Fundamental groups of positively curved manifolds with symmetry

We obtain a classification for the fundamental groups of closed $n$ -manifolds of positive sectional curvature which admit an isometric $T^k$ -action with $k \ge \frac{n}{6}+1 (n \ne 11, 15)$ .     

Topological properties of positively curved manifolds with symmetry

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.     

Weyl laws on open manifolds

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is purely discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume. The author has been partially supported by the Research and Training Network HPRN-CT-1999-00118 “Geometric Analysis” funded by the European Commission.     

Submersions on open symplectic manifolds

Let M be an open manifold with a symplectic form Ω, and N a manifold with dimN<dimM. We prove that submersions with symplectic fibres satisfy the h-principle. Such submersions define Dirac manifold structures on the given manifold. As an application to this result we show that CPn?CPk−1 admits a submersion into R2(2k−n) with symplectic fibres for n/2<k?n.     

Periodic trajectories with fixed energy on Riemannian and Lorentzian manifolds with boundary

We state some existence and multiplicity results for periodic solutions with prescribed energy of a Lagrangian system and for closed geodesics on Riemannian manifolds convex close to their boundary. As an application we get the existence and multiplicity of periodic trajectories with fixed energy on a class of physically important Lorentzian manifolds with boundary. Entrata in Redazione il 23 aprile 1998. Part of the Ph.D. thesis of this author is contained in this paper. Partially supported by MEC grant PB97-0784-C03-01.     

Piecewise-maximum closed Geodesic trajectories on bounded Toroidal manifolds

We use toroidal coordinates for the investigation of maximum geodesic trajectories for arbitrary parameters of a torus. Conditions under which trajectories are located in a bounded part of the toroidal manifold are considered. Using global invariants, we construct closed piecewise-maximum geodesic trajectories.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1139–1142, August, 2004.     

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D _infw,0 ^supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D _infw,0 ^supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program     

Covering of differentiable manifolds with open disks

Summary The aim of this paper is to prove that every open (i.e. noncompaet without boundary) manifold of dimensionn can be covered with exactlyn open disks. This is a generalization of a theorem of E. Luft [3] concerning the case of any open 2-dimensional manifold. It is then proved that every compact manifold of dimensionn with nonempty boundary can also be covered with exactlyn open disks. The proofs of the theorems are in the spirit of Morse theory [1].     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m5) of positive sectional curvature on which an (m–1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n8) of almost maximal symmetry rank Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China.Supported partially by NSF Grant DMS 0203164 and a research grant from Capital normal university.     

Rank two fundamental groups of positively curved manifolds

This paper deals with the construction of previously unknown fundamental groups for positively curved manifolds.