Intrinsic General Relativity and Clifford

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Submitted on 27 Aug 2021

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Thierry Socroun, Dominique Girardot

To cite this version:

Thierry Socroun, Dominique Girardot. Intrinsic General Relativity and Clifford. ICCA12, Aug 2020, Heifei (en ligne), China. �hal-03010517v2�

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and Clifford algebra

Thierry Socroun, Dominique Girardot

Abstract. This paper proposes to go beyond the Einstein General Rel- ativity theory.

The intrinsic writing of the General Relativity theory in a suitable framework allows us to naturally introduce Clifford algebra in a generalization that would enable a unification with the Dirac theory.

This is done starting by considering a submoduleV of functions from an abelian groupGto itself. In a second time, the groupGis replaced by a unitary ring that permits to add new properties as the Leibniz rule for derivations. At last, in the particular case when V is a free G-submodule, it becomes possible to write everything with coordinates and let appear the well-known writing of the classical General Relativity and potentially the Dirac operator.

Keywords. General Relativity, Einstein, Intrinsic, non commutative, Torsion, Clifford, Dirac.

1. Introduction

General relativity equations are based on the consideration of a four dimen- sional space-time manifold and for expressing the constraints on its curvature and torsion, the so-called “field equations”. For such an achievement, tangent vector fields on the space-time manifold are considered, covariant derivatives (connections), and further torsion, Riemann-Christoffel curvature and Ricci tensor fields [1]. These concepts can be defined intrinsically, i.e. without using a system of curved coordinates (parametrization...µν) of the space-time manifold, as well as in linear algebra, a general linear application can be defined axiomatically without using bases of vector spaces and matrices. But except for recent expositions [2], General Relativity courses always make use directly of coordinates for expressing Einstein field equations, leading to a profusion of indices, even in the intrinsic form of Penrose, where the use of the same indices (Latin instead of Greek), called “symbolic”, serves in fact only to express primal or dual nature of arguments of tensors. Historically,

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it was also the case for expressing Maxwell equations, but the development of vector analysis and the introduction of rotational and divergence opera- tors have enabled the standard intrinsic formulation of these equations, that have then evolved to tensorial or Clifford formulations [3, 4]. Curiously, intrinsic formulations of General Relativity are quite recent and not widely known [2, 5], yet they are much more elegant than the equivalents using coordinates or symbolic indices, and especially avoid the need to prove a posteriori the so-called “general covariance” or “diffeomorphism invariance”, that is equivalent to gauge invariance in this framework. A unique physi- cal reality is automatically captured by the intrinsic formulation (provided bounding complete conditions are also given). Note also that up today, to the authors knowledge, no intrinsic formulation (in the sense above) exists for Dirac equation.

In the spirit of non commutative geometry, the intrinsic axiomatic (commutative) formulation of Einstein’s field equations consists only in equations that constraint the torsion and curvature tensors fields ofM, simply defined as multi-linear functions of derivations (connections) of derivations (vector fields) of scalar fields on the space-time manifold. So these equations indi- rectly constraint the connections and eventually scalar and vectors fields.

• Scalar fields on a manifoldM are by definition functionsα:M →R, whose set is noted G, which form a commutative ring for the natural inheritance of laws ofR:

∀m∈ M, ∀α, β∈G, (α+β)[m] := α[m] +β[m]

∀m∈ M, ∀α, β∈G, (αβ)[m] := α[m]β[m]

• Tangent vector fieldsX are by definition derivations inG, i.e. additive functions fromG→Gsatisfying the Leibniz rule:

∀α, β∈G, X[α+β] = X[α] +X[β] (additivity)

∀α, β∈G, X[αβ] = α X[β] +X[α]β (Leibniz rule)

We note their setV, andV is a module over G (for this property, the commutativity of the product inRis required, see later).

• A connection or covariant derivative ∇ is by definition a 2-additive function fromV ×V → V, that is linear with respect to its first argument (the direction of derivation), and satisfies the Leibniz rule with respect to its second argument:

∀X, Y, Z∈ V, ∇[X+Y, Z] = ∇[X, Z] +∇[Y, Z] (additivity 1)

∀X, Y, Z∈ V, ∇[X, Y +Z] = ∇[X, Y] +∇[X, Z] (additivity 2)

∀α∈G, ∀X, Y ∈ V, ∇[αX, Y] = α∇[X, Y] (linearity 1)

∀β∈G,∀X, Y ∈ V, ∇[X, βY] = β∇[X, Y] +X[β]Y (Leibniz rule 2) In the literature, the notation∇XY est often used for∇[X, Y].

• A k-tensor is simply a k-additive function from V^k → V(orG). We make no use of arguments of tensors in the dual V^∗, so when passing to coordinates in a base, all indices of tensors will be lower, except one.

By convention, thei^st vectorei of the chosen base ofV is written with

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lower index i. In a base (ei)_1≤i≤n, we have T[ei₁, . . . , ei_k] = T_i^j

1...i_kej

(summation on j), by definition of T_i^j

1...i_k. We won’t raise and lower indices, then no metric is involved for these simple writing conventions.

• The torsion tensorT is the 2-tensor fromV × V → V defined by T[X, Y] :=∇[X, Y]− ∇[Y, X]− C_◦[X, Y]

whereC◦[X, Y] :=X◦Y −Y ◦X. It is easy to prove thatC◦[X, Y]∈ V.

In fact,V is a Lie algebra for the commutatorC_◦.

• The Riemann-Christoffel Einstein curvature tensorR_E (E for Einstein) is the tri-linear application fromV × V × V → V defined by:

RE[X, Y, Z] :=∇[X,∇[Y, Z]]− ∇[Y,∇[X, Z]]− ∇[C_◦[X, Y], Z] (1)

• The Ricci tensor Ric, bilinear application fromV × V → G, is defined by:

Ric[X, Z] := Tr[Y →RE[X, Y, Z]]

Tr, the trace linear form of an endomophismu, is intrinsically defined, up to a constant, by the characteristic algebraic property:

∀u, v, Tr[u◦v] = Tr[v◦u]

Note thatT,RE and Riconly depend on∇.

• For any 2-tensorgfromV ×V →G, the 3-tensor∇gfromV ×V ×V →G is implicitly defined by the following Leibniz-like rule:

X[g[Y, Z]] = (∇g)[X, Y, Z] +g[∇[X, Y], Z] +g[Y,∇[X, Z]]

then

(∇g)[X, Y, Z] :=X[g[Y, Z]]−g[∇[X, Y], Z]−g[Y,∇[X, Z]] (2)

• The Einstein’s field equations in empty space are:

T = 0, Ric= Λg, ∇g= 0

for the unknowns ∇ : V × V → V and g : V × V → G the so-called metric and Λ∈Rthe so-called cosmological constant.

Note that all this stuff remains unchanged if one replacesRby any commutative field. These few lines suffice to explain and fully describe gravity. Thanks to Einstein-Infeld-Hoffman (EIH) singularity theory [6, 7, 8], particles are just space-time singularity lines of solutions of field equations “in empty space”, i.e. without impulsion-energy second member term. The gravity interaction results only on the above field equations “in empty space”, that where also called pure field equations by Tonnelat [9]. In particular, the geodesic postulate for the motion of a test particle can be proved to be a consequence of the field equations, and then does not constitute an additional postulate [10]. The generaln-body motion problem for any mass bodies is also fully explained by the field equations “in empty space” only. Of course, in this theory, the expression “in empty space” is false. It means that the matter IS the singularity space-time lines of solutions and not a second source term.

The beauty and the economy of concepts of this theory open hope that all

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interactions could be described by an analogue theory. In order to extend Gen- eral Relativity towards a unified theory of all interactions, able to take into account quantum theory and especially the Dirac equation [11], we explore here a non commutative generalization of this classical axiomatic framework.

Note that by the simple fact of considering the commutative fieldCinstead ofR, vectors fields are spinors fields, whose components in any parametrization of the space-time manifold are, as usual, functions of space-time intoC⁴. However, this generalization is not sufficient to get the quantum nature to the resulting theory and we believe that we have to consider a non commutative generalization of this classical axiomatic framework: we will replaceR orCby any non commutative ring, in particular by a Clifford ringCl_p. Then G:={α:M → Clp} is a non commutative ring.

We think that the quantum phenomena could be explained by the resulting deterministic non linear dynamic properties of solutions of such non commutative theory, as Einstein and Schr¨odinger believed. The development of such a quantic behavior emergence is not achieved, and then not exposed here.

The energy-impulsion is believed also to be explainable by such an approach, especially because Maxwell equations should be a subproduct of the theory, as Einstein believed too.

We replaceR, the set of real numbers, by any (not necessarily commutative) ring. This is the general transcription of the Connes’ idea that “the coordinates no longer commute” [12], the stating point of his non commutative geometry theory. But difficulties appear:

The set of vector fields (if the definition above is maintained), is no longer a module overG: for any scalar fieldλ∈Gand any vector field X ∈ V, the applicationλX is not in general a vector field.

As a consequence, for connections, the axiom∇[αX, Y] =α∇[X, Y] has no sense because∇[αX, Y] is not defined (whenαX /∈ V).

Then to remain coherent, the connections can no longer be defined as before.

We generalize the definition of connections, and at the same time, we abstract the framework a step further.

It is useless to consider a manifold and the set of scalar fields on it, nor fi- brations. We directly start considering any additive groupGand distributive internal lawsτ (generalizing∇) in it. We establish in this minimal base the first and the second Bianchi identities, in the general case with any torsion.

The “differential” part of the second Bianchi identity is just in the definition ofτ[f, Rτ] which is calculated with a pseudo Leibniz rule whereτ[f, .] operates on the four termsRτ, g, h, kofRτ[g, h, k], the abelian group property of Gis sufficient.

Then we consider that G is a ring and V is any G-module, forgetting the

“derivation” nature of its elements (no additivity and no Leibniz rule). Addi- tivity and the Leibniz rule are reserved to the connections∇, and the linearity for the first argument must be replaced by a Leibniz-type rule, by adding a correcting term.

The partial derivative∂, classically identified to the application of a vector

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field on a scalar field on a point of the manifold (directional derivation), so corresponding intrinsically to vector fields, must also be axiomatized. An- other partial derivativeδ, and a vector productpmust be also introduced. A set of axioms that is coherent with the non commutativity of theGproduct is then established.

Commutators play a fundamental role in these equations.V becomes an algebra, and the present theory aims at being the coherent axiomatic theory of connected algebra of vector fields, dealing with spin and electroweak interactions in particular, and may be adapted to express the Dirac equation intrinsically.

2. Definitions

2.1. Definitions of structures

Let (G,+) be an abelian group. Without explicit precision, elements of G will be notedα, β, λ, µ, ν, with or without index. For exampleα1∈G.

LetF be the set of functions from Gto G, which naturally inherits the + law fromG.

F :={f :G→G} and (f +g)[λ] :=f[λ] +g[λ] (3) Without explicit precision, elements of F will be noted f, g, h, k, with or without index.

2.2. The 3-circular sum

For any functionF :F × F ×...× F

| {z }

n≥3

→ F, we defineF by F[f, g, h, k4, ..., kn] := +F[f, g, h, k4, ..., kn]

+F[g, h, f, k₄, ..., k_n]

+F[h, f, g, k4, ..., kn] (4) We defineF1 andF2 too, by

F[f, g, h, k1 4, ..., kn] := F[g, h, f, k4, ..., kn]

F[f, g, h, k2 4, ..., kn] := F[h, f, g, k4, ..., kn] (5) Then, of course

F[f, g, h, k4, ..., kn] = + F[f, g, h, k4, ..., kn] +F1 [f, g, h, k4, ..., kn]

+F2 [f, g, h, k₄, ..., k_n] (6) and

F[f, g, h, k4, ..., kn] = F[f, g, h, k1 4, ..., kn]

= F[f, g, h, k² 4, ..., kn] (7)

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3. Distributive internal laws

Let us takeτ, any F-internal law that is distributive over +.

τ is then 2-additive, that is

τ[f1+f2, g1+g2] =τ[f1, g1] +τ[f2, g1] +τ[f1, g2] +τ[f2, g2] (8) 3.1. FunctionsCτ andAτ

Let us define the commutatorCτ

Cτ[f, g] :=τ[f, g]−τ[g, f] (9) and the associatorAτ

A_τ[f, g, h] :=τ[f, τ[g, h]]−τ[τ[f, g], h] (10) Of course, we have

Cτ[f, g] =−Cτ[g, f] (11)

We get

Aτ[f, g, h] = Aτ[f, g, h] +A1 τ[f, g, h] +A2 τ[f, g, h]

= τ[f, τ[g, h]]−τ[τ[f, g], h] +τ[g, τ[h, f]]−τ[τ[g, h], f] +τ[h, τ[f, g]]−τ[τ[h, f], g]

= C_τ[f, τ[g, h]] +C_τ[g, τ[h, f]] +C_τ[h, τ[f, g]]

= Cτ[f, τ[g, h]] (12)

We will say that

τ is associative ⇔ Aτ = 0

τ is weakly-associative ⇔ Aτ = 0 (13) When consideringCτ instead ofτ, we get

A_C_τ[f, g, h] = Cτ[f,Cτ[g, h]]− Cτ[Cτ[f, g], h]

= Cτ[f,Cτ[g, h]] +Cτ[h,Cτ[f, g]]

= Cτ[f,Cτ[g, h]] +C² τ[f,Cτ[g, h]] (14) We deduce with equation (7):Cτ[f,Cτ[g, h]] =C2 τ[f,Cτ[g, h]], that

A_C_τ[f, g, h] = 2 C_τ[f,C_τ[g, h]] (15) Then,Cτ is weakly-associative if and only ifCτ satisfies the Jacobi identity

Cτ[f,Cτ[g, h]] = 0 (16)

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3.2. FunctionRτ

Let us defineR_τ by a mix of commutative and associative properties Rτ[f, g, h] :=Aτ[f, g, h]− Aτ[g, f, h] (17) Whenτ is associative, that isAτ= 0, we getRτ= 0.

Of course, in the general case, we have

R_τ[f, g, h] =−R_τ[g, f, h] (18) Using equation (12), we get

Rτ[f, g, h] = Aτ[f, g, h]− Aτ[g, f, h]

= C_τ[f, τ[g, h]]− C2 _τ[g, τ[f, h]]

= Cτ[f, τ[g, h]]− Cτ[f, τ[h, g]]

= Cτ[f,Cτ[g, h]] (19)

This is a generalization of the first Bianchi identity.

IfC_τ is weakly-associative (or satisfies the Jacobi identity), we get the well known equation

Rτ[f, g, h] = 0 (20)

and in the general case

(Rτ[f, g, h]− Cτ[f,Cτ[g, h]]) = 0 (21) Another writing ofR_τ is

Rτ[f, g, h] = τ[f, τ[g, h]]−τ[τ[f, g], h]−τ[g, τ[f, h]] +τ[τ[g, f], h]

= τ[f, τ[g, h]]−τ[g, τ[f, h]]−τ[Cτ[f, g], h] (22) Remark 1:In equation (1) the Riemann-Christoffel Einstein curvature tensor would be written as

R_E[f, g, h] = τ[f, τ[g, h]]−τ[g, τ[f, h]]−τ[f◦g−g◦f, h]

= Rτ[f, g, h] +τ[Cτ[f, g]− C_◦[X, Y], h]

= R_τ[f, g, h] +τ[T[f, g], h] (23) We get then the same writing when torsionT is null.

Remark 2:We don’t defineR_τ[f, g, h] =τ[f, τ[g, h]]−τ[g, τ[f, h]]−τ[C◦[f, g], h]

because later, when Gis a ring andF is replaced by aG-submoduleV, we haveCτ[f, g]∈ Vbut generallyC◦[f, g]∈ V/ . Moreover,R_τis obviously defined as the “commutator of the associator” ofτ.

3.3. The functionτ[f, R_τ]

From a pseudo Leibniz rule (τ[f, .] operates every term, evenRτ) onRτ[g, h, k]

τ[f, R_τ[g, h, k]] = +τ[f, R_τ][g, h, k] +R_τ[τ[f, g], h, k]

+Rτ[g, τ[f, h], k] +Rτ[g, h, τ[f, k]] (24) we defineτ[f, Rτ] the (3-additive) function fromF × F × F → F as

τ[f, Rτ][g, h, k] := +τ[f, Rτ[g, h, k]]−Rτ[g, h, τ[f, k]]

−Rτ[τ[f, g], h, k]−Rτ[g, τ[f, h], k] (25)

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Remark: Pay attention that we do not suppose any pseudo Leibniz rule for τ (G is only a group), it is just to explain why we define τ[f, Rτ] this way.

Of course, this definition is compatible with the usual∇[X, RE].

τ[f, Rτ] is additive with respect tof too, and we have τ[f, Rτ][g, h, k] =

+τ[f, τ[g, τ[h, k]]]−τ[f, τ[h, τ[g, k]]]−τ[f, τ[C_τ[g, h], k]]

−τ[g, τ[h, τ[f, k]]] +τ[h, τ[g, τ[f, k]]] +τ[Cτ[g, h], τ[f, k]]

−τ[τ[f, g], τ[h, k]] +τ[h, τ[τ[f, g], k]] +τ[Cτ[τ[f, g], h], k]

−τ[g, τ[τ[f, h], k]] +τ[τ[f, h], τ[g, k]] +τ[C_τ[g, τ[f, h]], k] (26) UsingCτ[g, τ[f, h]] =−Cτ[τ[f, h], g] and adding 0 written as

+τ[τ[g, f], τ[h, k]] −τ[τ[g, f], τ[h, k]]

+τ[g, τ[τ[h, f], k]] −τ[g, τ[τ[h, f], k]]

+τ[Cτ[τ[h, f], g], k] −τ[Cτ[τ[h, f], g], k] (27) we get

τ[f, Rτ][g, h, k] =

+τ[f, τ[g, τ[h, k]]]−τ[g, τ[h, τ[f, k]]]

+τ[h, τ[g, τ[f, k]]]−τ[f, τ[h, τ[g, k]]]

−τ[τ[f, g], τ[h, k]] +τ[τ[g, f], τ[h, k]] +τ[C_τ[g, h], τ[f, k]]

+τ[τ[f, h], τ[g, k]]−τ[τ[g, f], τ[h, k]]

+τ[h, τ[τ[f, g], k]]−τ[g, τ[τ[h, f], k]]

+τ[g, τ[τ[h, f], k]]−τ[g, τ[τ[f, h], k]]−τ[f, τ[C_τ[g, h], k]]

τ[Cτ[τ[f, g], h], k]−τ[Cτ[τ[h, f], g], k]

τ[C_τ[τ[h, f], g], k]−τ[C_τ[τ[f, h], g], k] (28) that is

τ[f, Rτ][g, h, k] = +τ[f, τ[g, τ[h, k]]]−τ[f, τ¹ [g, τ[h, k]]]

+τ[h, τ[g, τ[f, k]]]−τ[h, τ1 [g, τ[f, k]]]

−τ[Cτ[f, g], τ[h, k]] +τ[C1 τ[f, g], τ[h, k]]

+τ[τ[f, h], τ[g, k]]−τ[τ1 [f, h], τ[g, k]]

+τ[h, τ[τ[f, g], k]]−τ[h, τ2 [τ[f, g], k]]

+τ[g, τ[Cτ[h, f], k]]−τ[g, τ² [Cτ[h, f], k]]

+τ[Cτ[τ[f, g], h], k]−τ[C2 τ[τ[f, g], h], k]

+τ[Cτ[Cτ[h, f], g], k] (29)

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Using equation (19), we trivially deduce then τ[f, Rτ][g, h, k] = τ[C_τ[C_τ[f, g], h], k]

= τ[Cτ[Cτ[f, g], h], k] =−τ[Cτ[h, Cτ[f, g]], k]

= −τ[C_τ[f, C_τ[g, h]], k] =−τ[R_τ[f, g, h], k]

= − τ[Rτ[f, g, h], k] (30)

This is a generalization of the second Bianchi identityτ[f, R_τ][g, h, k] = 0 when R_τ[f, g, h] = 0, thanks to the Jacobi identity (whenC_τ is weakly- associative).

In the general case

(τ[f, Rτ][g, h, k] +τ[Rτ[f, g, h], k]) = 0 (31)

4. Unitary ring (G,+,∗)

From hereon, we add a product∗inG, and suppose that (G,+,∗) is a unitary ring. Then∀f, g∈ F and∀α∈G, we can define (by inheritance) the product functionsf g andαf, for any λ∈Gby

(f g)[λ] :=f[λ]∗g[λ] and (αf)[λ] :=α∗f[λ] (32) F is trivially a module overG.

LetV be anyG-submodule ofF. Then,∀X, Y ∈ V,∀λ∈Gwe have

X+Y ∈ V, λX ∈ V and λ(µX) = (λµ)X (33) Without explicit precision, elements of V will be notedX, Y, Z, W, with or without index.

InG, the productλ∗µwill be noted λµand we will restrict calculus onV (V could eventually beF but not necessarily).

5. Generalized connections∇

∇is anyτ with Leibniz-like properties.

Likeτ, let∇:V × V → V, be aV-internal law, distributive over +.

The aim is to remain the closest as possible from the usual ∇ and get, in a particular case (δ =C= 0, see later), what is already known (∂ plays its usual role in this case).

Remark: Because it is useless in this paper, we will not define a metric (a functiong :V × V →Gwith some properties) nor a associated Levi-Civita connection, i.e. with∇g= 0 that is with equation (2) the property:

X[g[Y, Z]] =g[∇[X, Y], Z] +g[Y,∇[X, Z]]

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5.1. Co-definitions of∇,∂,δandC

Associated to∇:V × V → V, we define ∂:V ×G→G,δ:G× V →Gand C : G×G→ G, 2-additive functions with Leibniz-like properties (for both arguments) and we introduce two 2-additive functions p: V × V → V and q:V × V → V such that

∇[X, βY] = β∇[X, Y] +∂[X, β]Y

∇[αX, Y] = α∇[X, Y] +δ[α, Y]X

∂[αX, β]Y = α ∂[X, β]Y +C[α, β]p[X, Y]

δ[α, βY]X = β δ[α, Y]X+C[α, β]q[Y, X] (34) In order to keep coherence in the non commutative framework, connections have to satisfy the Leibniz rule with respect to its first argument too, and the two added functionspandqare necessary, as we will see it.

Indeed, using these properties (34), we get

∇[αX, βY] = α∇[X, βY] +δ[α, βY]X

= αβ∇[X, Y] +α ∂[X, β]Y +βδ[α, Y]X+C[α, β]q[Y, X]

and (35)

∇[αX, βY] = β∇[αX, Y] +∂[αX, β]Y

= βα∇[X, Y] +βδ[α, Y]X+α∂[X, β]Y +C[α, β]p[X, Y] Thus, it is necessary to have

(αβ−βα)∇[X, Y] = C[α, β](p[X, Y]−q[Y, X]) (36) The most natural solution is to separateGandV constraints, then

C[α, β] =αβ−βα and q[Y, X] =p[X, Y]− ∇[X, Y] (37) We will then demand the following properties (withC[α, β] :=αβ−βα)

1. ∇[αX, Y] = α∇[X, Y] +δ[α, Y]X (38) 2. ∇[X, βY] = β∇[X, Y] +∂[X, β]Y (39) 3. ∂[αX, β]Y = α ∂[X, β]Y +C[α, β]p[X, Y] (40) 4. δ[α, βY]X = β δ[α, Y]X+C[α, β](p[X, Y]− ∇[X, Y]) (41) Contrary to∂ andδ, the commutatorC has a definition not linked to∇.

5.2. Leibniz properties of∂,δandC We have

∇[X, αβY] = ∇[X, α(βY)] =α∇[X, βY] +∂[X, α]βY

= αβ∇[X, Y] +α ∂[X, β]Y +∂[X, α]βY

and (42)

∇[X, αβY] = ∇[X,(αβ)Y] =αβ∇[X, Y] +∂[X, αβ]Y Then∂ verifies the Leibniz rule with respect to its second argument as usual

∂[X, αβ] =α ∂[X, β] +∂[X, α]β (43)

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We have

∇[αβX, Y] = ∇[α(βX), Y] =α∇[βX, Y] +δ[α, Y]βX

= αβ∇[X, Y] +αδ[β, Y]X+δ[α, Y]βX

and (44)

∇[αβX, Y] = ∇[(αβ)X, Y] =αβ∇[X, Y] +δ[αβ, Y]X Thenδverifies the Leibniz rule with respect to its first argument

δ[αβ, Y] =α δ[β, Y] +δ[α, Y]β (45) and of course, we already know thatC verifies the Leibniz rule with respect to its 2 arguments

C[αβ, γ] =C[α, γ]β+αC[β, γ] and C[α, βγ] =C[α, β]γ+βC[α, γ] (46) 5.3. Properties ofp

With equations (40) and (39), we have

C[α, β]p[X, Y] = ∂[αX, β]Y −α ∂[X, β]Y (47)

= ∇[αX, βY]−β∇[αX, Y]−α∇[X, βY] +αβ∇[X, Y] Moreover

∂[α²X, β]Y = α²∂[X, β]Y +C[α², β]p[X, Y]

= α²∂[X, β]Y + (αC[α, β] +C[α, β]α)p[X, Y]

and (48)

∂[α²X, β]Y = α∂[αX, β]Y +C[α, β]p[αX, Y]

= α(α∂[X, β]Y +C[α, β]p[X, Y]) +C[α, β]p[αX, Y] then (using “6expression” as “expressionis eliminated”)

6α²∂[X, β]Y+6αC[α, β]p[X, Y] +C[α, β]α p[X, Y]

= 6α²∂[X, β]Y+6αC[α, β]p[X, Y] +C[α, β]p[αX, Y] (49) So,pis left-linear

p[αX, Y] =α p[X, Y] (50)

We get too

δ[α, β²Y]X = β²δ[α, Y]X+C[α, β²] (p[X, Y]− ∇[X, Y])

= β²δ[α, Y]X+ (C[α, β]β+βC[α, β]) (p[X, Y]− ∇[X, Y])

and (51)

δ[α, β²Y]X = βδ[α, βY]X+C[α, β] (p[X, βY]− ∇[X, βY])

= +β(βδ[α, Y]X+C[α, β] (p[X, Y]− ∇[X, Y])) +C[α, β] (p[X, βY]− ∇[X, βY])

then

6β²δ[α, Y]X+C[α, β]β(p[X, Y]− ∇[X, Y]) +6βC[α, β] (p[X, Y]− ∇[X, Y])

=6β²δ[α, Y]X+6βC[α, β] (p[X, Y]− ∇[X, Y]) +C[α, β] (p[X, βY]− ∇[X, βY])

(13)

So, (p− ∇) is right-linear, thenpis right-Leibniz, as∇ p[X, βY]− ∇[X, βY] = β(p[X, Y]− ∇[X, Y])

p[X, βY] = β p[X, Y] +∂[X, β]Y (52) We deduce

C[α, β]p[p[X, Y], Z] = +p[C[α, β]p[X, Y], Z]

= +p[∂[αX, β]Y −α∂[X, β]Y, Z]

= +(∂[αX, β]−α∂[X, β])p[Y, Z]

= C[α, β]p[X, p[Y, Z]] (53) Thenpis associative. Moreoverpis 2-additive, thuspis distributive over the addition. Then (V,+, p) is a ring (ifp6= 0) andpis a product.

From hereon, we note

p[X, Y] =XY and p[X, p[Y, Z]] =p[p[X, Y], Z] =XY Z (54) We have with equation (41) and thepleft-linearity

C[α, β] ((XY)Z− ∇[XY, Z])

= +δ[α, βZ]XY −βδ[α, Z]XY = +(δ[α, βZ]X−βδ[α, Z]X)Y

= +(C[α, β](XZ− ∇[X, Z]))Y = +C[α, β]((XZ− ∇[X, Z])Y)

= +C[α, β] (XZY − ∇[X, Z]Y) (55)

With the commutator

Cp[X, Y] := XY −Y X (56)

we get

∇[XY, Z] =∇[X, Z]Y +XY Z−XZY =XCp[Y, Z] +∇[X, Z]Y (57) With equation (40) and the (p− ∇) right-linearity, we get

C[α, β] (X(Y Z)− ∇[X, Y Z])

= +XC[α, β]Y Z− ∇[X,C[α, β]Y Z]

= +X(∂[αY, β]Z−α∂[Y, β]Z)− ∇[X, ∂[αY, β]Z−α∂[Y, β]Z]

= +∂[αY, β]XZ−∂[αY, β]∇[X, Z]−α∂[Y, β]XZ+α∂[Y, β]∇[X, Z]

= +C[α, β] (Y XZ−Y∇[X, Z]) (58)

then

∇[X, Y Z] =Y∇[X, Z] +XY Z−Y XZ=Y∇[X, Z] +Cp[X, Y]Z (59) Moreover, we have, using equation (52) and the associativity ofp

0 = (XY)(αZ)−X(Y(αZ))

= α(XY)Z+∂[XY, α]Z−X(αY Z)−X(∂[Y, α]Z)

= 6α(XY)Z+∂[XY, α]Z− 6αX(Y Z)−∂[X, α]Y Z−X(∂[Y, α]Z)

= ∂[XY, α]Z−∂[X, α]Y Z−∂[Y, α]XZ−∂[X, ∂[Y, α]]Z (60)

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then

∂[XY, α]Z = ∂[X, α]Y Z+∂[Y, α]XZ+∂[X, ∂[Y, α]]Z (61) This leads to

R∂[X, Y, α] := ∂[X, ∂[Y, α]]−∂[Y, ∂[X, α]]−∂[XY −Y X, α] = 0 (62) Let us define 1∈ V as (if it exists)

p[1, X] =p[X,1] =X (63)

In general, p 6= ◦ (the law of composition of applications). Nevertheless, if p=◦ (i.e.XY =p[X, Y] :=X ◦Y), we have 1 = Id. In this case, because pis 2-additive, to get X◦(Y +Z) = X ◦Y +X◦Z, elements of V must necessarily be additive (X[λ+µ] = X[λ] +X[µ]) which is not supposed a priori.

Then, using equations (57) and (59) withX =Z= 1, we get

∇[Y,1] = ∇[1,1]Y

∇[1, Y] = Y∇[1,1] (64) 5.4. Special case whereV=G

This case has sense becauseGis a module on itself.

Equations (38) and (40) are

∇[αX, Y]−α∇[X, Y]−δ[α, Y]X = 0

(∂[αX, β]−α∂[X, β]− C[α, β]X)Y = 0 (65) They must be the same whenβ=Y, then∇=∂andδ=C.

Equation (41) becomes

δ[α, βY]X−βδ[α, Y]X− C[α, β](XY − ∇[X, Y]) = 0 C[α, βY]X−βC[α, Y]X− C[α, β](XY − ∇[X, Y]) = 0 6βC[α, Y]X+C[α, β]Y X− 6βC[α, Y]X− C[α, β](XY − ∇[X, Y]) = 0

C[α, β](Y X−XY +∇[X, Y]) = 0 (66) Then

∇[X, Y] =XY −Y X=C_p[X, Y] (67) Finally, we have∇=∂=δ=C=Cp.

We get then R_∇[X, Y, Z]

=C[X,C[Y, Z]]− C[Y,C[X, Z]]− C[C[X, Y], Z] +C[C[Y, X], Z]

= +6XY Z− 6XZY− 6Y ZX+6ZY X− 6Y XZ+6Y ZX+6XZY− 6ZXY

− 6XY Z+Y XZ+ZXY− 6ZY X+6Y XZ−XY Z−ZY X+6ZXY

= +Y XZ+ZXY −XY Z−ZY X

=−C[C[X, Y], Z] (68)

then of course

R∇[X, Y, Z] = 0 and R∇[X, Y, Z] =−R∇[Y, X, Z] (69)