On the fundamental group of compact homogeneous manifolds carrying an invariant fat distribution

We show that a homogeneous space of a compact Lie group carrying an invariant fat distribution must have finite fundamental group.     

Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions

For generic families of vector fields or transformations, normally hyperbolic invariant products of spheres appear near partially elliptic rest points. To cite this article: M. Kammerer-Colin de Verdière, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Birth of attracting compact invariant submanifolds diffeomorphic to moment-angle manifolds in generic families of dynamics

All the compact intersections of quadrics known as moment-angle manifolds appear as attractors in generalized Hopf bifurcations. To cite this article: M. Chaperon, S. López De Medrano, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Explicit construction of graph invariant for strongly pseudoconvex compact 3-dimensional rational CR manifolds

Let X be a strongly pseudoconvex compact 3-dimensional CR manifolds which bounds a complex variety with isolated singularities in some C^N. The weighted dual graph of the exceptional set of the minimal good resolution of V is a CR invariant of X; in case X has a tranversal holomorphic S¹ action, we show that it is a complete topological invariant of except for two special cases. When X is a rational CR manifolds, we give explicit algebraic algorithms to compute the graph invariant in terms of the ring structure of _k=0 m^k/m^k+1, where m is the maximal ideal of each singularity. An example is computed explicitly to demonstrate how the algorithms work.     

Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let \((M,g_0)\) be a smooth compact manifold of dimension \(n\ge 3\) with boundary. Given any smooth functions f in M and h on \(\partial M\), does there exist a conformal metric of \(g_0\) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant \(Q(M,\partial M)\) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.     

Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.     

On the Conley index in the invariant manifolds

Let ? be a flow on a manifold M and assume that N⊂M is an invariant manifold. The aim of this note is to compare the Conley indices of an isolated invariant set S⊂N with respect to the flow ? and the flow ? restricted to N.     

Pompeiu sets in compact Heisenberg manifolds

A necessary and sufficient condition is presented for a set to be a Pompeiu subset of any compact homogeneous space with a finite invariant measure. The condition, which is expressed in terms of the intertwining operators of each primary summand of the quasi-regular representation, is then interpreted in the case of the compact Heisenberg manifolds. Examples are presented demonstrating that the condition to be Pompeiu in these manifolds is quite different from the corresponding condition for a torus of the same dimension. This provides a contrast with the existing comparison between the Heisenberg group itself and Euclidean space in terms of Pompeiu sets. In addition, the closed linear span of all translates of any square integrable function on any compact homogeneous space is determined.     

Stabilization of DAEs and invariant manifolds

Summary. Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions. Received September 1992/Revised version received May 13, 1993     

An invariant of multiply connected manifolds

The conditions under which a multiply connected open manifold has the homotopic type of a finite complex are studied. Examples are analyzed.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 155–164, August, 1968.