Linear Liftings of Skew-Symmetric Tensor Fields to Weil Bundles

We define equivariant tensors for every non-negative integer p and every Weil algebra A and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type (p, 0) on an n-dimensional manifold M to tensor fields of type (p, 0) on T ^A M if 1 ≤ p ≤ n. Moreover, we determine explicitly the equivariant tensors for the Weil algebras , where k and r are non-negative integers.     

Some liftings of poisson structures to Weil bundles

We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.     

Natural Operators Lifting Vector Fields to Bundles of Weil Contact Elements

Let A be a Weil algebra. The bijection between all natural operators lifting vector fields from m-manifolds to the bundle functor K ^A of Weil contact elements and the subalgebra of fixed elements SA of the Weil algebra A is determined and the bijection between all natural affinors on K ^A and SA is deduced. Furthermore, the rigidity of the functor K ^A is proved. Requisite results about the structure of SA are obtained by a purely algebraic approach, namely the existence of nontrivial SA is discussed.     

Invariants of Lagrangians on Weil Bundles and Their Classifications

Let Q:BA be an algebra epimorphism, where A, B are Weil algebras. Let Q M:T B M T AM denote the canonical extension of Q, from the bundle of B-velocities onto the bundle of A-velocities, over a manifold M. Classifications of all natural operators transforming real-valued functions on T AM into real-valued functions on T BM or into 1-forms on T BM of finite order with respect to Q M are given, provided the dimension of M is sufficiently large.     

Tangent and normal bundles in almost complex geometry

We define and study pseudoholomorphic vector bundle structures, particular cases of which are tangent and normal bundle almost complex structures. As an application we deduce normal forms of almost complex structures along a pseudoholomorphic submanifold.In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the question of non-deformation and persistence of pseudoholomorphic tori.     

Variational sequences in mechanics

Let be a fibered manifold over a base manifold . A differential 1-form , defined on the -jet prolongation of , is said to be contact, if it vanishes along the -jet prolongation of every section of . The notion of contactness is naturally extended to -forms with . The contact forms define a subsequence of the De Rham sequence on . The corresponding quotient sequence is known as the rth order variational sequence. In this paper, the case of 1-dimensional base is considered. A simple proof is given of the fact that the rth order variational sequence is an acyclic resolution of the constant sheaf. Then the 1st order variational sequence is studied in detail. The quotient sheaves, as well as the quotient mappings, are determined explicitly, and their relationship to the standard concepts of the 1st order calculus of variations is discussed. The following is shown: a) the lagrangians in the 1st order variational sequence (classes of 1-forms) coincide with 2nd order lagrangians, affine in the second derivative variables, b) the concept of the Euler-Lagrange form is extended to 2-forms which are not necessarily variational, c) the concept of the Helmholtz-Sonin form is introduced as the class of an arbitrary 3-form, d) the well-known fundamental notions such as the Euler-Lagrange, and Helmholtz-Sonin mappings are represented by two arrows at the beginning of the variational sequence; this relates the global structure of the Euler-Lagrange mapping to the cohomology of , e) all the remaining classes of -forms with , as well as the quotient mappings, are determined explicitly, f) a locally variational form is defined as a generalization of a symplectic form; locally variational forms, associated to a fixed Euler-Lagrange form, are characterized, and g) distributions associated with a locally variational form are described, and their relation to the Euler-Lagrange equations is studied. These results illustrate differences between finite order variational sequences and variational bicomplexes, based on infinite jet constructions. Received February 18, 1996 / In revised form December 1996 / Accepted December 2, 1996     

Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash sets  X⊂M$X\subset M$whose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space, and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k   semialgebraically diffeomorphic, for k>m²$k>m^2$, are also Nash diffeomorphic.     

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c ₁/{r ² ln r} and below by −c ₂ r ², where c ₁ and c ₂ are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M. Received: 19 September 1997     

The overdetermined Cauchy problem in some classes of ultradifferentiable functions

In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of ultradifferentiable functions of class (M _p ). We show that evolution is equivalent to the validity of a Phragmén-Lindel?f principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties, and make applications in different situations. We find necessary and sufficient conditions for well posedness, and relate the hyperbolicity of a given system to that of its principal part. Received: January 19, 1999?Published online: May 10, 2001     

Indice local pour des ensembles symétriques par arcs

Let M be a complete Riemannian manifold with sectional curvature and dimension . Given a unit vector and a point we prove the existence of a complete geodesic through x whose tangent vector never comes close to v. As a consequence we show the existence of a bounded geodesic through every point in a complete negatively pinched manifold with finite volume and dimension . Received April 13, 1998; in final form July 23, 1999 / Published online October 11, 2000     

KAM theorem of symplectic algorithms for Hamiltonian systems

Summary. In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given. Received December 27, 1997 / Revised version received July 15, 1998 / Published online: July 7, 1999     

The geometry of the bundle of connections

A generalized symplectic structure on the bundle of connections of an arbitrary principal G-bundle is defined by means of a -valued differential 2-form on C(P), which is related to the generalized contact structure on . The Hamiltonian properties of are also analyzed. Received August 31, 1999; in final form January 4, 2000 / Published online February 5, 2001     

Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces

We present a theory of harmonic maps where the target is a complete geodesic space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov and the domain is a measure space with a symmetric Markov kernel p on it. Our theory is a nonlinear generalization of the theory of symmetric Markov kernels and reversible Markov chains on M. It can also be regarded as a particular case of the theory of generalized (= nonlinear) Dirichlet forms and energy minimizing maps between singular spaces, initiated by Jost (1994) and Korevaar, Schoen (1993) and developed further by Jost (1997a), (1998) and Sturm (1997). We investigate the discrete and continuous time heat flow generated by p and show that various properties of the linear heat flow carry over to this nonlinear heat flow. In particular, we study harmonic maps, i.e. maps which are invariant under the heat flow. These maps are identified with the minimizers of the energy. Received April 2, 2000 / Accepted May 9, 2000 /Published online November 9, 2000     

Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs

Using non-Archimedian integration over spaces of arcs of algebraic varieties, we define stringy Euler numbers associated with arbitrary Kawamata log-terminal pairs. There is a natural Kawamata log-terminal pair corresponding to an algebraic variety V having a regular action of a finite group G. In this situation we show that the stringy Euler number of this pair coincides with the physicists’ orbifold Euler number defined by the Dixon-Harvey-Vafa-Witten formula. As an application, we prove a conjecture of Miles Reid on the Euler numbers of crepant desingularizations of Gorenstein quotient singularities. Received March 19, 1998