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The natural operators of general affine connections into general affine connections

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doi: 10.17951/a.2017.71.1.61

A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXXI, NO. 1, 2017 SECTIO A 61–64

JAN KUREK and WŁODZIMIERZ M. MIKULSKI

The natural operators of general affine connections into general affine connections

Abstract. We reduce the problem of describing all Mfm-natural operators transforming general affine connections on m-manifolds into general affine ones to the known description of all GL(Rm)-invariant maps Rm∗⊗ Rm

kRm∗⊗ ⊗kRmfor k = 1, 3.

Introduction. All manifolds considered in this paper are assumed to be finite dimensional, without boundaries, second countable, Hausdorff and smooth (of class C). Maps between manifolds are assumed to be smooth (of class C). The category of m-dimensional manifolds and their embed- dings is denoted by Mfm.

A classical linear connection on a manifold M is a right invariant connec- tion Γ on the principal fiber bundle LM of linear frames of M . It can be considered equivalently as the corresponding R-bilinear map ∇ : X (M ) × X (M ) → X (M ) such that ∇f XY = f ∇XY and ∇Xf Y = X(f )Y + f ∇XY for any map f : M → R and any vector fields X, Y ∈ X (M ) on M , see [2].

A general affine connection on M is a right invariant connection Γ on the principal fiber bundle AM of affine frames of M . It can be equivalently considered as the corresponding pair (∇, K) consisting of a classical linear connection ∇ on M and a tensor field K of type (1, 1) on M , see [2].

The general concept of natural operators can be found in [3].

2010 Mathematics Subject Classification. 58A32.

Key words and phrases. General affine connection, natural operator.

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62 J. Kurek and W. M. Mikulski

In the present note, we study the problem of finding all Mfm-natural operators B : Qgen-af Qgen-af transforming general affine connections (∇, K) on m-manifolds M into general affine connections B(∇, K) on M .

Given an Mfm-natural operator B : Qgen-af Qgen-af, we define an Mfm-natural operator ∆ : Qgen-af (⊗2T⊗ T ) ⊕ (T⊗ T ) by

B(∇, K) = (∇, K) + ∆(∇, K)

for all general affine connections (∇, K) on m-manifolds M , and vice versa.

So, to find all Mfm-natural operators B : Qgen-af Qgen-af it is sufficient to find all Mfm-natural operators ∆ : Qgen-af (⊗2T ⊗ T ) ⊕ (T⊗ T ) transforming general affine connections (∇, K) on m-manifolds M into pairs

∆(∇, K) = (∆1(∇, K), ∆2(∇, K)) of tensor fields ∆1(∇, K) of type (1, 2) and ∆2(∇, K) of type (1, 1) on M .

In the present note, we prove that the above problem of finding all Mfm- natural operators B : Qgen-af Qgen-af (or ∆ : Qgen-af (⊗2T ⊗ T ) ⊕ (T ⊗ T )) can be reduced to the one of describing all GL(Rm)-invariant maps Rm∗⊗ Rm → ⊗kRm∗⊗ ⊗kRm for k = 1, 3.

This “reduction” is satisfactory, because the GL(Rm)-invariant maps Rm∗⊗ Rm → ⊗kRm∗⊗ ⊗kRm for k = 1, 2, 3 are described in [1].

1. The crucial lemma. We prove the following lemma.

Lemma 1. There is the bijection between the set C of all Mfm-natural operators ∆ : Qgen-af (⊗2T⊗T )⊕(T⊗T ) and the set D of all GL(Rm)- invariant maps (V2

Rm∗⊗Rm)⊕(Rm∗⊗Rm)⊕(⊗2Rm∗⊗Rm) → (⊗2Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm).

Proof. We define a map Φ : C → D as follows.

Any ∆ ∈ C is determined by the values

∆(∇, K)(x) = (∆1(∇, K)(x), ∆2(∇, K)(x))

∈ (⊗2TxM ⊗ TxM ) ⊕ (TxM ⊗ TxM )

for all m-manifolds M , all linear connections ∇ on M , all tensor fields K of type (1, 1) on M and all x ∈ M . Because of the Mfm-invariance of ∆, we may assume that M = Rm, x = 0. We can even assume that idRm is

∇-normal with center 0 (then ∇(0) ∈V2

Rm∗⊗ Rm because the Christoffel symbols ∇ijk of ∇ satisfy ∇ijk(0) + ∇ikj(0) = 0). Then using the invariance of ∆ with respect to the homotheties at = t idRm for t > 0, we obtain the homogeneity condition

∆((at)∇, (at)K)(0) = (t∆1(∇, K)(0), ∆2(∇, K)(0)) .

Because of the homogeneous function theorem [3], this type of the homo- geneity implies that ∆(∇, K)(0) depends on ∇(0) and j01K (only). Let (Λ, τ0, τ1) ∈ (V2

Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) ⊕ (⊗2Rm∗⊗ Rm) ˜=(V2

T0Rm

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The natural operators of general affine connections... 63

T0Rm) ⊕ J01(TRm⊗ T Rm), where ˜= is the usual GL(Rm)-invariant iden- tification. We put

Φ(∆)(Λ, τ0, τ1) := ∆(∇, K)(0) ∈ (⊗2Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) (modulo the usual GL(Rm)-invariant identification), where ∇ is the linear connection on Rm such that the Christoffel symbols of ∇ with respect to the chart idRm are constant maps and ∇(0) = ∇o(0)+Λ and ∇ois the usual flat torsion free connection on Rm and K is the tensor field of type (1, 1) on Rm such that the coefficients of K in the chart idRm are polynomials of degree not more than 1 and j01K = (τ0, τ1).

Since ∆ is determined by Φ(∆), Φ is injective.

It remains to show that Φ is surjective. Let c : (V2Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) ⊕ (⊗2Rm∗⊗ Rm) → (⊗2Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) be a GL(Rm)- invariant map (an element from D). Using the usual GL(Rm)-invariant identification Rm= T0Rm, we have the GL(Rm)-invariant map

c : (

2

^T0Rm⊗ T0Rm) ⊕ (J01(TRm⊗ T Rm)) →

→ (⊗2T0Rm⊗ T0Rm) ⊕ (T0Rm⊗ T0Rm) . Let (∇, K) be a general connection on an m-manifold M . Using c, we define a pair ∆c(∇, K) consisting of tensor fields ∆1c(∇, K) of type (1, 2) and ∆2c(∇, K) of type (1, 1) on M as follows. Let x ∈ M . Consider a normal coordinate system ϕ of ∇ with center x. Then (ϕ∇)0 ∈V2T0Rm⊗ T0Rm modulo the obvious GL(Rm)-invariant identification and j01K) ∈ J01(TRm⊗ T Rm). We put

c(∇, K))0:= c((ϕ∇)0, j01K)) .

If ψ is another normal coordinate system of ∇ with center x, then ψ = η ◦ ϕ for a GL(Rm)-map η. Then (ψc(∇, K))0 = (ϕc(∆, K))0 because of the GL(Rm)-invariance of c. That is why, the definition of ∆c(∇, K) is correct. Thus we have the Mfm-natural operator ∆c : Qgen-af (⊗2T

T ) ⊕ (T⊗ T ). Clearly, Φ(∆c) = c. 

2. The main result. The main result of the note is the following “reduc- tion” theorem.

Theorem 1. The problem of finding all Mfm-natural operators B : Qgen-af Qgen-af can be reduced to the one of describing all GL(Rm)-invariant maps Rm∗⊗ Rm → ⊗kRm∗⊗ ⊗kRm for k = 1, 3.

Proof. Any GL(Rm)-invariant map c : (V2

Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) ⊕ (⊗2Rm∗⊗ Rm) → (⊗2Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) is the system of GL(Rm)- invariant maps

c1: (

2

^Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) ⊕ (⊗2Rm∗⊗ Rm) → ⊗2Rm∗⊗ Rm

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64 J. Kurek and W. M. Mikulski

and c2 : (

2

^Rm∗⊗ Rm) ⊕ (Rm∗⊗ Rm) ⊕ (⊗2Rm∗⊗ Rm) → Rm∗⊗ Rm. Using the invariance of ci with respect to the homotheties at = tidRm

for t > 0, we obtain the respective homogeneity conditions. Then (by the homogeneous function theorems) c1(Λ, τ0, τ1) is linear in Λ and τ1 and not necessarily linear in τ0. Then c1can be treated as the sum of GL(Rm)-linear maps

c01: Rm∗⊗ Rm → (

2

^Rm∗⊗ Rm)⊗ (⊗2Rm∗⊗ Rm) ⊂ ⊗3Rm∗⊗ ⊗3Rm and

c001 : Rm∗⊗ Rm → (⊗2Rm∗⊗ Rm)⊗ (⊗2Rm∗⊗ Rm) ˜= ⊗3Rm∗⊗ ⊗3Rm. By the same arguments, c2(Λ, τ0, τ1) is independent of Λ and τ1. Then c2 : Rm∗⊗ Rm → Rm∗⊗ Rm is a GL(Rm)-invariant map.

Now, Theorem 1 is an immediate consequence of Lemma 1.  References

[1] Dębecki, J., The natural operators transforming affinors to tensor fields of type (3, 3), Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 39 (2000), 37–49.

[2] Kobayashi, S., Nomizu, K., Foundations of Differential Geometry. Vol. I, J. Wiley- Interscience, New York–London, 1963.

[3] Kol´r, I., Michor, P. W., Slov´ak, J., Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.

Jan Kurek Włodzimierz M. Mikulski

Institute of Mathematics Institute of Mathematics Maria Curie-Skłodowska University Jagiellonian University pl. M. Curie-Skłodowskiej 1 ul. S. Łojasiewicza 6

Lublin Cracow

Poland Poland

e-mail: kurek@hektor.umcs.lublin.pl e-mail: Wlodzimierz.Mikulski@im.uj.edu.pl Received December 31, 2016

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