Complete Foliation Research Articles

Progress in the theory of singular Riemannian foliations

A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action.In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions in particular. We also review the solution of the Palais–Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their relations, are treated. A characterization of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molinoʼs conjecture, are presented.

Complete foliations of space forms by hypersurfaces

We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of r-minimal foliations of the Euclidean space, possibly with a singular set, we are able to invoke a theorem of Ferus to give conditions under which the non- singular leaves are foliated by hyperplanes.

INVARIANT MANIFOLD WITH COMPLETE FOLIATIONS AND CHAOTIFICATION ANALYSIS FOR A KIND OF PDES WITH BOUNDARY COUPLING

In this paper, we are concerned with the existence of invariant manifolds and complete foliations for a class of PDEs with boundary coupling. Some new form of gap relative coupling and inequality conditions are obtained. Further, we prove the topological equivalence of the flows in the respective attractors between the system and its spatial discretization system (an ODE system). Finally, the chaotification of the system is discussed through an example and simulation is generated to illustrate the theoretical results.

A Complete Foliation of Schwarzschild Spacetime by Free FallingHypersurfaces

Free falling hypersurfaces in the Schwarzschild geometry have beenstudied to provide a complete foliation of spacetime. The hypersurfacesdo not cross into the maximally extended spacetime and are well behavedeverywhere except at the singularity r = 0 the mean extrinsic curvature becomes infinity.

FOLIATION OF THE SCHWARZSCHILD AND REISSNER–NORDSTRÖM SPACE–TIMES BY FLAT SPACE-LIKE HYPERSURFACES

It is known that spherically symmetric space–times admit flat space-like foliations. In this paper a simple procedure for the complete foliation of the Schwarzschild space–time by flat space-like hypersurfaces is developed, using the fact that unforced geodesics are orthogonal to such hypersurfaces. The method is then extended to obtain a complete foliation of the Reissner–Nordström space–time by such hypersurfaces. In this case, as there is a barrier beyond which the geodesics do not go, a complete foliation is obtained by analytically continuing the hypersurfaces beyond the barrier. A duality between imaginary time-like geodesics and space-like hypersurfaces is noted, which also provides complete foliation — but the hypersurfaces are not flat. The features of our foliation are discussed.

COMPLETELY AND PARTIALLY INTEGRABLE HAMILTONIAN SYSTEMS IN THE NONCOMPACT CASE

Action-angle coordinates are shown to exist around an instantly compact invariant submanifold of a time-dependent completely integrable Hamiltonian system. Partially integrable Hamiltonian systems are also considered in the noncompact case; a comparison is made with other possible approaches. Results on symplectically complete foliations contained in the Appendix A can be used to give alternative proofs of some propositions.

K slicing the Schwarzschild and the Reissner–Nordstrom spacetimes

A K slicing of the Reissner–Nordstrom (RN) spacetime is shown to provide a complete foliation of the region up to the inner horizon, and the horizon to correspond to the limit as K→±∞. The implications of this foliation in the context of York time are discussed.

Bott’s vanishing theorem for regular Lie algebroids

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example: — differential groupoids, — principal bundles, — vector bundles, — actions of Lie groups on manifolds, — transversally complete foliations, — nonclosed Lie subgroups, — Poisson manifolds, — some complete closed pseudogroups. We carry over the idea of Bott’s Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

Foliation by constant-mean-curvature hypersurfaces of the Schwarzschild spacetime.

In this Brief Report a procedure for the complete foliation of the Schwarzschild spacetime by spacelike hypersurfaces of constant York time is proposed.

On completeness of transversely projective foliations

A CODIMENSION-I foliation on a differentiable n-manifold is called complete if the manifold is covered by the product of the real line and an (possibly noncompact) (n I)-manifold C, such that the leaves of the foliation are the images of the hypersurfaces constant x 1. Much of the study of a complete foliation reduces to the study of a group of diffeomorphisms of the real line. Is completeness stable under small perturbations of the foliation? The aim of this paper is to answer this question in the rcstrictcd frame of codimcnsion-I foliations which carry a transverse projective geometry. on 3-manifolds. In this paper, all manifolds, smooth maps and foliations arc of class C’. A trun.scerse projectiw utfus for a foliation 9, is a covering of the underlying manifold by a collection of 9-distinguished charts, for which the transverse coordinate transformations are projective, i.c. of the form .Y H ux + b/cx + d. A trunsuerse projeclioe structure is a maximal transverse projective atlas. A trunscersely projective fdiution is a foliation endowed with a transverse projective structure. We shall exclude from this work the following particular case. We call a transversely projective foliation reducible if it admits an atlas in which all the transverse coordinate transformations, considered as elements of PSL(2, [w), belong to some proper closed connected Lie subgroup. For example, let 9 be a transversely projective foliation which admits an atlas in which all the transverse coordinate transformations are of the form x H (IX + b (this is called a trunsversely uJfine foliurion). Then 9 is reducible. Here is our main result. We shall endow the set of all foliations on a tixcd manifold, with the C’fine topology in the sense of [3]. Roughly speaking: two foliations are close to each other when they admit C’-close atlases of distinguished charts.

On the Lie algebra of a transversally complete foliation

On the Lie algebra of a transversally complete foliation

Sequence and style of major post-nappe structures, Simplon—Pennine Alps

In the Upper Pennine nappe complex of the Simplon—Pennine Alps (Switzerland and Italy), at least three phases of major post-nappe folding (in places associated with thrusting) can be distinguished. These are superimposed on an earlier-formed, partly chaotic, complex of tectonic units, including the Bernhard and Monte Rosa continental flakes and the Zermatt—Saas and Antrona ophiolite complexes. The earliest post-nappe folds were essentially isoclinal throughout the whole region and were accompanied by a strong schistosity which is the main foliation in most areas. Later, two successive phases of back-folding led to the present overall structure. Both phases typically show rapid variations in style from open folds lacking axial planar schistosity to very tight structures with complete foliation transposition. This has been demonstrated by systematically mapping the major axial traces over the whole region. Successively removing the major structures in reverse order shows that the ophiolite complexes were originally part of a continuous unit marking an important suture between the Bernhard and Monte Rosa nappes.