Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{|ℳfₙ}⇝T*T^{r*}$ transforming vector fields on n-manifolds M to 1-forms on $T^{r*}M=J^r(M,ℝ)₀$ are classified. For n ≥ 3 all natural operators $T_{|ℳfₙ}⇝\Lambda ²T*T^{r*}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r*}M$ are completely described.

We prove that the problem of finding all $ℳf_m$-natural operators $B:Q⟿Q{T}^A$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_M\left(\nabla \right)$ on the Weil bundle $T^{A}M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳf_m$-natural operators $C:Q⟿(T{¹}_{p-1},T*\otimes T*\otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_M\left(\nabla \right):T{¹}_{p-1}M={⨁}_M^{p-1}TM\to T*M\otimes T*M\otimes TM$.

All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r}E$ of E are given. Similar results are deduced for the r-jet prolongations $J_v^{r}E$ and $J^{\left[r\right]}E$ in place of $J^{r}E$.

We give a classification of all linear natural operators transforming $p$-vectors (i.e., skew-symmetric tensor fields of type $(p,0)$) on $n$-dimensional manifolds $M$ to tensor fields of type $(q,0)$ on $T^{A}M$, where $T^A$ is a Weil bundle, under the condition that $p\ge 1$, $n\ge p$ and $n\ge q$. The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$-vectors to tensor fields of type $(q,0)$ on $T^A$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

We describe all ${}_m\left(G\right)$-gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^{r}P=P^{r}M{\times }_M{J}^{r}P$ of P → M. In other words, we classify all ${}_m\left(G\right)$-natural transformations $J^{r}LP{\times }_M{W}^{r}P\to T{W}^{r}P=L{W}^{r}P{\times }_M{W}^{r}P$ covering the identity of $W^{r}P$, where $J^{r}LP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all ${}_m\left(G\right)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^{r}LP=L{W}^{r}P$. We formulate axioms which...

We describe all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D\left(\nabla \right)$ on cotangent bundles $T^{*}Y$ of $Y$. We show that this problem can be reduced to finding ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{*},{\otimes }^p{T}^{*}\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.

We classify all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\nabla$ on fibered-fibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into $r$th order Lagrangians $A\left(r\right)$ on the fibered-fibered linear frame bundle $L^{fib-fib}\left(Y\right)$ on $Y$. Moreover, we classify all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into Euler morphism $B\left(\nabla \right)$ on $L^{fib-fib}\left(Y\right)$. These classifications can be expanded on...

We establish in this paper a lower bound for the volume of a unit vector field $\overrightarrow{v}$ defined on ${𝐒}^n\setminus \{\pm x\}$, $n=2,3$. This lower bound is related to the sum of the absolute values of the indices of $\overrightarrow{v}$ at $x$ and $-x$.

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{*}M$ 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 $T^{\left(r\right)}M\widetilde{=}{T}^{r*}M$ between the $r$-th order vector tangent $T^{\left(r\right)}M={\left(J^r{(M,R)}_0\right)}^{*}$ 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 $C_M\left(g\right):T^{\left(r\right)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This...

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

Let $X$ be a smooth projective curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p\>0$. Given a semistable vector bundle $E$ over $X$, we show that its direct image $F_{*}E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank-$p$ vector bundles $F_{*}L$, where $L$ is a line bundle over $X$.

Let $M$ be an $m$-dimensional manifold and $A={𝔻}_k^r/I=ℝ\oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T_x^{A}f\in T_x^{A*}M$, $x\in M$ is determined by its values over arbitrary $max\{\mathrm{width}A,m\}$ regular and under the first jet projection linearly independent elements of $T_x^{A}M$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M\simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial...

Let ${\overline{L}}_i\to X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\overline{L}}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\overline{L}}_i$ are equivariant ${\left({ℂ}^{*}\right)}^{n_i}$-bundles satisfying some additional conditions....

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi }_V$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^m$-measurable subset of ℝⁿ with ${}^m\left(A\right)<\infty$. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi }_V\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for...

General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}\left({\mathrm{F}}_p^2\left(K\right)\right|K)$, that is the Galois group of the second Hilbert $p$-class field ${\mathrm{F}}_p^2\left(K\right)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group ${\mathrm{Cl}}_p\left(K\right)$, the position of $G$ is restricted to certain admissible branches...

Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮*$..................................................................................................................... 5 § 3. $𝒮*$-algebras.....................................................................................................................

Let m and r be natural numbers and let $P^r:ℳf_m\to ℱℳ$ be the rth order frame bundle functor. Let $F:ℳf_m\to ℱℳ$ and $G:ℳf_k\to ℱℳ$ be natural bundles, where $k=dim\left(P^r{ℝ}^m\right)$. We describe all $ℳf_m$-natural operators A transforming sections σ of $FM\to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G\left(P^{r}M\right)\to P^{r}M$. We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

Let $\mathbb{Q}\left(\alpha \right)$ and $\mathbb{Q}\left(\beta \right)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\mathbb{Q}\left(\beta \right)\to \mathbb{Q}\left(\alpha \right)$. The algorithm is particularly efficient if there is only one isomorphism.

If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${𝒟}_X^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${𝒟}_X^{\left(\infty \right)}$ on an ${𝒪}_X$-module $ℰ$ is equivalent to giving an infinite sequence of ${𝒪}_X$-modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it...