Iterated Differential Forms II: Riemannian Geometry Revisited by

The Diffiety Institute Preprint Series Preprint DIPS–3/2006

September 11, 2006

Iterated Differential Forms II: Riemannian Geometry Revisited by A. M. Vinogradov and L. Vitagliano

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The Diffiety Institute

DIPS-3/2006 arXiv:math.DG/0609287

Iterated Differential Forms II: Riemannian Geometry Revisited A. M. VINOGRADOV AND L. VITAGLIANO Abstract. A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated form formalism developed in [1] and an application to general relativity is given.

In this note the formalism of iterated differential forms developed in the preceding note [1] (see also [2]) is used to decipher the conceptual meaning of Riemannian geometry in the context of differential calculus over (graded) commutative algebras. This is indispensable in order to extend Riemannian geometry to algebraic geometry and Secondary Calculus and to clarify some aspects in general relativity and, more generally, in field theory. In particular, we show that a Levi-Civita-like connection is naturally associated with a generic covariant 2-tensor whose symmetric part is nondegenerate and on this basis present new Einstein-like equations in general relativity. The super-case is briefly sketched. In the sequel the notation and definitions of [1] (see also [3, 4, 5, 6]) are used. 1. Differential form calculus and connections Let G = (G, μ) be a grading group, k be a field of zero–characteristic. In what follows a graded k–algebra will be always assumed associative, unitary and graded commutative (see [1]). Accordingly, a graded morphism of algebras will be always assumed unitary. The necessary facts concerning differential forms over G–graded k–algebras we shall need are summarized below along the lines of [7].k Let B be a G–graded k–algebra, Λ(B) = k Λ (B) be the G ⊕ Z–graded k–algebra of differential forms over B and dB : Λ(B) −→ Λ(B) the exterior differential. If P is a graded B–module, denote by D(B, P ) the totality of all P –valued graded derivations of B. In a natural way D(B, P ) is a graded B–module. Let ϕ : B −→ C be a morphism of G–graded k–algebras. It allows to supply any C–module Q with a structure of graded B–module which will be denoted by Qϕ . ϕ 2000 Mathematics Subject Classification. 53B20, 58A10, 58A50. Key words and phrases. Differential Calculus, Graded Algebras, Differential Forms, Riemannian Geometry. 1

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A. M. VINOGRADOV AND L. VITAGLIANO

induces a morphism of G ⊕ Z–graded k–algebras Λ(ϕ) : Λ(B) −→ Λ(C). Denote by Λ1 (ϕ) : Λ1 (B) −→ Λ1 (C)ϕ its first homogeneous component (with respect to the Z– grading). Λ1 (ϕ) is a morphism of B–modules. Let X ∈ D(B, Λl (C)ϕ ), l ≥ 0. The insertion of X into a differential form σ ∈ Λ(B), denoted by iX σ ∈ Λ(C), is defined inductively on the degree of σ by the formula iX (da ∧ σ) = X(a) ∧ (Λ(ϕ)(σ)) − (−1)X·a+l ((dC ◦ ϕ)(a)) ∧ iX σ,

a ∈ B.

The insertion operator iX : Λ(B) −→ Λ(C)ϕ is a derivation of the algebra Λ(B) of def bi–degree (|X|, l − 1). The Lie derivative along X is defined as LX = [iX , d] = iX ◦ dB + (−1)l dC ◦ iX : Λ(B) −→ Λ(C)ϕ. LX is a derivation of the algebra Λ(B) of bi–degree (|X|, l). Let Λ1 (B) be a projective and finitely generated B–module. Then, for any X, Y ∈ D(B, Λ(B)) there exists a unique Z ∈ D(B, Λ(B)) such that [LX , LY ] = LZ . Put def Z = [X, Y ]fn . The bracket [·, ·]fn : D(B, Λ(B)) × D(B, Λ(B))  (X, Y ) −→ [X, Y ]fn ∈ D(B, Λ(B)) is called Fr¨olicher–Nijenhuis (F–N) bracket and supplies D(B, Λ(B)) with a structure of G ⊕ Z–graded k–Lie–algebra. Now let ϕ : B1 −→ B2 be a morphism of G–graded k–algebras. Consider the functor D(B1 , • ϕ ) : P −→ D(B1 , P ϕ), P being a G–graded B2 –module. It is represented by the graded B2 –module Λ1 (B1 )⊗B1 B2ϕ , i.e., for any graded B2 –module P and X ∈ D(B1 , P ϕ) there exists a unique graded B2 –module homomorphism ψX : Λ1 (B1 ) ⊗B1 B2ϕ −→ P such that X = ψX ◦ dϕ , where dϕ : B1 −→ Λ1 (B1 ) ⊗B1 B2ϕ is a natural derivation defined by def

dϕ b = dB1 b ⊗ 1 ∈ Λ1 (B1 ) ⊗B1 B2ϕ ,

b ∈ B1 .

Note that Λ1 (ϕ) ◦ dB1 : B1 −→ Λ1 (B2 ) is a graded derivation, i.e., Λ1 (ϕ) ◦ dB1 ∈ D(B1 , Λ1 (B2 )ϕ ). Denote ψ = ψΛ1 (ϕ)◦dB1 : Λ1 (B1 ) ⊗B1 B2ϕ −→ Λ1 (B2 ) and put Λ1ϕ (B1 , B2 ) = im ψ ⊂ Λ1 (B2 ) (or, simply, Λ1 (B1 , B2 ) if the context does not allow a confusion). Definition 1. A connection in ϕ is a graded derivation ∇ : B2 −→ Λ(B1 ) ⊗B1 B2ϕ such that, ∇(B2 ) ⊂ Λ1 (B1 ) ⊗B1 B2ϕ and ∇ ◦ ϕ = dϕ . Let ∇ be a connection in ϕ and i∇ : Λ1 (B2 ) −→ Λ1 (B1 ) ⊗B1 B2ϕ the unique graded B2 –module homomorphism such that ∇ = i∇ ◦ dB2 . By definition i∇ is a left inverse of ψ. This proves that ψ is injective and Λ1 (B1 ) ⊗B1 B2ϕ Λ1ϕ (B1 , B2 ). A connection ∇ in ϕ defines a natural transformation of functors ∇• : D(B1 , • ϕ ) −→ D(B2 , • ) (see [7]). Namely, def

∇P : D(B1 , P ϕ )  X −→ ∇PX = ψX ◦ ∇ ∈ D(B2 , P ), P being a G–graded B2 –module. By definition ∇PX ◦ ϕ = X. There are many situations when a new connection can be constructed on the basis of other ones. Some of them we need are described below.

ITERATED DIFFERENTIAL FORMS II: RIEMANNIAN GEOMETRY REVISITED

3

Let ∇ : B2 −→ Λ(B1 ) ⊗B1 B2ϕ be a connection in ϕ : B1 −→ B2 and κ : B2 −→ B2 an automorphism of the algebra B2 . Obviously, κ : B2ϕ −→ B2κ◦ϕ is B1 –linear and, so, the map idΛ(B1 ) ⊗ κ : Λ(B1 ) ⊗B1 B2ϕ −→ Λ(B1 ) ⊗B1 B2κ◦ϕ .

(1)

is well-defined. The composition ∇κ = (idΛ(B1 ) ⊗ κ) ◦ ∇ ◦ κ−1 : B2 −→ Λ(B1 ) ⊗κ◦ϕ B1 B2 def

is a connection in κ ◦ ϕ : B1 −→ B2 . Let ϕ and ∇ be as above. ∇ extends naturally to a connection in Λ(ϕ) : Λ(B1 ) −→ Λ(B2 ) as follows. Consider the bi–differential algebra (Λ2 (B1 ), d1 , d2 ) of doubly iterated differential forms over B1 (see [1]). Note that the map ν : B2  b −→ 1 ⊗ b ∈ Λ(B1 ) ⊗B1 B2ϕ is a morphism of G–graded algebras and ∇ ∈ D(B2 , (Λ(B1 ) ⊗B1 B2ϕ )ν ). According to the above construction ∇ extends to a derivation L∇ : Λ(B2 ) −→ Λ(Λ(B1) ⊗B1 B2ϕ ). There exists a canonical isomorphism of graded algebras, φ : Λ(Λ(B1) ⊗B1 B2ϕ ) −  → Λ2 (B1 ) ⊗Λ(B1 ) Λ(B2 )Λ(ϕ) , given by the formula (φ ◦ dΛ(B1 )⊗B2 )(ω ⊗ b) = d2 ω ⊗ b + ω ⊗ dB2 b,

ω ∈ Λ(B1 ), b ∈ B2 .

Put def

Λ∇ = φ ◦ L∇ : Λ(B2 ) −→ Λ2 (B1 ) ⊗Λ(B1 ) Λ(B2 )Λ(ϕ) . Then Λ∇ is a connection in Λ(ϕ) : Λ(B1 ) −→ Λ(B2 ). Finally, let B1 , B2 , B3 be G–graded algebras, ϕ : B2 −→ B3 and ϕ : B1 −→ B2 be homomorphisms of graded algebras and ∇ : B3 −→ Λ(B2 ) ⊗B2 B3ϕ , ∇ : B2 −→  Λ(B1 )⊗B1 B2ϕ be connections in ϕ, ϕ , respectively. Similarly as above (see (1)) consider the homomorphism 



idΛ(B1 ) ⊗ ϕ : Λ(B1 ) ⊗B1 B2ϕ −→ Λ(B1 ) ⊗B1 B3ϕ◦ϕ . By the universal property of Λ1 (B2 ) ⊗B2 B3ϕ there exists a unique homomorphism of  graded B3 –modules ψ : Λ1 (B2 ) ⊗B2 B3ϕ −→ Λ(B1 ) ⊗B1 B3ϕ◦ϕ such that ψ ◦ dϕ = (idΛ(B1 ) ⊗ ϕ) ◦ ∇ . Put 

 ≡ ψ ◦ ∇ : B3 −→ Λ(B1 ) ⊗B1 B3ϕ◦ϕ .

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A. M. VINOGRADOV AND L. VITAGLIANO

Then  is a connection in ϕ ◦ ϕ : B1 −→ B3 . B2

∇

/ Λ(B ) ⊗ B ϕ 1 2

/ Λ(B ) ⊗ B ϕ◦ϕ 3 3 7 1 ggggg oo

id⊗ϕ

g oo ψ gggggg ooo gg g o g o g g gggg ooo  ggggg ooo o o  oo Λ1 (B2 ) ⊗ B3ϕ ooo o O o oo ooo o o ∇ oo ooo o o o dϕ

B3

Definition 2.  is called the composition of connections ∇ and ∇ . Abusing the notation we also denote this connection by ∇ ◦ ∇ . 2. Levi-Civita-like connections over smooth algebras To simplify the exposition and for pedagogical reasons we shall limit to the case when the basic smooth algebra A is the smooth function algebra on a smooth manifold. Consider an n–dimensional smooth manifold M, a local chart (x1 , . . . , xn ) on it and put A ≡ C ∞ (M). Let Λ = Λ1 be the Z–graded algebra of (geometric) differential forms over M and Λk = Λ(Λk−1) be the Zk –graded algebra of k–times iterated (geometric) differential forms over M (see [8], where iterated differential forms were first introduced, and [9] for an alternative approach to iterated differential forms on super–manifolds). Denote by κ : Λ2 −→ Λ2 the canonical involution (see [1, 2]). Recall that κ ◦ d1 ◦ κ = d2 , κ ◦ d2 ◦ κ = d1 and d1 is a natural extension of the de Rham differential on M to Λ2 . In the following we shall denote simply by ωω  (rather than ω ∧ ω  ) the product of the iterated forms ω and ω . Since Λ12 ≡ Λ1 (Λ) is a projective and finitely generated Λ–module the F–N bracket in D(Λ, Λ2) is well-defined. The injective A–homomorphism ι2 : T20 (M) → Λ2 (see [1]) allows one to interpret covariant 2–tensors, in particular, (pseudo-)metric tensors, as iterated forms. This way pseudo–Riemannian geometry becomes a subject of the differential calculus over the algebra of iterated forms Λ∞ . This interpretation allows to associate a linear connection in the tangent bundle of M with a (possibly non–symmetric) 2–tensor whose symmetric part is non–degenerate. According to the standard approach, the Levi-Civita connection associated with a pseudo-metric is defined implicitly as the unique one satisfying certain properties. On the contrary, we define it directly by applying to the considered 2– tensor a natural operator in the algebra Λ∞ . This is one of numerous examples showing potentialities of the calculus of iterated forms. Proposition 3. Let g ∈ Λ2 be a second iterated form of bi–degree (1, 1). Then the following two assertions are equivalent:

ITERATED DIFFERENTIAL FORMS II: RIEMANNIAN GEOMETRY REVISITED

(2)

5

(1)

(1) The map D(M) × D(M)  (X, Y ) −→ (iY ◦ iX )(g) ∈ A ⊂ Λ is A–bilinear, (2) κ(g) = g and the A–homomorphism g : D(M)  X −→ iX g ∈ Λ1 ⊂ Λ2 is bijective. (2) g = ι2 (g  ) for a Riemannian or pseudo–Riemmannian metric g  over M. Proposition 3 is a corollary of proposition 4 from [1] and motivates the following new point of view on what, in reality, the Riemannian geometry is. Definition 4. A metric over a smooth algebra A, for instance, A = C ∞ (M), is an iterated 1–form g ∈ Λ2 satisfying hypothesis 1 of proposition 3. Locally a metric g ∈ Λ2 looks as g = gμν d1 xμ d2 xν with gμν = gνμ ∈ A, 1 ≤ μ, ν ≤ n. This way the Riemannian geometry is embedded in the context of iterated differential forms and, after that, it is generalized rather directly to more general 2–tensors. Namely, let τ ∈ Λ2 be an iterated form of bi–degree (1, 1) such that τ = ι2 (τ  ) for a covariant 2–tensor τ  over M. Locally τ looks as τ = τμν d1 xμ d2 xν with τμν ∈ A, 1 ≤ μ, ν ≤ n. τ can be uniquely written as τ = g + ω where κ(g) = g and κ(ω) = −ω. Clearly, g = ι2 (g ) and ω = ι2 (ω  ), where g  and ω  are symmetric and skew–symmetric parts of τ  , respectively. From now on, suppose that g is nondegenerate, i.e., a pseudo-metric. Consider the iterated 2–form γ = −d2 d1 τ . It is of bi–degree (2, 2) and its local expression is γ = ∂ν ∂β ταμ d1 xβ d1 xα d2 xμ d2 xν + γβαμ d1 xα d2 xμ d2 d1 xβ + ταβ d2 d1 xα d2 d1 xβ . with ∂μ = ∂x∂μ and γβαμ = ∂μ ταβ + ∂α τβμ − ∂β ταμ . Note that if ω = 0, then the γβ αμ ’s are doubled first kind Christoffel symbols of g . By this reason γ is naturally baptized to be the (generalized) Christoffel form. Next, the homomorphism g = 12 −1 : Λ1 −→ D(M) extends to Λ as an A–linear, g D(A, Λ)–valued derivation of Λ of degree −1 denoted by g1 : Λ −→ D(A, Λ). It should be noted that D(A, Λ) is a Λ–module and g1 ∈ D(Λ, D(A, Λ)). Recall that with any X ∈ D(A, Λ) the insertion operator iX ∈ D(Λ, Λ) is associated and consider the map def

Δ : Λ  σ −→ Δ(σ) = ig1 (σ) ∈ D(Λ, Λ). Δ is an A–linear, D(Λ, Λ)–valued derivation of Λ of degree −2. Define the map g2 : Λ2 −→ D(Λ, Λ2), by putting for any Ω ∈ Λ2 and ω ∈ Λ g2 (Ω)(ω) ≡ (−1)Ω·ω i(2) Δ(ω) Ω. Note that D(Λ, Λ2) possesses a natural Λ2 –module structure and g2 is a D(Λ, Λ2)–valued derivation of Λ2 of bi–degree (−2, −1). Locally it looks as g2 (Ω)(ω) = (−1)|Ω|·(0,−1) 21 g μν (ii∂μ Ω)(i∂ν ω) ∈ Λ2 , (2)

where g μν  = gμν −1 .

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A. M. VINOGRADOV AND L. VITAGLIANO

Definition 5. The Λ2 –valued derivation Γ =g2 (γ) = −g2 (d1 d2 τ ) of the algebra Λ is called the Levi-Civita connection form of τ . This terminology is motivated by the fact that the local expression of Γ is Γ = (d1 d2 xα + Γμ αβ d1 xβ d2 xμ ) i∂α , with Γμ αβ ≡ 12 g αδ γδβμ and in the case when ω = 0 Γρ αγ ’s are nothing but Christoffel symbols of g . So, with any covariant 2–tensor whose symmetric part is nondegenerate a linear connection in the tangent bundle is associated (see below). (2) Now consider the insertion into 2-iterated forms operator iΓ ∈ D(Λ2, Λ2 ) correspond(2) ing to the graded form–valued derivation Γ. The composition κ ◦ iΓ ◦ κ is again a derivation of Λ2 . Put def

(2)

κ(Γ) = [κ ◦ iΓ ◦ κ, d2 ]|Λ ∈ D(Λ, Λ2). The graded form–valued derivation T = Γ − κ(Γ) has the following local expression T = Tμ αβ d1 xβ d2 xμ i∂α ,

Tμ αβ = Γμ αβ − Γβ αμ = 3g αδ ∂[μ ωβδ] d1 xβ d2 xμ i∂α ,

where the ωβδ ’s are local components of ω, i.e., locally ω = ωβδ d1 xβ d2 xδ , and indexes in square bracket are skew–symmetrized. Definition 6. The Λ2 –valued derivation T of the algebra Λ is called the torsion of τ . The graded F–N square of Γ R = [Γ, Γ]fn ∈ D(Λ, Λ2) is well defined and has the following local expression R = [Γ, Γ]fn = Rσμ αδ d1 xδ d2 xσ d2 xμ i∂α , where Rσμ αδ = ∂σ Γμ αδ − ∂μ Γσ αδ + Γσ αβ Γμ βδ − Γμ αβ Γσ βδ . Definition 7. The Λ2 –valued derivation R of the algebra Λ is called the Riemann tensor of τ . The introduced above concepts of generalized Levi-Civita connection, torsion and Riemann tensor associated with the 2–tensor τ have standard counterparts. Namely, recall that τ = ι2 (τ  ), τ  ∈ T20 (M) being a “standard” covariant 2–tensor on M. The above computations show that there is a standard connection Γ in the tangent bundle of M associated with τ  . Indeed, the covariant derivative along X ∈ D(M) with respect to Γ ∇X is well defined by (∇X Y )(f ) = (iX ◦ iY ◦ Γ)(df ) ∈ A ⊂ Λ2 . (1)

(2)

Now, denote by T  and R the “standard” torsion and Riemann curvature tensor of Γ , respectively.

ITERATED DIFFERENTIAL FORMS II: RIEMANNIAN GEOMETRY REVISITED

7

Proposition 8. For any X, Y, Z ∈ D(M) and f ∈ A T  (X, Y )(f ) = (iX ◦ iY ◦ T )(df ) ∈ A ⊂ Λ2 . (1)

(2)

R (X, Y )(Z)(f ) = (iZ ◦ iX ◦ iY ◦ R)(df ) ∈ A ⊂ Λ2 . (1)

(2)

(2)



Proof. Obvious from local expression.

In new terms geodesics of Γ are described as follows. Let dtd : C ∞ (R) −→ C ∞ (R) be the “time current” field. Consider a smooth curve χ : I −→ M on M, I ⊂ R being an interval, the algebra Λ2 (I) of second iterated forms on I and the pull–back homomorphism χ∗ : Λ2 → Λ2 (I). The geodesic curvature vector field (along χ) is the composition def (2) Kχ = id/dt ◦ id/dt ◦ χ∗ ◦ Γ : Λ −→ Λ(I). Proposition 9. χ is an affinely parameterized geodesic of the metric g  if and only if Kχ = 0. This is obvious from the local expression ρ

Kχ (dxμ ) = Γρ μα dχ dt

dχα dt

+

d2 χμ . dt2

We now describe covariant derivatives of covariant tensors in new terms. First of all note that for any l < k, Λ(Λl ) ⊗Λl Λk is naturally isomorphic to Λ(Λl , Λk ). Therefore, a connection in the inclusion Λl ⊂ Λk (in the sense of section 1) may be understood as a derivation Λk −→ Λ(Λl , Λk ). In the following we will adopt this point of view. Define a new graded form-valued derivation ∇ = d2 − Γ : Λ −→ Λ2 .∇ has local expression ∇ = d2 xμ (∂μ − Γμ αβ d1 xβ i∂α ). Note that im ∇ ⊂ Λ(A, Λ). Proposition 10. ∇ : Λ −→ Λ(A, Λ) is a connection in the natural inclusion ϕ : A → Λ in the sense of section 1. As shown in section 1, in its turn ∇ induces a connection Λ∇ in Λ(ϕ) : Λ → Λ2 . Note that Λ(ϕ) = κ|Λ . Therefore, ∇2 ≡ Λ(κ) ◦ ∇ ◦ κ : Λ2 −→ Λ(Λ, Λ2) is a connection in the natural inclusion Λ ⊂ Λ2 . Iterating the procedure we get, at the k–th step, a connection ∇k in the inclusion Λk−1 ⊂ Λk . Note that the existence of the connections ∇k enriches noteworthy the standard calculus in Riemannian geometry k

(consequences of this remark will be explored elsewhere). Denote by ∇ the composition k

∇ = ∇k ◦ ∇k−1 ◦ · · · ◦ ∇2 ◦ ∇1 : Λk −→ Λ(A, Λk ). k

∇ is a connection in the inclusion A ⊂ Λk . Accordingly, we can lift A–linearly a k

k

derivation X ∈ D(A, Λk ) to a derivation ∇X = i(k+1) ◦ ∇ ∈ D(Λk , Λk ). X

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A. M. VINOGRADOV AND L. VITAGLIANO

k

Definition 11. ∇ is called the k–th covariant derivative associated to Γ. 2

As an example, we report local expression of ∇: 2

(2)

(2)

∇ = d3 xμ (∂μ − Γμ αβ d1 xβ i∂α − Γμ αβ d2 xβ i∂α − d1 (Γμ αβ d2 xβ ) ii∂α ). The terminology of definition 11 is motivated by the fact that the standard covariant derivatives with respect to Γ of a covariant k–tensor s ∈ Tk0 (M) are naturally recovered k

from ∇. Namely, we have Proposition 12. Let s = ιk (s ) ∈ Λk and X, Y1 , . . . , Yk ∈ D(M). Then (∇X s )(Y1 , . . . , Yk ) = (iY1 ◦ · · · ◦ iYk ◦ iX (1)

(k)

(k+1)

k

◦ ∇)(s) ∈ A ⊂ Λk .

It is worth mentioning that, on the contrary to the the classical Levi-Civita connections, i.e., associated with metrics, the covariant derivative of a 2–tensor τ with respect 2

to the associated connection is, generally, different from zero, i.e., ∇τ = 0. Namely, let k

Γg : Λ1 −→ Λ2 be the (generalized) Levi-Civita connection of g and ∇g the associated k–th covariant derivative. Then, 2

2

2

∇τ = ∇ω = ∇g ω + T (ω), where T (ω) is the iterated form locally given by β T (ω) = Tμ[ν ωα]β d1 xα d2 xν d3 xμ .

3. Natural equations in general relativity The possibility to associate in a canonical way a Levi-Civita-like connection with a 2–tensor τ with non–degenerate symmetric part suggests an immediate application to general relativity. Indeed, it is natural to interpret the skew-symmetric part ω of τ as a matter field. Then, it is natural as well to assume that the space, i.e., the symmetric part g of τ , is shaped by the matter in such a way that the Ricci tensor Ric(τ ) of the associated with τ connection vanishes. This way one gets the natural equations in general relativity. We use here the word “natural” in order to stress that in the proposed approach the credit is definitively given to the naturalness of the background mathematical language, but not to the “physical intuition” which could be easily misleading in this context. In coordinates these natural equations look as follows. Let R = Rμσ αδ dxδ ⊗dxσ ⊗dxμ ⊗ ∂ be the associated with τ standard Riemann tensor. The Ricci tensor Ric(τ ) of τ ∂xα is defined as the 2–tensor locally given by Ric(τ ) = Rμδ dxμ ⊗ dxδ , where Rμδ = Rμα αδ , μ, δ = 1, . . . , n (the interpretation of the Ricci tensor in terms of iterated differential forms will be discussed separately). Then the natural equations for τ read Ric(τ ) = 0.

ITERATED DIFFERENTIAL FORMS II: RIEMANNIAN GEOMETRY REVISITED

9

g

Now denote by ∇ and Ric(g) the covariant differential and the Ricci tensor of the g

pseudo-metric g, respectively, and put Ric(g) = Rμδ dxμ ⊗ dxδ . Then in terms of g and ω the natural equations for τ read g

Rμν +

9 γρ δσ g g ∂[δ ωμρ] ∂[γ ωνσ] 16 g ∇γ (∂[μ ωνγ] ) = 0

=0

.

(2)

System (2) may be understood as Einstein-like equations for g in presence of a matter field (represented by ω) satisfying a suitable constitutive equation. Indeed, it can be shown that (2) are Einstein equations in presence of a perfect irrotational fluid with a definite equation of state (details will be discussed in a separate publication). It is worth stressing that the matter in the above natural equations is treated in a simplified manner. More rich and exact versions of natural equations in the frames of the developed approach will be discussed separately. 4. Levi-Civita connections over smooth super-algebras The previous results are straightforwardly generalized to supermanifolds. In this section, we shall sketch how Riemannian supergeometry (see, for instance, [10] and references therein) can be developed in the framework of iterated differential forms along the lines of section 2. Consider a super–manifold S, a local chart (θ1 , . . . , θn ) on it and put A = C ∞ (S). A is a Z2 –graded commutative algebra. Put α ¯ = |θα | ∈ Z2 . Let Λ = Λ1 be the Z2 ⊕Z–graded algebra of (geometric) differential forms over S and Λ2 = Λ(Λ) be the Z2 ⊕Z2 –graded algebra of doubly iterated geometric differential forms over S (see also [8]). Let g be an even doubly iterated form of bi–degree (1, 1) over A such that the map (2) (1) D(S) × D(S)  (X, Y ) −→ (iY ◦ iX )(g) ∈ A ⊂ Λ is A–bilinear, κ(g) = g and the (2) A–homomorphism g : D(M)  X −→ iX g ∈ Λ1 ⊂ Λ2 is bijective. Definition 13. g is called a supermetric over S. Locally, g = gμν d1 θμ d2 θν , gμν = (−1)μ¯ν¯ gνμ ∈ A. Moreover, the matrix gνμ  is invertible in A. Given a metric g over S the derivation g2 : Λ2 −→ DΛ(Λ2 ) is defined exactly as above. Its local expression is g2 (Ω)(ω) = (−1)|Ω|·(¯ν ,0,−1) 21 g μν (ii∂μ Ω)(i∂ν ω) ∈ Λ2 , (2)

ν , 0, −1, ) ∈ Z2 ⊕Z2 is the multi–degree of i∂ν , ∂ν = where Ω ∈ Λ2 , ω ∈ Λ, (¯ g μν ∈ A, (−1)γ¯·α¯ gνα g νγ = δαγ .

∂ ∂θν

and

Definition 14. The Λ2 –valued derivation Γ =g2 (γ) = −g2 (d1 d2 g) of the algebra Λ is called the Levi-Civita connection form of g.

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A. M. VINOGRADOV AND L. VITAGLIANO

The local expression of Γ is Γ = (d1 d2 θα + Γμ αβ d1 θβ d2 θμ ) i∂α , ¯

¯

with Γμ αβ = 12 g γα [(−1)γ¯ ·¯γ +¯μ(¯μ+¯γ ) ∂μ gβγ + (−1)γ¯·¯γ +β(¯μ+β+¯γ ) ∂β gμγ − ∂γ gβμ ]. Similarly, the Riemann tensor and the covariant derivative are introduced. For instance, R = [Γ, Γ]fn = Rγβ να d1 θα d2 θγ d2 θβ i∂ν where ¯

¯ ¯

¯ ¯

¯ β+¯ γ )+β(β+¯ ν) ∂β Γα νγ − (−1)β(β+¯γ +¯ν ) ∂γ Γα νβ Rγβ να = (−1)α( ¯

¯

¯γ + (−)μ¯β+α¯ Γμ νβ Γα μγ − (−1)β·(¯α+¯γ )+¯μγ¯ Γμ νγ Γα μβ .

References [1] A. M. Vinogradov, L. Vitagliano, Diffiety Inst. Preprint Series, DIPS–1/06, http://diffiety.ac.ru/preprint/2006/01-06abs.htm , see also arXiv:math.DG/0605113 . [2] A. M. Vinogradov, L. Vitagliano, Dokl. Math. 73, n◦ 2 (2006) 169. [3] A. M. Vinogradov, Soviet Math. Dokl. 13 (1972) 1058. [4] A. M. Vinogradov, J. Soviet Math. 17 (1981) 1624. [5] M. M. Vinogradov, Russian Math. Surveys 44 (1989) no. 3, 220. [6] J. Nestruev, Smooth Manifolds and Observables, Springer–Verlag (New York) 2003. [7] I. S. Krasil’shchik, A. M. Verbovetsky, Homological Methods in Equation of Mathematical Physics Open Education (Opava) 1998. See also Diffiety Inst. Preprint Series, DIPS 7/98, http://diffiety.ac.ru/preprint/98/07_98abs.htm . [8] D. Kochan, P. Severa, math.DG/0307303. [9] P. Severa, arXiv:math.DG/0606645 . [10] O. Goertsches, arXiv:math.DG/0604143 . ` di Scienze MM. FF. NN. Universita ` degli Studi Alexandre Vinogradov, DMI, Facolta di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy E-mail address: [email protected] ` di Scienze MM. FF. NN. Universita ` degli Studi di Luca Vitagliano, DMI, Facolta Salerno, Via Ponte don Melillo, 84084 Fisciano (SA) and INFN, sez. Napoli–Salerno, Italy E-mail address: luca [email protected]