-symmetric submanifolds
All natural affinors on the -th order cotangent bundle are determined. Basic affinors of this type are the identity affinor id of and the -th power affinors with defined by the -th power transformations of . An arbitrary natural affinor is a linear combination of the basic ones.
Let be such that . Let be a fibered manifold with -dimensional basis and -dimensional fibers. All natural affinors on are classified. It is deduced that there is no natural generalized connection on . Similar problems with instead of are solved.
We deduce that for and , every natural affinor on over -manifolds is of the form for a real number , where is the identity affinor on .
For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle over an (m,n)-dimensional fibered manifold Y.
We consider several explicit examples of solutions of the differential equation Φ₁’²(z) + Φ₂’²(z) + Φ₃’²(z) = d²(z) of meromorphic curves in ℂ³ with preset infinitesimal arclength function d(z) by nonlinear differential operators of the form (f,h,d) → V(f,h,d), V = (Φ₁,Φ₂,Φ₃), whose arguments are triples consisting of a meromorphic function f, a meromorphic vector field h, and a meromorphic differential 1-form d on an open set U ⊂ ℂ or, more general, on a Riemann surface Σ. Most of them are natural...
Let and be two natural bundles over -manifolds. We prove that if is of type (I) and is of type (II), then any natural differential operator of into is of order 0. We give examples of natural bundles of type (I) or of type (II). As an application of the main theorem we determine all natural differential operators between some natural bundles.
We determine all natural functions on and .
We determine all first order natural operators transforming –tensor fields on a manifold into –tensor fields on .
Natural liftings are classified for . It is proved that they form a 5-parameter family of operators.
We prove, that -th order gauge natural operators on the bundle of Cartan connections with a target in the gauge natural bundles of the order (“tensor bundles”) factorize through the curvature and its invariant derivatives up to order . On the course to this result we also prove that the invariant derivations (a generalization of the covariant derivation for Cartan geometries) of the curvature function of a Cartan connection have the tensor character. A modification of the theorem is given for...
For natural numbers n ≥ 3 and r ≥ 1 all natural operators transforming functions from n-manifolds into affinors (i.e. tensor fields of type (1,1)) on the r-cotangent bundle are classified.
The complete description of all natural operators lifting real valued functions to bundle functors on fibered manifolds is given. The full collection of all natural operators lifting projectable real valued functions to bundle functors on fibered manifolds is presented.
Let be a Weil algebra. The bijection between all natural operators lifting vector fields from -manifolds to the bundle functor of Weil contact elements and the subalgebra of fixed elements of the Weil algebra is determined and the bijection between all natural affinors on and is deduced. Furthermore, the rigidity of the functor is proved. Requisite results about the structure of are obtained by a purely algebraic approach, namely the existence of nontrivial is discussed.
Let us consider two closed surfaces , of class and two functions , of class , called measuring functions. The natural pseudodistance between the pairs , is defined as the infimum of as varies in the set of all homeomorphisms from onto . In this paper we prove that the natural pseudodistance equals either , , or , where and are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and critical values...