic mean curvature, where Bochner technique and vanishing results are known to work. Beside classical conditions concerning the positivity of some curvature terms we obtain new rela- tions between the mean curvature form and the kernel of the basic Dirac operator

The automorphism groups of foliations with transverse linear connection

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Nina Zhukova ◽  
Anna Dolgonosova
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Automorphism Groups ◽  
Linear Connection ◽  
The Other ◽  
Riemannian Foliations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Group

AbstractThe category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.

Lie superalgebras with a set grading

2020 ◽  
pp. 2150028 ◽  
Author(s):  
Valiollah Khalili
Keyword(s):  
Linear Subspace ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Minimal Set ◽  
Direct Sum ◽  
Simple Set ◽  
The Family

This paper studies Lie superalgebras graded by an arbitrary set [Formula: see text] (set grading). We show that the set-graded Lie superalgebra [Formula: see text] decomposes as the sum of well-described set-graded ideals plus a certain linear subspace. Under certain conditions, the simplicity of [Formula: see text] is characterized and it is shown that the above decomposition is exactly the direct sum of the family of its minimal set-graded ideals, each one being a simple set-graded Lie superalgebra.

scholarly journals Riemannian foliations with parallel curvature

1983 ◽  
Vol 90 ◽  
pp. 145-153
Author(s):  
Robert A. Blumenthal
Keyword(s):  
Tangent Bundle ◽  
Normal Bundle ◽  
Compact Manifold ◽  
Riemannian Metric ◽  
Riemannian Foliation ◽  
Riemannian Foliations ◽  
Smooth Compact Manifold

Let M be a smooth compact manifold and let be a smooth codimension q Riemannian foliation of M. Let T(M) be the tangent bundle of M and let E ⊂ T(M) be the subbundle tangent to . We may regard the normal bundle Q = T(M)/E of as a subbundle of T(M) satisfying T(M) = E ⊕ Q. Let g be a smooth Riemannian metric on Q invariant under the natural parallelism along the leaves of .

Geometric characterization of linearisable second-order differential equations

1996 ◽  
Vol 119 (2) ◽  
pp. 373-381 ◽  
Author(s):  
Eduardo Martínez ◽  
José F. Cariñena
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Sufficient Conditions ◽  
Linear Connection ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Ehresmann Connection ◽  
Local Coordinates ◽  
Necessary And Sufficient

AbstractGiven an Ehresmann connection on the tangent bundle τ: TM → M we define a linear connection on the pull-back bundle τ*(TM). With the aid of this tool, necessary and sufficient conditions are derived for the existence of local coordinates in which a system of second-order differential equations is linear.

scholarly journals ON CLOSED GEODESICS IN THE LEAF SPACE OF SINGULAR RIEMANNIAN FOLIATIONS

2011 ◽  
Vol 53 (3) ◽  
pp. 555-568 ◽  
Author(s):  
MARCOS M. ALEXANDRINO ◽  
MIGUEL ANGEL JAVALOYES
Keyword(s):  
Conjugate Points ◽  
Closed Geodesics ◽  
Riemannian Foliations ◽  
Singular Riemannian Foliations ◽  
Leaf Space

AbstractIn this paper we prove the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.fs. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.

scholarly journals PERTURBATIONS OF BASIC DIRAC OPERATORS ON RIEMANNIAN FOLIATIONS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350072 ◽  
Author(s):  
IGOR PROKHORENKOV ◽  
KEN RICHARDSON
Keyword(s):  
Dirac Operator ◽  
Dirac Operators ◽  
Riemannian Foliation ◽  
Riemannian Foliations ◽  
Basic Index ◽  
Witten Deformation ◽  
Foliated Bundle

Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.

scholarly journals The weighted mixed curvature of a foliated manifold

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1097-1105
Author(s):  
Vladimir Rovenski
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Foliation ◽  
Foliated Manifold ◽  
Totally Geodesic ◽  
Geodesic Foliation ◽  
New Concepts ◽  
Sectional And Ricci Curvatures ◽  
Totally Geodesic Foliation ◽  
Foliated Riemannian Manifold

We introduce the weighted mixed curvature of an almost product (e.g. foliated) Riemannian manifold equipped with a vector field. We define several qth Ricci type curvatures, which interpolate between the weighed sectional and Ricci curvatures. New concepts of the ?mixed-curvature-dimension condition? and ?synthetic dimension of a distribution? allow us to renew the estimate of the diameter of a compact Riemannian foliation and splitting results for almost product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. We also study the Toponogov?s type conjecture on dimension of a totally geodesic foliation with positive weighted mixed sectional curvature.

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