Lifting connections to the r-jet prolongation of the cotangent bundle

We show that the problem of finding all Mfm -natural operators C : Q "M QJ r T ∗ lifting classical linear connections ∇ on m-manifolds M into classical linear connections CM (∇) on the r-jet prolongation J r T ∗M of the cotangent bundle T ∗M p of M can be reduced to that of finding all Mfm -natural operators D : Q "M® T ⊗ q ® T ∗ sending classical linear connections ∇ on M into tensor fields DM (∇) of type (p, q) on M

The natural operators similar to the twisted Courant bracket one

Given natural numbers m≥3 and p≥3, all Mfm-natural operators AH sending p-forms H∈Ωp(M) on m-manifolds M into bilinear operators AH:(X(M)⊕Ω1(M))×(X(M)⊕Ω1(M))→X(M)⊕Ω1(M) transforming pairs of couples of vector fields and 1-forms on M into couples of vector fields and 1-forms on M are founded. If m≥3 and p≥3, then that any (similar as above) Mfm-natural operator A which is defined only for closed p-forms H can be extended uniquely to the one A which is defined for all p-forms H is observed. If p=3 and m≥3, all Mfm-natural operators A (as above) such that AH satisfies the Leibniz rule for all closed 3-forms H on m-manifolds M are extracted. The twisted Courant bracket [−,−]H for all closed 3-forms H on m-manifolds M gives the most important example of such Mfm-natural operator A

On the twisted Dorfman-Courant like brackets

There are completely described allVBm,n-gauge-natural operatorsCwhich, like tothe Dorfman–Courant bracket, send closed linear3-forms $H∈Γl−closE(∧3T∗E)$on a smooth(C∞) vector bundleEintoR-bilinear operatorsCH: $ΓlE(T E⊕T∗E)×ΓlE(T E⊕T∗E)→ΓlE(T E⊕T∗E)$ transforming pairs of linear sections of $T E⊕T∗E→E$ into linear sections of $T E⊕T∗E→E.$ Then all suchCwhich also, like to the twisted Dorfman–Courant bracket, satisfy both some“restricted” condition and the Jacobi identity in Leibniz form are extracted

On regular local operators on smooth maps

Let X, Y, Z, W be manifolds and π : Z → X be a surjective submersion. We characterize π-local regular operators A : C∞(X,Y) → C∞(Z,W) in terms of the corresponding maps à : J∞(X,Y) ×XZ → W satisfying the so-called local finite order factorization property

The gauge-natural bilinear operators similar to the Dorfman-Courant bracket

All gauge-natural bilinear operators A:ΓlE(TE⊕T∗E)×ΓlE(TE⊕T∗E)→ΓlE(TE⊕T∗E) transforming pairs of linear sectionsof the “doubled” tangent bundleTE⊕T∗Eof a vector bundleEintolinear sections ofTE⊕T∗Eare completely described. Then, all suchAwith the Jacobi identity in Leibniz form are extracted