Let F:ℱ ℳ → ℬ be a vector bundle functor. First we classify all natural operators $T_{proj|ℱ{ℳ}_{m,n}}⇝T^{(0,0)}\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ transforming projectable vector fields on Y to functions on the dual bundle (FY)* for any $ℱ{ℳ}_{m,n}$-object Y. Next, under some assumption on F we study natural operators $T{*}_{hor|ℱ{ℳ}_{m,n}}⇝T*\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ lifting horizontal 1-forms on Y to 1-forms on (FY)* for any Y as above. As an application we classify natural operators $T{*}_{hor|ℱ{ℳ}_{m,n}}⇝T*\left(F_{|ℱ{ℳ}_{m,n}}\right)*$ for some vector bundle functors F on fibered manifolds.

Admissible fiber product preserving bundle functors F on $ℱ{ℳ}_m$ are defined. For every admissible fiber product preserving bundle functor F on $ℱ{ℳ}_m$ all natural operators $B:T_{proj|ℱ{ℳ}_{m,n}}\to TF$ lifting projectable vector fields to F are classified.

For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{\left(r\right),a}$ over $n$-manifolds such that $T^{\left(r\right),0}$ is the (classical) vector tangent bundle $T^{\left(r\right)}$ of order $r$. For integers $r\ge 1$ and $n\ge 3$ and a real number $a<0$ we classify all natural operators $T_{|M_n}⇝T{T}^{\left(r\right),a}$ lifting vector fields from $n$-manifolds to $T^{\left(r\right),a}$.

For integers $r\ge 2$ and $n\ge 2$ a complete classification of all natural operators $A:T_{|M_n}⇝T{\left(J^r{T}^{*}\right)}^{*}$ lifting vector fields to vector fields on the natural bundle ${\left(J^r{T}^{*}\right)}^{*}$ dual to $r$-jet prolongation $J^r{T}^{*}$ of the cotangent bundle over $n$-manifolds is given.

All natural operators T ↝ T(T ⊗ T*) lifting vector fields X from n-dimensional manifolds M to vector fields B(X) on the bundle of affinors ™ ⊗ T*M are described.

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A\left(f\right):T{T}^{\left(r\right)}M\to T{T}^{\left(r\right)}M$ on the vector r-tangent bundle $T^{\left(r\right)}M=\left(J^r(M,ℝ)₀\right)*$ over M. This problem is reflected in the concept of natural operators $A:T_{|ℳfₙ}^{(0,0)}⇝T^{(1,1)}{T}^{\left(r\right)}$. For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ${}^{\infty }\left(T^{\left(r\right)}ℝ\right)$ and we construct explicitly a basis of this module.

If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) 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 ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in...

If (M,g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T* M given by v → g(v,−) between the tangent TM and the cotangent T* M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrTM between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT M of cotangent TM bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT* M depending on a Riemannian...

For natural numbers $r\ge 2$ and $n$ a complete classification of natural transformations $A:T{T}^{\left(r\right)}\to T{T}^{\left(r\right)}$ over $n$-manifolds is given, where $T^{\left(r\right)}$ is the linear $r$-tangent bundle functor.

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C*-algebra theory.