On almost polynomial structures from classical linear connections

Let Mfm be the category of m-dimensional manifolds and local diffeomorphisms and let T be the tangent functor on Mfm. Let V be the category of real vector spaces and linear maps and let Vm be the category of m-dimensional real vector spaces and linear isomorphisms. Let w be a polynomial in one variable with real coefficients. We describe all regular covariant functors F : Vm → V admitting Mfm-natural operators P̃ transforming classical linear connections ∇ on m-dimensional manifolds M into almost polynomial w-structures P̃ (∇) on F (T )M = ⋃ x∈M F (TxM).


Introduction.
All manifolds considered in the paper are assumed to be Hausdorff, finite dimensional, second countable, without boundaries and smooth (i.e. of class C ∞ ).Maps between manifolds are assumed to be of class C ∞ .
The category of m-dimensional manifolds and local diffeomorphisms is denoted by Mf m .The category of vector bundles and vector bundle homomorphisms between them is denoted by VB.The category of m-dimensional real vector spaces and linear isomorphisms is denoted by V m .The category of finite dimensional real vector spaces and linear maps is denoted by V.
Let w be a polynomial in one variable.A tensor field P of type (1, 1) on a manifold N is called an almost polynomial w-structure on N if w(P ) = 0 (i.e.w(P |x ) = 0 for any x ∈ N ).
In the present paper we solve the following problem.If w(t) = t 2 + 1, then we reobtain the result from [5] on the characterization of covariant regular functors F : V m → V admitting Mf m -natural operators J transforming classical linear connections ∇ on m-manifolds M into almost complex structures J(∇) on F (T )M .
If w(t) = t 2 − 1, then we characterize covariant regular functors F : V m → V admitting Mf m -natural operators J transforming classical linear connections ∇ on m-manifolds M into almost para-complex structures J(∇) on F (T )M .

Basic definitions.
The concept of natural bundles and natural operators can be found in the fundamental monograph [3].
Let F : V m → V be a covariant regular functor.The regularity of the functor F means that F transforms smoothly parametrized families of isomorphisms into smoothly parametrized families of linear maps.Let T : Mf m → VB be the tangent functor sending any m-dimensional manifold M into the tangent bundle T M of M and any Mf m -map ϕ : M 1 → M 2 into the tangent map T ϕ : T M 1 → T M 2 .Applying F to fibers T x M of T M , one can define a natural vector bundle F (T ) of order 1 over m-manifolds by for any m-manifold M and any Mf m -map ϕ : M 1 → M 2 between mmanifolds M 1 and M 2 .In particular, if F is the identity functor, then are any vector fields on M and f, f 1 , f 2 : M → R are any smooth functions on M .Equivalently, a classical linear connection on M is a right invariant decomposition T LM = H ∇ ⊕ V LM of the tangent bundle T LM of LM , where LM is the principal bundle with the structural group GL(m) of linear frames over M and V LM is the vertical bundle of LM , see [2].
+ a 0 be the polynomial in one variable with real coefficients a m−1 , . . ., a 0 .
A polynomial w-structure on a real vector space W is a linear endomorphism P : W → W such that w(P ) = P m +a m−1 P m−1 +• • •+a 1 P +a 0 I = 0, where P k denotes the composition and I denotes the identity map on W .
An almost polynomial w-structure on manifold N is a tensor field P : T N → T N on N of type (1, 1) (affinor) such that P x : T x N → T x N is a polynomial w-structure on T x N for any x ∈ N .In other words, an almost polynomial w-structure is a tensor field P of type (1, 1) on manifold N satisfying a polynomial equation P m + a m−1 P m−1 + • • • + a 1 P + a 0 I = 0, where a m−1 , . . ., a 0 are real numbers, at every point of N .
The general concept of natural operators can be found in the fundamental monograph [3].In particular, we have the following definition.Let F : V m → V be as above.A V m -canonical polynomial w-structure on V ⊕ F V is a V m -invariant system P of polynomial w-structures

The main result.
The main result of the present note is the following theorem.
Theorem 1.Let F : V m → V be a covariant regular functor and w be a polynomial in one variable with real coefficients.The following conditions are equivalent: (i) There exists an Mf m -natural operator P : Q (AwS)F (T ).(ii) There exists a V m -canonical polynomial w-structure P on V ⊕ F V .
Proof.(i) ⇒ (ii).Let P : Q (AwS)F (T ) be an Mf m -natural operator in question.Let V be an m-dimensional vector space from the category V m and let ∇ V be the V m -canonical torsion free flat classical linear connection on V .Then the almost polynomial w-structure P (∇ V ) : T F (T )V → T F (T )V on F (T )V restricts to the polynomial w-structure is the polynomial w-structure on V ⊕ F V for any V m -object V .Because of the canonical character of the construction of P , the structure Then we have the identification ) canonically depending on ∇, where the equality is the connection decomposition, the identification ∼ = is the usual one (namely, H∇ v = T x M modulo the tangent of the projection of F (T )M and V v F (T )M = T v (F (T ) x M ) = F (T ) x M modulo the standard identification) and the second equality is by the definition of F (T )M .We define P (∇) |v : By the canonical character of P (∇), the resulting family P : 4. An application to para-complex structures.Let w(t) = t 2 −1.Let J be a polynomial w-structure on a vector space W . Then W = W + ⊕ W − , where W ± = {v ∈ W : J(v) = ±v}.If additionally dim(W + ) = dim(W − ), then J is called a para-complex structure on W , see [6].
An almost para-complex structure on a manifold N is an affinor J : T N → T N on N such that J x : T x N → T x N is a para-complex structure on T x N for any x ∈ N .In other words, an almost para-complex structure is a smooth (1, 1)-tensor field on the manifold N of even dimension m, if the following conditions are satisfied: (1) J 2 = id T N (2) for each point x ∈ N , the eigenspaces T + x N and T − x N of J x (the value of J at x) are both m 2 -dimensional subspaces of the tangent space T x N at x, [1], [7].
Corollary 1.Let F : V m → V be a regular covariant functor.The following conditions are equivalent: (a) There is an Mf m -natural operator J : Proof.This is a simple consequence of Theorem 1.

Lemma 1.
Let p be a positive integer.Let F : V m → V be a covariant regular functor given by F Proof.If p is even, we have the V m -canonical para-complex structure on V × • • • × V (p times of V ) .Namely, we have the p 2 copies of the canonical para-complex structure on V × V given by (v, w) → (v, −w).
A Weil algebra A is a finite dimensional, commutative, associative and unital algebra of the form A = R × N , where N is the ideal of all nilpotent elements of A.
Lemma 2 (Lemma 5.1 in [4]).Let A be a p-dimensional Weil algebra and let T A be the corresponding Weil functor.For any classical linear connection ∇ on an m-manifold M , we have the base-preserving fibred diffeomorphism I A ∇ : T A M → T M ⊗ R p−1 canonically depending on ∇.We see that T M ⊗ R p−1 = T M × M • • • × M T M ((p − 1) times of T M ) = F (T )M , where F :

Problem 1 .
Let w be a polynomial in one variable with real coefficients.We characterize all covariant regular functors F : V m → V admitting Mf mnatural operators P transforming classical linear connections ∇ on m-manifolds M into almost polynomial w-structures P (∇) on F (T )M = x∈M F (T x M ), where T : Mf m → VB denotes the tangent functor on the category Mf m .

− 1 )
times of ϕ).So, from Corollary 1, Lemma 1 and Lemma 2 we obtain Proposition 1.Let A be a Weil algebra.If A is even dimensional, there exists an Mf m -natural operator J : Q (AP C)T A sending classical linear connections ∇ on m-manifolds M into almost para-complex structures J(∇) on T A M .