Notre étude porte sur une catégorie de structures de Poisson singulières holomorphes au voisinage de $0\in {ℂ}^n$ et admettant une forme normale formelle polynomiale i.e. un nombre fini d’invariants formels. Les séries normalisantes sont divergentes en général. On montre l’existence de transformations normalisantes holomorphes sur des domaines sectoriels de la forme $a\<\arg x^R\<b$, où $x^R$ est un monôme associé au problème. Il suit une classification analytique.

Consider a (1,1) tensor field J, defined on a real or complex m-dimensional manifold M, whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions $f_1,...,f_m$, defined around p, such that $(d{f}_1\wedge ...\wedge d{f}_m)\left(p\right)\ne 0$ and $d\left(d{f}_j\left(J\left(\right)\right)\right)\left(p\right)=0$, j = 1,...,m. Then there exists a dense open set such that we can find coordinates, around each of its points, on which J is written with affine coefficients. This result is obtained by associating to J a bihamiltonian structure on T*M.

For the family of degree at most 2 polynomial self-maps of C3 with nowhere vanishing Jacobian determinant, we give the following classification: for any such map f, it is affinely conjugate to one of the following maps:(i) An affine automorphism;(ii) An elementary polynomial autormorphismE(x, y, z) = (P(y, z) + ax, Q(z) + by, cz + d),where P and Q are polynomials with max{deg(P), deg(Q)} = 2 and abc ≠ 0.(iii)⎧ H1(x, y, z) = (P(x, z) + ay, Q(z) + x, cz + d)⎪ H2(x, y, z) = (P(y, z) + ax, Q(y)...

This paper deals with the classification of hyperbolic Monge-Ampère equations on a two-dimensional manifold. We solve the local equivalence problem with respect to the contact transformation group assuming that the equation is of general position nondegenerate type. As an application we formulate a new method of finding symmetries. This together with previous author's results allows to state the solution of the classical S. Lie equivalence problem for the Monge-Ampère equations.

A semi-algebraic analytic manifold and a semi-algebraic analytic map are called a Nash manifold and a Nash map respectively. We clarify the category of Nash manifolds and Nash maps.

We consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. Let $K$ be a principal connection on $PE$ and let $\Lambda$ be a linear connection on $M$. We classify all principal connections on $W^{2}PE=P^{2}M{\times }_M{J}^{2}PE$ naturally given by $K$ and $\Lambda$.

Cotangent type functions in Rn are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds Rn/Zk where 1 < = k ≤ M. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this...

[For the entire collection see Zbl 0742.00067.]For the purpose of providing a comprehensive model for the physical world, the authors set up the notion of a Clifford manifold which, as mentioned below, admits the usual tensor structure and at the same time a spin structure. One considers the spin space generated by a Clifford algebra, namely, the vector space spanned by an orthonormal basis $\{e_j:j=1,\cdots ,n\}$ satisfying the condition $\{e_i,e_j\}\equiv e_i{e}_j=e_j{e}_i=2I{\eta }_{ij}$, where $I$ denotes the unit scalar of the algebra and (${\eta }_{ij}$) the nonsingular Minkowski...