We classify all F2Mm1, m2, n1, n2-natural operators Atransforming projectable-projectable torsion-free classical linear connections ∇ on fibered-fibered manifolds Y of dimension (m1,m2, n1, n2) into rth order Lagrangians A(∇) on the fibered-fibered linear frame bundle Lfib-fib(Y) on Y. Moreover, we classify all F2Mm1, m2, n1, n2-natural operators B transforming projectable-projectable torsion-free classical linear connections ∇ on fiberedfibered manifolds Y of dimension (m1, m2, n1, n2) into Euler...

Let $r,n$ be fixed natural numbers. We prove that for $n$-manifolds the set of all linear natural operators $T^{*}\to T^{*}{T}^{\left(r\right)}$ is a finitely dimensional vector space over $R$. We construct explicitly the bases of the vector spaces. As a corollary we find all linear natural operators $T^{*}\to T^{r*}$.

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A}M$, where $T^A$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

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 $1\le p\le n$. Moreover, we determine explicitly the equivariant tensors for the Weil algebras ${𝔻}_k^r$, where $k$ and $r$ are non-negative integers.

Nous étudions les métriques riemanniennes holomorphes sur les variétés complexes compactes de dimension $3$. Nous montrons que, contrairement au cas réel, une métrique riemannienne holomorphe possède un “grand” pseudo-groupe d’isométries locales. Ceci implique qu’une telle métrique n’existe pas sur les variétés complexes compactes simplement connexes de dimension $3$.

Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....

In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.

All natural affinors on the $r$-th order cotangent bundle $T^{r*}M$ are determined. Basic affinors of this type are the identity affinor id of $T{T}^{r*}M$ and the $s$-th power affinors $Q_M^s:T{T}^{r*}M\to V{T}^{r*}M$ with $s=1,\cdots ,r$ defined by the $s$-th power transformations $A_s^{r,r}$ of $T^{r*}M$. An arbitrary natural affinor is a linear combination of the basic ones.

Let $r,s,q,m,n\in \mathbb{N}$ be such that $s\ge r\le q$. Let $Y$ be a fibered manifold with $m$-dimensional basis and $n$-dimensional fibers. All natural affinors on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are classified. It is deduced that there is no natural generalized connection on ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$. Similar problems with ${\left(J^{r,s}{(Y,ℝ)}_0\right)}^{*}$ instead of ${\left(J^{r,s,q}{(Y,{ℝ}^{1,1})}_0\right)}^{*}$ are solved.

We deduce that for $n\ge 2$ and $r\ge 1$, every natural affinor on $J^{r}T$ over $n$-manifolds is of the form $\lambda \delta$ for a real number $\lambda$, where $\delta$ is the identity affinor on $J^{r}T$.

For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle $T^{r,s,q*}Y$ over an (m,n)-dimensional fibered manifold Y.