Considering different finite sets of maps generating a pseudogroup G of locally Lipschitz homeomorphisms between open subsets of a compact metric space X we arrive at a notion of a Hausdorff dimension of G. Since , the dimension loss can be considered as a “topological price” one has to pay to generate G. We collect some properties of and (for example, both of them are invariant under Lipschitz isomorphisms of pseudogroups) and we either estimate or calculate for pseudogroups arising...
We consider the energy of a unit vector field defined on a compact Riemannian manifold M except at finitely many points. We obtain an estimate of the energy from below which appears to be sharp when M is a sphere of dimension >3. In this case, the minimum of energy is attained if and only if the vector field is totally geodesic with two singularities situated at two antipodal points (at the 'south and north pole').
The Hausdorff dimension of the holonomy pseudogroup of a codimension-one foliation ℱ is shown to coincide with the Hausdorff dimension of the space of compact leaves (traced on a complete transversal) when ℱ is non-minimal, and to be equal to zero when ℱ is minimal with non-trivial leaf holonomy.
We classify surfaces in 3-dimensional space forms which have all the local conformal invariants constant and show that compact 3-manifolds of nonzero constant sectional curvature admit no foliations by such surfaces.
A distality property for pseudogroups and foliations is defined. Distal foliated bundles satisfying some growth conditions are shown to have zero geometric entropy in the sense of É. Ghys, R. Langevin and P. Walczak [Acta Math. 160 (1988)].
CONTENTSIntroduction.................................................51. Preliminaries...........................................6 1. A. Foliations...........................................7 1. B. Geometry of submanifolds.................92. The characteristic form..........................113. Stability of minimal foliations..................184. A metric on the space of foliations.........245. Jacobi fields on leaves..........................276. The Gauss mapping of a foliation.........37References...............................................45...
Decomposing the space of k-tensors on a manifold M into the components invariant and irreducible under the action of GL(n) (or O(n) when M carries a Riemannian structure) one can define generalized gradients as differential operators obtained from a linear connection ∇ on M by restriction and projection to such components. We study the ellipticity of gradients defined in this way.
We deal with locally connected exceptional minimal sets of surface homeomorphisms. If the
surface is different from the torus, such a minimal set is either finite or a finite
disjoint union of simple closed curves. On the torus, such a set can admit also a
structure similar to that of the Sierpiński curve.
We study the global behavior of foliations of ellipsoids by curves making a constant angle with the lines of curvature.
Given some geometric bounds for the base space and the fibres, there is a finite number of conjugacy classes of Riemannian submersions between compact Riemannian manifolds.
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