We study geometrical properties of natural transformations $T^{A}T*\to T*T^A$ depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations $T^{A}T*\to T*T^A$ can be described in a uniform way by means of a simple geometrical construction.

In this paper are determined all natural transformations of the natural bundle of $(g,r)$-covelocities over $n$-manifolds into such a linear natural bundle over $n$-manifolds which is dual to the restriction of a linear bundle functor, if $n\ge q$.

A classification of natural transformations transforming functions (or vector fields) to functions on such natural bundles which are restrictions of bundle functors is given.

A classification of natural transformations transforming vector fields on $n$-manifolds into affinors on the extended $r$-th order tangent bundle over $n$-manifolds is given, provided $n\ge 3$.

Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)u^H+\beta \left(h\left(u\right)\right)u^V,$$ where $u^V$ is the vertical lift of $u\in T_{x}M$, $u^H$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta$ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma }_1\left(h\left(u\right)\right)\Lambda (g,K)+{\gamma }_2\left(h\left(u\right)\right)u^H\wedge u^V,$$ where ${\gamma }_1$, ${\gamma }_2$ are smooth real functions defined...

We establish new estimates for the Laplacian, the div-curl system, and more general Hodge systems in arbitrary dimension $n$, with data in $L^1$. We also present related results concerning differential forms with coefficients in the limiting Sobolev space $W^{1,n}$.

New versions of Slovák’s formulas expressing the covariant derivative and curvature of the linear connection ${}_A\Gamma$ are presented.

This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.

It is well-known that the Fundamental Identity (FI) implies that Nambu brackets are decomposable, i.e. given by a determinantal formula. We find a weaker alternative to the FI that allows for non-decomposable Nambu brackets, but still yields a Darboux-like Theorem via a Nambu-type generalization of Weinstein’s splitting principle for Poisson manifolds.