Results in Mathematics: Volume 60, Issue 1 (2011), Page 213-223. The final publication is available at springerlink.com http://www.springerlink.com/content/207470434443386j/
In this work we investigate the relation between the fundamental group of a complete Riemannian manifold $M$ and the quotient between the Weyl group and reflection group of a polar action on $M$, as well as the relation between the fundamental group of $M$ and the quotient between the lifted Weyl group and lifted reflection group. As applications we give short alternative proofs of two results. The first one, due to the author and Töben, states that there is non-exceptional orbit, if $M$ is simply connected. The second result, due to Lytchak, states that the orbits are closed and embedded if $M$ is simply connected.
All results are proved in the more general case of polar foliations.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
We prove that, up to isometric congruence, there are exactly 2n+1 homogeneous polar foliations of the complex hyperbolic space. We also give an explicit description of each of these foliations.
For a singular Riemannian foliation $\mathcal{F}$ on a Riemannian manifold, a curve is called horizontal if it meets the leaves of $\mathcal{F}$ perpendicularly. For a singular Riemannian foliation $\mathcal{F}$ on a unit sphere $\mathbb{S}^{n}$, we show that if $\mathcal{F}$ is a polar foliation or if $\mathcal{F}$ is given by the orbits of an infinitesimally polar action, then the horizontal diameter of $\mathbb{S}^{n}$ is $π$, i.e., any two points in $\mathbb{S}^{n}$ can be connected by a horizontal curve of length $\leqπ$.
We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb H P^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is even or $n=1$). This shows the existence of inhomogeneous examples of codimension one and higher.
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres…
A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $Σ$, called section, that passes through $p$ and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of $M$ into the orbits of a polar action, i.e., an isometric action with sections. In this work we prove that the leaves of $F$ coincide with the level sets of a smooth map $H: M\to Σ$ if $M$ is simply connected. In particular, we have that the orbits of a polar action on a…
A 2-web in the plane is given by two everywhere transverse 1-foliations. In this paper we introduce the study of singular 2-webs, given by any two foliations, which may be tangent in some points. We show that such two foliations are tangent along a curve, which will be called the polar curve of the 2-web, and we study the relationship between the contact order of leaves of both foliations and the singularities of the polar curve.
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 give a new construction of the holonomy and fundamental groupoids of a singular foliation. In contrast with the existing construction of Androulidakis and Skandalis, our method proceeds by taking a quotient of an infinite dimensional space of paths. This strategy is a direct extension of the classical construction for regular foliations and mirrors the integration of Lie algebroids via paths (per Crainic and Fernandes). In this way, we obtain a characterization of the holonomy and fundamental groupoids of a singular foliation that more clearly reflects the homotopic character of these invar…
In this paper, we establish a number of results about the topology of the leaves of a closed singular Riemannian foliation $(M,\fol)$. If $M$ is simply connected, we prove that the leaves are finitely covered by nilpotent spaces, and characterize the fundamental group of the generic leaves. If $M$ has virtually nilpotent fundamental group, we prove that the leaves have virtually nilpotent fundamental group as well.
Let (M,F) be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino's commuting sheaf of (M,F) vanish if (M,F) is developable and the fundamental group of M is of polynomial growth. By theorems of Álvarez López, our result implies that (M,F) is minimizable under the same conditions. As a corollary, we show that (M,F) is minimizable if F is of codimension 2 and the fundamental group of M is of polynomial growth.
This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; that is, they are (at least locally) warped products of constant curvature Lorentz manifolds by Riemannian manifolds.
We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyl's First Fundamental Theorems, provides a characterization of the recently discovered 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.
The main purpose of these lecture notes is to provide a concise introduction to Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostly at advanced undergraduate and graduate students. In addition, the connection between such classic theories and the research area of the first author is explored. Namely, generalizations to isoparametric submanifolds, polar actions and singular Riemannian foliations with sections (s.r.f.s.) are mentioned. The first chapters cover basic concepts, giving results on adjoint representation, closed subgroups, bi-invariant metrics, Killing forms an…
