The natural operators lifting -projectable vector fields to product-preserving bundle functors on -fibered manifolds.
Admissible fiber product preserving bundle functors F on are defined. For every admissible fiber product preserving bundle functor F on all natural operators lifting projectable vector fields to F are classified.
For natural numbers and and a real number we construct a natural vector bundle over -manifolds such that is the (classical) vector tangent bundle of order . For integers and and a real number we classify all natural operators lifting vector fields from -manifolds to .
For integers and a complete classification of all natural operators lifting vector fields to vector fields on the natural bundle dual to -jet prolongation of the cotangent bundle over -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 on the vector r-tangent bundle over M. This problem is reflected in the concept of natural operators . 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 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 and a complete classification of natural transformations over -manifolds is given, where is the linear -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.