Abstract
Let be a smooth manifold and a Weil algebra. We discuss the differential forms in the Weil bundles , and we established a link between differential forms in and as well as their cohomology. We also discuss the cohomology in.
1. Introduction
The theory of bundles of infinitely near points was introduced in 1953 by Andre Weil in [1] and has become a subject of significant interest in differential geometry. A commutative, associative, unitary real algebra is called Weil algebra if it is a finite-dimensional local algebra of the form (i.e., ) where is its only maximal ideal (see [2], from page 625). As an example, one defines the algebra of dual numbers whose the maximal ideal is .
Let be a smooth manifold and . Given a Weil algebra with maximal ideal and basis , one defines a morphism of algebrassuch thatwhere and . Such a morphism is called -point of near to , and one denotes by the set of all A-points of near to . There is a functor from the category of smooth manifolds to itself sending a smooth manifold to the bundle which is known as the bundle of -points near to points in ; in this case, can be regarded as a manifold with (see [3]). One of the questions that draw researcher’s attention is the prolongation of geometric structures from to (see [4], chap. 4 for the general theory). This approach consists of sending a geometric structure from to (regarded as an manifold, i.e., ) as developed in [5–10] where the authors studied the prolongations of vector fields and differential forms, linear connections, symplectic structures, and pseudo-Riemannian structures. Many directions have been developed from the last decades for these manifolds such as affine manifold structures studied in [2] and principal fiber bundles studied in [11], and nice applications to Grassmann bundles can be found in [12].
Instead of regarding as an manifold, we discuss in this paper differential forms and de Rham cohomology on without any prolongation. This approach consists of regarding as an manifold (i.e., ) (see [13]). More specifically, if denotes the space of forms in , we introduce the mapsending a form from to a form in . Conversely, we introduce the mapsending a form from to a form in . These two maps are central and enable to extend the de Rham complex in by introducing the operatorin where is the de Rham operator in , and we prove that defines indeed the de Rham cohomology operator in .
2. Basic Notions
Definition 1. (Weil functor)
Let be a Weil algebra with maximal ideal , be a smooth manifold. Denote by the category of smooth manifolds. By the Weil functor of , we mean a functor such that(1)for any,with projection and fibers for any .(2)for anyand, we havesuch that for anyand the following diagram commutes
Remark 1. (1)Denote byand(2)Ifis a function, thensuch that for any , we have(3)Claim: if is a function and , then . This is a very important claim and will be widely used thoughout this paper.(4)Letbe a basis forandbe a manifold such thatis a system of local coordinate around, then there existsand functionssuch that for The functionsare a system of local coordinate around. It is clear that.(5)Ifandare smooth manifolds anda smooth map (resp. diffeomorphism) then such that is a smooth map (resp. diffeomorphism).(6)Given a Weil bundlewithand, define a special sectionofsuch that for any.
Lemma 1. Let be a function on and be the special section of the Weil bundle (i.e., ). Then, .
Proof. Let , then there exists such that . For this, .since are both points near to (see the claim on Remark 2).
3. Revisiting Tangent Spaces
Let be a smooth manifold and be the ring of dual numbers, then can be identified with the tangent . Let , then the tangent space can be identified with the space of points of near to by: if and , then
Let be a Weil algebra, then the tangent bundle on can be identified as . Ifis the external multiplication of , then one can see in [3], Definition 1 that the mapgives to the structure of module. Since , then one can define naturally the multiplicationwhich gives to the structure of vector space.
Definition 2. By a tangent vector on , we mean a linear mapsatisfying the Leibniz rule, i.e., Such a map is called a derivation. We denote
Remark 2. Let be a system of local coordinates around a neighborhood of and be a system of local coordinate around . Denote by a basis of whereis the derivation introduced in [6] (page 4).
Since is a system of local coordinate of around , define the tangent vectorof around such that then we claim that
Remark 3. Let , i.e.,is a derivation. Definesuch that for any , we haveSince and , one can write as an linear combination of basis elements of .
Definition 3. Definewhere .
Remark 4. Denote by the set of functions from to , by those of functions from to . Definesuch thatsuch thatwhere is the linear form such that is the dual basis of a basis of . Also,such thatThe above maps play a very important role in our approach and satisfy the following results, proven in [13].(i)If is a vector field on , regarded as a derivation from to , then so it is for regarded as a derivation from to (ii)If is a vector field on , then so it is for in regarded as a derivation from to
Proposition 1. The mapsuch thatis surjective.
Proof. The linearity of is straightforward. Let us prove that is a tangent vector at . Let , thenThis shows that is well-defined. It remains to prove that is surjective. Let , then andWe need to prove that . Let , thenThus,
Remark 5. For definesuch thatwhere , and for any , definewhereis a linear form. We claim that is the dual basis for andThe mapis surjective. For any , denote by
4. Differential Form and Cohomology
We denote by the space of sections of the bundle .
Definition 4. By a form on , we mean the multilinear skew-symmetric map
Proposition 2. The mapsuch thatis well-defined for all and . In other words, forms of give rise to forms of .
Proof. We need to prove that is a form, i.e., a multilinear which is skew-symmetric. The additivity and the skew-symmetric condition are straightforward. Let , thenObserve that for any , we havethenSince is a form, then . Thus,
Remark 6. If is a vector field on , it is proven in [13] (Proposition 3.5) that is a vector field on where and are the maps introduced in the Remark 2. Define the mapthen we have the following result.
Proposition 3. The mapsuch thatis well-defined for any .
Proof. The additivity and the skew-symmetry are straightforward. Let , thenObserve that for any , we havethat is,Observe that if , thenthen
Definition 5. For any , define the operatorsuch that where is the cohomology operator on .
With notations as above, we have the following:
Theorem 1. For any and , we have
Proof. By definitionSet , thenObserve that for each ,Observe that for anyLet , thenThus,Observe also thatAlso,Observe thatLet , thenIt is not difficult to see thatThen,Thus,Replacing (1) and (2) in (), we obtain
Remark 7. (Conclusion). The previous result shows that is the exterior derivative in and satisfies , which makes the sequence to a complex of differential forms on , and we write for the de Rham cohomology on and denote by the cohomology associated to the complex . This gives the possibility to extend this area in different directions of differential geometry with applications.
Data Availability
The data supporting the current study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.