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 $C_M\left(\nabla \right):T{¹}_{p-1}M={⨁}_M^{p-1}TM\to T*M\otimes T*M\otimes TM$.

We classify all $ℱ{ℳ}_{m,n}$-natural operators $:J²⟿J²V^A$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $\left(\Gamma \right):V^{A}Y\to J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y\to M$ corresponding to a Weil algebra A.

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 $M$.

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 $p=3$, $q=0$.

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 $X$.

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 first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic...

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...

Let $𝒜={\left\{A_t\right\}}_{t\in G}$ and $ℬ={\left\{B_t\right\}}_{t\in G}$ be $C^{*}$-algebraic bundles over a finite group $G$. Let $C={⨁}_{t\in G}{A}_t$ and $D={⨁}_{t\in G}{B}_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we show that if there is an equivalence $𝒜-ℬ$-bundle over $G$ with some properties, then the unital inclusions of unital $C^{*}$-algebras $A\subset C$ and $B\subset D$ induced by $𝒜$ and $ℬ$ are strongly Morita equivalent. Also, we suppose that $𝒜$ and $ℬ$ are saturated and that $A^{\text{'}}\cap C=𝐂1$. We show that...

Some properties and applications of natural vector bundle morphisms $T_A\circ {⨂}_s^q\to {⨂}_s^q\circ T_A$ over $i{d}_{T_A}$ are presented.

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 $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$. Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d=2$ or $3$, we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$); for $d=2$, we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

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 $J^{r}E$.

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 $F\left(T\right)M={\bigcup }_{x\in M}F\left(T_{x}M\right)$.

We prove that any first order ${ℱ}_2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operator transforming projectable general connections on an $(m_1,m_2,n_1,n_2)$-dimensional fibred-fibred manifold $p=(p,p):(p_Y:Y\to Y)\to (p_M:M\to M)$ into general connections on the vertical prolongation $VY\to M$ of $p:Y\to M$ is the restriction of the (rather well-known) vertical prolongation operator $𝒱$ lifting general connections $\overline{\Gamma }$ on a fibred manifold $Y\to M$ into $𝒱\overline{\Gamma }$ (the vertical prolongation of $\overline{\Gamma }$) on $VY\to M$.

We classify classical linear connections $A(\Gamma ,\Lambda ,\Theta )$ on the total space $Y$ of a fibred manifold $Y\to M$ induced in a natural way by the following three objects: a general connection $\Gamma$ in $Y\to M$, a classical linear connection $\Lambda$ on $M$ and a linear connection $\Theta$ in the vertical bundle $VY\to Y$. The main result says that if $\dim \left(M\right)\ge 3$ and $\dim \left(Y\right)-\dim \left(M\right)\ge 3$ then the natural operators $A$ under consideration form the $17$ dimensional affine space.

It is well known that the concept of holonomic $r$-jet can be geometrically characterized in terms of the contact of individual curves. However, this is not true for the semiholonomic $r$-jets, [5], [8]. In the present paper, we discuss systematically the semiholonomic case.

Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^k(D,𝒪)=0$ for $2\le k\le n-1$. We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$.

We consider a convexity notion for complex spaces $X$ with respect to a holomorphic line bundle $L$ over $X$. This definition has been introduced by Grauert and, when $L$ is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if $H^0(X,L)$ separates each point of $X$, then $X$ can be realized as a Riemann domain over the complex projective...

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...

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....

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...

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.