Compact Heisenberg manifolds as CR manifolds

Let M be the quotient of the Heisenberg group by a discrete co-compact subgroup, with the natural strongly pseudoconvex CR structure. We identify the eigenvalues and eigenforms of the Kohn Laplacians on M and show how to realize M as the boundary of a bounded domain in a line bundle over an Abelian variety.     

Center manifolds for smooth invariant manifolds

We study dynamics of flows generated by smooth vector fields in in the vicinity of an invariant and closed smooth manifold . By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of ) based on the information of the linearization along , which contains every locally bounded solution and is persistent under small perturbations.

     

Total curvature of manifolds in self-immersed manifolds

Chern-Lashof [3] and Kuiper [5] showed the total absolute curvature of a manifold in Euclidean space equals the mean value of the number of critical points of height functions. Teufel [10] proved that a similar result holds for the total absolute curvature of a manifold in a unit sphere. The purpose of this paper is to extend Teufel's result to a relation between the total absolute curvature of some manifolds in self-immersed manifolds and the mean value of the number of zeros of certain vector fields.     

On complex Banach manifolds similar to Stein manifolds

We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.     

Plateau-Stein manifolds

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all ?∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.     

Degenerate manifolds

A geometric structure of manifolds with singular semi-Riemannian metrics is studied.     

Dissipative-asymptotic manifolds

Asymptotic analogs are considered for the fundamental concepts of differential geometry, with the role of the ring of smooth functions being played by the ring of dissipative-asymptotic functions. Among the tools developed are asymptotic changes of variables and asymptotic compositions.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 31, pp. 3–92, 1987.     

Moishezon manifolds

Let X be a compact Moishezon manifold which becomes projective after blowing up a smooth subvariety . We assume also that there exists a proper map onto a projective variety with a point, such that and is -big. We prove some inequalities between the dimensions of Y andX and we construct examples which shows the optimality of the inequalities. In the last section we discuss some differential geometry properties of these examples which lead to a conjecture. Received December 19, 1997     

Incidence manifolds

It is shown that there are differentiable manifolds which can be characterized only by the geometric and topological structure of their linear subspaces. More precisely, such a characterization is possible for Grassmann and C. Segre's product manifolds. The starting point of this investigation was a paper by J. Misfeld, G. Tallini and the author [10], the ideas in which were further developed in [15,16,17].Dedicated to Professor Giuseppe Tallini on his 60^th birthday     

Semi-cosymplectic manifolds

SoitM(Ω, η, ξ,g) une variété à (2m+1)-dimensions presque cosymplectique (i. e. Ω∈Λ² M est de rang 2m et Ω ^m Λη≠0). On définitM comme étant une variété semi-cosymplectique si en termes ded ^ω-cohomologie la paire (Ω, η) satisfait àdη=0,d ^−cη Ω=Ψ∈Λ³ M,c=constant. Dans ce cas le champ vectoriel de structure ξ=b ⁻¹(η) est un champ conforme horizontal et siM est une forme-espace elle est nécessairement du type hyperbolique. Différentes propriétés de cette structure sont étudiés et le cas oùM est une variété para Sasakienne dans le sens large est discuté.     

Exploded manifolds

This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which is a union of convex polytopes glued along faces. Exploded manifolds are useful for defining and computing Gromov–Witten invariants relative to normal crossing divisors, and using tropical curve counts to compute Gromov–Witten invariants.     

Bicompact manifolds

The paper is an introduction to the theory of bitopological manifolds. It contains the basic definitions and assertions of the theory and also some examples of bitopological manifolds.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 26–68, 1985.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Bordism classes represented by multiple point manifolds of immersed manifolds

We present a geometrical version of Herbert’s theorem determining the homology classes represented by the multiple point manifolds of a self-transverse immersion. Herbert’s theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert’s paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.