182

ISSN 1064–5624, Doklady Mathematics, 2006, Vol. 73, No. 2, pp. 182–184. © Pleiades Publishing, Inc., 2006.Original Russian Text © A.M. Vinogradov, L. Vitagliano, 2006, published in Doklady Akademii Nauk, 2006, Vol. 407, No. 2, pp. 151–153.

We briefly describe how to construct (pseudo-)Rie-mannian geometry over arbitrary smooth algebras in apurely algebraic way by interpreting metric tensors asnondegenerate (1,1)-iterated forms in the sense of [1].In particular, we show that the Levi-Cività connectionassociated with a metric

g

is the derivation of the alge-bra of iterated form that is the infinitesimal

g

-inversionof the second mixed differential of

g

.

In this paper, the formalism developed in [1] is usedto determine the conceptual meaning of the basic con-structions of Riemannian geometry. This is needed forthe extension of Riemannian geometry to algebraicgeometry and secondary differential calculus, whichwill be done in forthcoming publications. We use thenotation and definitions from [1] (see also [2–5]).

1. THE FRÖLICHER–NIJENHUIS BRACKET OVER GRADED ALGEBRAS

Suppose that

is a graded group,

k

is a field of char-

acteristic zero, and

B

∈

(see [1]). Below, werecall the necessary facts on the Frölicher–Nijenhuis(FN) formalism over the

-graded algebra

B

as they aregiven in [6].

First, consider the

B

-module

Λ

(

B

)

of differentialforms over

B

.

It is endowed with the structure of a(

⊕

)-graded

k

-algebra. Let

D

p

(

B

)

denote the

B

-module of skew-symmetric graded

p

-multideriva-tions of the algebra

B

. Recall that the insertion (convo-lution)

i

∇

ω ∈ Λ

k

–

p

(

B

)

of any

p

-multiderivation

∇ ∈

D

p

(

B

)

into any

k

-form

ω ∈ Λ

k

(

B

)

is well defined.

Let

D

B

(

Λ

(

B

))

denote the graded

B

-module of deriva-tions of the algebra

B

with values in

Λ

(

B

).

It has the nat-ural structure of a (

⊕

)-graded

Λ

(

B

)

-module. Thesubstitution of a graded vector field

X

∈

D

B

(

Λ

k

(

B

))

into

Algk

a differential form

ω ∈ Λ

l

(

B

)

, which is denoted by

i

X

ω ∈ Λ

k

+

l

– 1

(

B

)

, is defined as follows. For

l

= 0, we set

i

X

ω

= 0. If

l

> 0, then

ω

=

a

i

∧

σ

i

for some

a

i

∈

B

and

σ

i

∈ Λ

l

– 1

(

B

)

, and

i

X

ω

is defined by induction on

l

as

The substitution operator

i

X

:

Λ

(

B

)

→ Λ

(

B

)

is a deriva-tion of bidegree

⟨

X

⟩

+ (0, –1)

of the algebra

Λ

(

B

)

. TheLie derivative along

X

is defined by

X

≡

[

i

X

,

d

]:

Λ

(

B

)

→Λ

(

B

)

; it is a derivation of

Λ

(B) of bidegree ⟨X⟩.Let Λ1(B) be a projective finitely generated B-mod-

ule. Then, for any X, Y ∈ DB(Λ(B)), there exists aunique element Z ∈ DB(Λ(B)) such that [X, Y] = Z.We set Z ≡ [X, Y]fn. The bracket [·, ·]fn: DB(Λ(B)) ×DB(Λ(B)) (X, Y) [X, Y]fn ∈ DB(Λ(B)) is called theF–N bracket; it endows DB(Λ(B)) with the structure ofa ( ⊕ )-graded Lie k-algebra.

2. RIEMANNIAN GEOMETRYOVER SMOOTH ALGEBRAS

For simplicity and from pedagogical considerations,we consider only the case where the base smooth alge-bra A is the algebra of smooth functions on a smoothmanifold.

Suppose that M is a smooth n-manifold, (x1, x2, …, xn)is a local chart on it, and A ≡ C∞(M). Let Λ = Λ1 be the-graded algebra of differential forms over M, and letΛ2 = Λ(Λ) be the 2-graded algebra of second iteratedgeometric differential forms over M. By κ: Λ2 → Λ2 wedenote the involution that interchanges the roles of thedifferentials d1: Λ2 → Λ2 and d2: Λ2 → Λ2. Recall thatd1 is the natural extension over Λ∞ of the de Rham dif-

ferential on M. It is easy to see that ≡ Λ1(Λ) is a pro-jective finitely generated Λ-module. This guaranteesthe existence of a FN bracket on DΛ(Λ2).

The injective A-homomorphism ι2: (M) Λ2

makes it possible to interpret covariant 2-tensors,including (pseudo-)metric tensors, as iterated forms. Inthis way, pseudo-Riemannian geometry becomes an

d∑

iXω X a( ) σ∧ 1–( ) X⟨ ⟩ a⟨ ⟩⋅ da– iXσ.∧=

Λ21

T20 →

MATHEMATICS

Iterated Differential Forms: Riemannian Geometry RevisitedA. M. Vinogradov and L. Vitagliano

Presented by Academician V.V. Kozlov June 28, 2005

Received August 16, 2005

DOI: 10.1134/S1064562406020074

Department of Engineering Mathematics and Informatics, University of Salerno, Fisciano, 84084 ItalyNational Institute of Nuclear Physics, Napoli–Salerno, Italye-mail: vinograd@unisa.it; luca_vitagliano@fastwebnet.it

DOKLADY MATHEMATICS Vol. 73 No. 2 2006

ITERATED DIFFERENTIAL FORMS: RIEMANNIAN GEOMETRY REVISITED 183

object of differential calculus over the algebra of iter-ated forms Λ∞.

Proposition 1. Let g ∈ Λ2 be a twice iterated formof bidegree (1, 1). Then, the following conditions areequivalent.

(i) The mapping D(M) × D(M) (X, Y) ( °

)(g) ∈ A ⊂ Λ1 is A-bilinear, κ(g) = g, and the

A-homomorphism ⎦g: D(M) X g ∈ Λ1 ⊂ Λ2 isbijective;

(ii) g = ι2(g') for some pseudo-Riemannian metric g'over M.

Proposition 1, which is a special case of Proposi-tion 4 from [1], is a motivation for the following newpoint of view on the nature of Riemannian geometry.

Definition 1. A metric over a smooth algebra A [e.g.,A = C∞(M)] is an iterated (1, 1)-form g ∈ Λ2 satisfyingcondition (i) in Proposition 1.

Locally, a metric g ∈ Λ2 has the form g = gµνd1xµ ∧d2xν, where gµν = gνµ ∈ A for 1 ≤ µ, ν ≤ n.

An important advantage of this new approach is thatit makes it possible to construct objects associated witha metric g, such as the Levi-Cività connection or theRiemannian curvature tensor, by directly applying thecorresponding natural operations defined on Λ∞ to g. Inwhat follows, we describe this in more detail.

Consider the iterated 2-form γ = – d2d1g. It has

bidegree (2, 2) and the local expression

where ∂µ ≡ and the Γβµα ≡ (∂µgαβ + ∂αgµβ – ∂βgµα)

are the Christoffel symbols of the first kind. For thisreason, it is natural to refer to the form γ as the Christ-offel form.

The homomorphism ⎤g = : Λ1 → D(M) can beextended over Λ as an A-linear derivation of Λ of

degree –1 with values in DA(Λ); we denote it by : Λ →

DA(Λ). Note that DA(Λ) is a Λ-module and ∈DΛ(DA(Λ)). Recall that each X ∈ DA(Λ) is associatedwith the substitution operator iX ∈ DΛ(Λ). The mapping

is an A-linear derivation of Λ of degree –2 with valuesin DΛ(Λ). In particular, ∇ ∈ DΛ(DΛ(Λ)). Consider themapping

iY2( )

iX1( )

iX2( )

12---

γ 12---∂ν∂βgαµd1xβ– d1xα d2xν d2xµ∧∧∧=

+ Γβµαdx1α d2xµ d2d1xβ 1

2---gαβd2d1xα d2d1xβ,∧+∧∧

∂∂xµ-------- 1

2---

⎦g1–

⎤1g

⎤1g

∇: Λ σ ∇ σ( ) i⎤1

g σ( )DΛ Λ( )∈≡

⎤2g: Λ2 DΛ Λ2( )→

defined by

for any Ω ∈ Λ2 and ω ∈ Λ. Note that DΛ(Λ2) has the nat-

ural structure of a Λ2-module, and is a derivation ofΛ2 of bidegree (–2, –1) with values in DΛ(Λ2). Locally,it has the form

where ||gµν|| = ||gµν||–1.

Definition 2. The derivation Γ = (γ) = – (d1d2g)

of the algebra Λ with values in Λ2 is called the Levi-Cività connection of the metric g.

This definition is motivated, in particular, by thecoordinate description of Γ, which is

where the ≡ gαβ(∂ρgγβ + ∂γgρβ – ∂βgργ) are the

Christoffel symbols of g.Since Γ is a graded derivation with values in forms,

its graded FN square R = [Γ, Γ]fn ∈ DΛ(Λ2) is welldefined; its local form is

where = ∂µ – ∂σ + – .

Definition 3. The derivation R of the algebra Λ withvalues in Λ2 is called the Riemann tensor of the metric g.

The notions of the Levi-Cività connection and Rie-mann tensor associated with a metric g, which are intro-duced above, differ conceptually from the classicalnotions. The classical version is obtained as follows.

Suppose that g = ι2(gcl), where gcl ∈ (M), is a classicalmetric on M, ∇X is the classical covariant derivativealong X ∈ D(M) with respect to the Levi-Cività connec-tion associated with gcl, and Rcl is the classical Rieman-nian curvature tensor of the metric gcl.

Proposition 2. For any X, Y, Z ∈ D(M) and f ∈ A,

In the new terminology, the geodesics of the metricgcl are described as follows. Consider the time flow

field : C∞() → C∞(); a smooth curve χ: I → M on

M, where I ⊂ is an interval; the algebra Λ2(I) of twiceiterated forms on I; and the induced homomorphism χ*:

⎤2g Ω( ) ω( ) 1–( ) Ω⟨ ⟩ ω⟨ ⟩⋅ i∇ ω( )

2( ) Ω≡

⎤2g

⎤2g Ω( ) ω( ) 1–( ) Ω⟨ ⟩ 0 1–,( )⋅ gµν ii∂µ

2( )Ω( ) i∂νω( ) Λ2,∈∧=

⎤2g 1

2---⎤2

g

Γ Γργα d1xγ d2xρ d1d2xα+∧( ) i∂α

,∧=

Γργα 1

2---

R Γ Γ,[ ]fn Rµσ δαd1xδ d2xσ d2xµ i∂α

,∧∧∧= =

Rµσ δα Γσδ

α Γµδα Γµβ

α Γσδβ Γσβ

α Γµδβ

T20

∇XY( ) f( ) iX2( ) Γ iY,[ ]fn fd( )( )– A Λ2,⊂∈=

Rcl X Y,( ) Z( ) f( )

= iZ ° iY2( )

° iX2( )( ) R fd( )( ) A∈ Λ2.⊂

ddt-----

184

DOKLADY MATHEMATICS Vol. 73 No. 2 2006

VINOGRADOV, VITAGLIANO

Λ2 → Λ2(). The geodesic curvature of the curve χ isthe composition

Proposition 3. The curve χ is an affinely parameter-ized geodesic of the metric gcl if and only if

This is seen from, e.g., the local expression

REFERENCES

1. A. M. Vinogradov and L. Vitagliano, Dokl. Akad. Nauk407 (1) (2006) [Dokl. Math. 71 (2), (2006)].

2. A. M. Vinogradov, Dokl. Akad. Nauk 205 (5), 1058–1062 (1972).

3. A. M. Vinogradov, Itogi Nauki Tekh., Ser.: Probl.Geometrii 11, 89–134 (1980).

4. M. M. Vinogradov, Usp. Mat. Nauk 44 (3), 151–152(1989).

5. Dzh. Nestruev, Smooth Manifolds and Observables(MTsNMO, Moscow, 2000) [in Russian].

6. I. S. Krasil’shchik and A. M. Verbovetsky, HomologicalMethods in Equation of Mathematical Physics (OpenEducation, Opava, 1998).

i ddt-----

° i ddt-----

2( ) ° χ* ° Γ: Λ Λ ( ).→

i ddt-----

° i ddt-----

2( ) ° χ* ° Γ 0.=

i ddt-----

° i ddt-----

2( ) ° χ* ° Γ⎝ ⎠⎛ ⎞ dxµ( ) Γργ

µ dχρ

dt---------dχγ

dt--------=

d2χµ

dt2-----------.+