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Criteria for a diffeomorphism of a smooth manifold $M$ to be lifted to a linear automorphism of a given real vector bundle $p\colon V\rightarrow M$, are stated. Examples are included and the metric and complex vector-bundle cases are also considered. more...

Differential $p$-forms and $q$-vector fields with constant coefficients are studied. Differential $p$-forms of degrees $p=1,2,n-1,n$ with constant coefficients on a smooth $n$-dimensional manifold $M$ are characterized. In the contravariant case, the obstruction for a $q$-vector field $V_q$ to have constant coefficients is proved to be the Schouten-Nijenhuis bracket of $V_q$ with itself. The $q$-vector fields with constant coefficients of degrees $q=1,2,n-1,n$ are also characterized. The notions of differential $p$-forms and $q$-vector fields with conformal constant coefficients are introduced. For arbitrary degrees $p$ and $q$, such differential $p$-forms and $q$-vector fields are seen to be the solutions to two second-order partial differential systems on $J^2(M,\mathbb{R}^n)$, which are reducible to two first-order partial differential systems by adding variables. Computational aspects in solving these systems are discussed and examples and applications are also given. more...

Let $\mathbb{F}$ be a field of characteristic $\neq 2$ and $3$, let $V$ be a $\mathbb{F}$-vector space of dimension $6$, and let $\Omega \in \wedge ^2V^\ast $ be a non-degenerate form. A system of generators for polynomial invariant functions under the tensorial action of the group $Sp(\Omega )$ on $\wedge ^3 V^\ast $, is given explicitly. Applications of these results to the normal forms of De Bruyn-Kwiatkowski and Popov are given., Comment: Preprint more...

The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below we tackle one of these problems: The existence of a finite system of generators of gauge-invariant Lagrangians and how to compute them. More precisely, if $p\colon C\to M$ is the bundle of connections on a principal $G$-bundle $\pi\colon P\to M$, then a finite number $L_1,\dotsc,L_{N^\prime }$ of gauge-invariant Lagrangians defined on $J^1C$ is proved to exist such that for any other gauge-invariant Lagrangian $L\in C^\infty (J^1C)$ there exists a function $F\in C^\infty (\mathbb{R}^{N^\prime })$ such that $L=F(L_1,\dotsc,L_{N^\prime})$. Several examples are dealt with explicitly. more...

Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved to be isomorphic to the algebra of smooth $Sp(\Omega_{x})$-invariant functions on $\wedge^{3}T_{x}^{\ast}M$, $x$ being a fixed point in $M$, and $\Omega_{x}$ a fixed element in $(O_{M})_{x}$. The maximum number of functionally independent invariants is computed. more...

The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $\Omega$ is 4, then the equivalence under diffeomorphisms of $\Omega$ is reduced to the equivalence of a symplectic linear frame functorially attached to $\Omega$. As the equivalence problem for linear parallelisms is known, the present work solves generically the equivalence problem under diffeomorphisms of germs of 2-forms in 4 variables., Comment: 16 pages more...

Let $M$ be a connected smooth manifold, let $\operatorname{Aut}(p)$ be the group automorphisms of the bundle $p\colon \mathbb{R}\times M\to \mathbb{R}$, and let $q\colon J^1(\mathbb{R},M)\times \mathbb{R\to }J^1(\mathbb{R},M)$ be the canonical projection. Invariant functions on $J^r(q)$ under the natural action of $\operatorname{Aut}(p)$ are discussed in relationship with the Lagrangian equivalence problem. The second-order invariants are determined geometrically as well as some other higher-order invariants for $\dim M\geq 2$. more...

A first-order Lagrangian $L^\nabla $ variationally equivalent to the second-order Einstein-Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by $L^\nabla $ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to $\nabla $. more...

Let $C\to M$ be the bundle of connections of a principal bundle on $M$. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian density $\Lambda $ on $C$ satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler-Lagrange equations for $\Lambda $. This structure is also studied for the Jacobi fields and for the moduli space of extremals. more...

Two examples of $\mathrm{Diff}^+S^1$-invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on $H^2(\mathfrak{X}(S^1))$. more...

The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions $n=4r-2$. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the symplectic reduction corresponding to the pre-symplectic form and its moment map attached to the first Pontryagin form, is proved to coincide with the Teichm\"{u}ller space endowed with the Weil-Petersson symplectic form. more...

Let $FM,\mathcal{M}_M$ be the bundles of linear frames and Riemannian metrics of a manifold $M$, respectively. The existence of a unique $\mathrm{Diff}M$-invariant connection form on $J^1\mathcal{M}_M\times_MFM\to J^1\mathcal{M}_M$, which is Riemannian with respect to the universal metric on $J^1\mathcal{M}_M\times_MTM$, is proved. Aplications to the construction of universal Pontryagin and Euler forms, are given. more...