The natural brackets on couples of vector fields and 1-forms

All natural bilinear operators transforming pairs of couples of vector fields and 1-forms into couples of vector fields and 1-forms are found. All natural bilinear operators as above satisfying the Leibniz rule are extracted. All natural Lie algebra brackets on couples of vector fields and 1-forms are collected

On lifting of 2-vector fields to rr-jet prolongation of the tangent bundle

If $m \geq 3$ and $r \geq 1$, we prove that any natural linear operator $A$ lifting 2-vector fields $\Lambda \in \Gamma (\bigwedge^2 TM)$ (i.e., skew-symmetric tensor fields of type (2,0)) on $m$-dimensional manifolds $M$ into 2-vector fields $A(\Lambda)$ on $r$-jet prolongation $J^rTM$ of the tangent bundle $TM$ of $M$ is the zero one

Natural operations of Hamiltonian type on the cotangent bundle

summary:The authors study some geometrical constructions on the cotangent bundle $T^*M$ from the viewpoint of natural operations. First they deduce that all natural operators transforming functions on $T^*M$ into vector fields on $T^*M$ are linearly generated by the Hamiltonian vector field with respect to the canonical symplectic structure of $T^*M$ and by the Liouville vector field of $T^*M$. Then they determine all natural operators transforming pairs of functions on $T^*M$ into functions on $T^*M$. In this case, the main generator is the classical Poisson bracket

On canonical constructions on connections

We study  how a projectable general connection $\Gamma$ in a 2-fibred manifold $Y^2\to Y^1\to Y^0$  and a general vertical connection $\Theta$ in $Y^2\to Y^1\to Y^0$ induce a general connection $A(\Gamma,\Theta)$ in $Y^2\to Y^1$

Lifting vector fields from manifolds to the rr-jet prolongation of the tangent bundle

If m≥3 and r≥0, we deduce that any natural linear operator lifting vector fields from an m-manifold M to the r-jet prolongation JrTM of the tangent bundle TM is the composition of the flow lifting Jr corresponding to the r-jet prolongation functor Jr with a natural linear operator lifting vector fields from M to TM. If 0≤s≤r and m≥3, we find all natural linear operators transforming vector fields on M into base-preserving fibred maps JrTM→JsTM

The twisted gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms

We completely describe all gauge-natural operators $C$ which send linear $(p+2)$-forms $H$ on vector bundles $E$ (with sufficiently large dimensional bases) into $\mathbf{R}$-bilinear operators $C_H$ transforming pairs $(X_1\oplus\omega_1,X_2\oplus\omega_2)$ of couples of linear vector fields and linear $p$-forms on $E$ into couples $C_H(X_1\oplus\omega_1, X_2\oplus\omega_2)$ of linear vector fields and linear $p$-forms on $E$. Further, we extract all $C$ (as above) such that $C_0$ is the restriction of the well-known Courant bracket and $C_H$ satisfies the Jacobi identity in Leibniz form for all closed linear $(p+2)$-forms $H$

On the existence of connections with a prescribed skew-symmetic Ricci tensor

We study the so-called inverse problem. Namely, given a prescribed skew-symmetric Ricci tensor we find (locally) a respective linear connection

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\mathrel{\tilde=}T^*M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^*M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{(r)}M\mathrel{\tilde=} T^{r*}M$ between the $r$-th order vector tangent $T^{(r)}M=(J^r(M,R)_0)^*$ and the $r$-th order cotangent $T^{r*}M=J^r(M,R)_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M(g):T^{(r)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$

The natural operators of general affine connections into general affine connections

We reduce the problem of describing all $\mathcal{M} f_m$-natural operators  transforming general affine connections on $m$-manifolds into general affine ones to the known description of all $GL(\mathbf{R}^m)$-invariant maps $\mathbf{R}^{m*}\otimes \mathbf{R}^m\to \otimes^k\mathbf{R}^{m*}\otimes\otimes ^k\mathbf{R}^m$ for $k=1,3$

On almost complex structures from classical linear connections

Let $\mathcal{M} f_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and  let $T$ be the tangent functor on $\mathcal{M} f_m$. Let $\mathcal{V}$ be the category of real vector spaces and linear maps and let $\mathcal{V}_m$ be the category of $m$-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F:\mathcal{V}_m\to\mathcal{V}$ admitting $\mathcal{M} f_m$-natural operators $\tilde J$ transforming classical linear connections $\nabla$ on $m$-dimensional manifolds $M$ into almost complex structures $\tilde J(\nabla)$ on $F(T)M=\bigcup_{x\in M}F(T_xM)$