Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category ${\mathrm{ℱℳ}}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$ to be a $GL\left(m\right)$-invariant algebra homomorphism $\nu :A\to A^1$ with $t^1=\nu \circ t$. The main result is that there exists an ${\mathrm{ℱℳ}}_m$-natural transformation $FY\to F^{1}Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization)...

We prove that the problem of finding all $ℳf_m$-natural operators $C:Q⇝Q{T}^{r*}$ lifting classical linear connections $\nabla$ on $m$-manifolds $M$ into classical linear connections $C_M\left(\nabla \right)$ on the $r$-th order cotangent bundle $T^{r*}M=J^r{(M,ℝ)}_0$ of $M$ can be reduced to the well known one of describing all $ℳf_m$-natural operators $D:Q⇝{⨂}^{p}T\otimes {⨂}^q{T}^{*}$ sending classical linear connections $\nabla$ on $m$-manifolds $M$ into tensor fields $D_M\left(\nabla \right)$ of type $(p,q)$ on $M$.

Let $𝒳\to S$ be a smooth proper family of complex curves (i.e. family of Riemann surfaces), and $ℱ$ a $G$-bundle over $𝒳$ with connection along the fibres $𝒳\to S$. We construct a line bundle with connection $({ℒ}_{ℱ},{\nabla }_{ℱ})$ on $S$ (also in cases when the connection on $ℱ$ has regular singularities). We discuss the resulting $({ℒ}_{ℱ},{\nabla }_{ℱ})$ mainly in the case $G={ℂ}^{*}$. For instance when $S$ is the moduli space of line bundles with connection over a Riemann surface $X$, $𝒳=X\times S$, and $ℱ$ is the Poincaré bundle over $𝒳$, we show that $({ℒ}_{ℱ},{\nabla }_{ℱ})$ provides a prequantization...

Let $ℳ=(M,{𝒪}_{ℳ})$ be a smooth supermanifold with connection $\nabla$ and Batchelor model ${𝒪}_{ℳ}\cong {\Gamma }_{\Lambda E^{*}}$. From $(ℳ,\nabla )$ we construct a connection on the total space of the vector bundle $E\to M$. This reduction of $\nabla$ is well-defined independently of the isomorphism ${𝒪}_{ℳ}\cong {\Gamma }_{\Lambda E^{*}}$. It erases information, but however it turns out that the natural identification of supercurves in $ℳ$ (as maps from ${ℝ}^{1|1}$ to $ℳ$) with curves in $E$ restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for...

We consider a function $u:\Omega \to {ℝ}^N$, $\Omega \subset {ℝ}^n$, minimizing the integral ${\int }_{\Omega }\left(\right|D_1{u|}^2+\cdots +|D_{n-1}{u|}^2+|D_{n}u{|}^p)dx$, $2(n+1)/(n+3)\le p<2$, where $D_{i}u=\partial u/\partial x_i$, or some more general functional with the same behaviour; we prove the existence of second weak derivatives $D\left(D_{1}u\right),\cdots ,D\left(D_{n-1}u\right)\in L^2$ and $D\left(D_{n}u\right)\in L^p$.