Sur la continuité de Hölder du semi-groupe de la chaleur sur les variétés coniques

In this Note, we study initially the heat kernel, p_t, on conic manifolds of dimension 2. Then, we improve the upper bound of p_t obtained in Li (Bull. Sci. Math. 124 (2000) 365–384) on conic manifolds of dimension ?3. Finally, we study the Hölder continuity of the heat semigroup on conic manifolds. Some new phenomenons are found on conic manifolds, in particular, on conic manifolds of dimension 2. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Approximately holomorphic geometry and estimated transversality on 2-calibrated manifolds

The notion of 2-calibrated structure, generalizing contact structures, smooth taut foliations, etc., is defined. Approximately holomorphic geometry as introduced by S. Donaldson for symplectic manifolds is extended to 2-calibrated manifolds. An estimated transversality result that enables to study the geometry of such manifolds is presented. To cite this article: A. Ibort, D. Martínez Torres, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Corrigendum to the Note “Symplectic capacities of toric manifolds and combinatorial inequalities” [C. R. Acad. Sci. Paris,Ser. I 334 (10) (2002) 889–892]

In this Note we correct some results in Lu, Symplectic capacities of toric manifolds and combinatorial inequalities [C. R. Acad. Sci. Paris, Ser. I 334 (10) (2002) 889–892] on (pseudo) symplectic capacities for toric manifolds. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Une caractérisation des variétés complexes compactes parallélisables admettant des structures affines

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections. To cite this article: S. Dumitrescu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Gromov–Witten invariants of noncompact symplectic manifolds

This is a short survey about our Gromov–Witten invariant theory for noncompact geometrically bounded symplectic manifolds. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Recurrence and genericity

We prove a C¹-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C¹-generic diffeomorphisms. For instance, C¹-generic conservative diffeomorphisms are transitive. To cite this article: C. Bonatti, S. Crovisier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Differentiability of the isoperimetric profile and topology of analytic Riemannian manifolds

We show that smooth isoperimetric profiles are exceptional for real analytic Riemannian manifolds. For instance, under some extra assumptions, this can happen only on topological spheres. To cite this article: R. Grimaldi et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Mesonic Differential Forms

When M is a differentiable manifold, the exterior differential k -forms on M are the alternate k -linear forms on the tangent bundle T(M) . The mesonic differential k -forms are the k -linear forms on T(M) that are alternate with respect to the variables of odd rank, and also alternate with respect to the variables of even rank. After a reminder about meson algebras, and after the presentation of elementary properties of mesonic forms, this article introduces the mesonic differentiation of mesonic forms, which can be partially compared to the exterior differentiation of exterior forms. Some applications to riemannian manifolds and flat manifolds follow.     

Parallel pure spinors and holonomy

We characterize the spin pseudo-Riemannian manifolds which admit parallel pure spinors by their holonomy groups. In particular, we study the Lorentzian case. To cite this article: A. Ikemakhen, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $ \mathfrak{n} $ -reductive. They are orbits of minimal dimension of a compact Lie group K ₀ in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.     

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).     

Symplectic capacities of toric manifolds and combinatorial inequalitiesCapacités symplectiques de variétés toriques et inéqualités combinatoires

We shall give concrete estimations for the Gromov symplectic width of toric manifolds in combinatorial data. As by-products some combinatorial inequalities in the polytope theory are obtained. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 889–892.     

On the compactification of hyperconcave ends

We find a class of manifolds whose ‘pseudoconcave holes’ can be filled in, even in dimension two. To cite this article: G. Marinescu, T.-C. Dinh, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

More compact invariant manifolds appearing in the non-linear coupling of oscillators

Near partially elliptic rest points of generic families of vector fields or transformations, many types of normally hyperbolic invariant compact manifolds can appear, diffeomorphic to intersections of quadrics. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Sur les variétés sous-riemanniennes de contact isotropes

In this Note, we show that contrary to the dimension 3 case, isotropic contact sub-Riemannian manifolds of dimension greater than 3 do not exist. To cite this article: A.-R. Mansouri, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Generalized Bergman kernels on symplectic manifolds

We study the asymptotic of the generalized Bergman kernels of the renormalized Bochner–Laplacian on high tensor powers of a positive line bundle on compact symplectic manifolds. To cite this article: X. Ma, G. Marinescu, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

The topology of corank 1 multi-singularities of stable smooth mappings of equidimensional manifolds

We study conditions for the coexistence of singularities of a stable smooth mapping of a closed manifold into a manifold of the same dimension n. Assuming that this mapping has only singularities of corank 1, we find universal linear relations between the Euler characteristics of the manifolds of multi-singularities in the image of the considered mapping. To cite this article: V.D. Sedykh, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Milnor–Wood inequalities for manifolds locally isometric to a product of hyperbolic planes

This Note describes sharp Milnor–Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern–Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to some other locally symmetric spaces they do admit interesting flat vector bundles in the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number. To cite this article: M. Bucher, T. Gelander, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The main invariants of a complex finsler space

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T'M: two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with Kähler spaces, in the two – dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kähler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0. Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.