Any $r$-dimensional subbundle of the cotangent bundle on an $n$-dimensional manifold $M$ partitions $M$ into subsets $M_0,...,M_m$ ($m$ being the minimum of $r$ and $C(n-r,2)$, the combinations of $n-r$ things taken 2 at a time). $M_i$ is the set on which the first derived systems of the subbundle has codimension $i$.In this paper we prove the following:Theorem. Let $s\ge 2$ and let $Q$ be a generic $C^s$$r$-dimensional subbundle...

In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curve $X$ and a stratified Morse function $f$. In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of $f$.

It is proved that a closed r-form ω on a manifold M defines a cohomology (called ω-coeffective) on M. A general algebraic machinery is developed to extract some topological information contained in the ω-coeffective cohomology. The cases of 1-forms, symplectic forms, fundamental 2-forms on almost contact manifolds, fundamental 3-forms on $G_2$-manifolds and fundamental 4-forms in quaternionic manifolds are discussed.

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework...

We show how the theory of ${\mathbb{Z}}_2^n$-manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for $n=2$. We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3.

Let $r$, $s$, $m$, $n$, $q$ be natural numbers such that $s\ge r$. We prove that any $2$-$ℱ{𝕄}_{m,n,q}$-natural operator $A:T_{2-proj}⇝T{J}^{(s,r)}$ transforming $2$-projectable vector fields $V$ on $(m,n,q)$-dimensional $2$-fibred manifolds $Y\to X\to M$ into vector fields $A\left(V\right)$ on the $(s,r)$-jet prolongation bundle $J^{(s,r)}Y$ is a constant multiple of the flow operator ${𝒥}^{(s,r)}$.

For natural numbers $r$ and $n\ge 2$ a complete classification of natural affinors on the natural bundle ${\left(J^r{T}^{*}\right)}^{*}$ dual to $r$-jet prolongation $J^r{T}^{*}$ of the cotangent bundle over $n$-manifolds is given.

We classify all natural affinors on vertical fiber product preserving gauge bundle functors $F$ on vector bundles. We explain this result for some more known such $F$. We present some applications. We remark a similar classification of all natural affinors on the gauge bundle functor $F^{*}$ dual to $F$ as above. We study also a similar problem for some (not all) not vertical fiber product preserving gauge bundle functors on vector bundles.

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s)(Y, R)0)*→R for any Y as above.

For natural numbers n ≥ 3 and r a complete description of all natural bilinear operators $T*{\times }_{ℳfₙ}{T}^{(0,0)}⇝T^{(0,0)}{T}^{\left(r\right)}$ is presented. Next for natural numbers r and n ≥ 3 a full classification of all natural linear operators $T{*}_{|ℳfₙ}⇝T{T}^{\left(r\right)}$ is obtained.