Open access peer-reviewed chapter

# Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold

By Francisco Bulnes

Submitted: April 16th 2020Reviewed: May 21st 2020Published: July 7th 2020

DOI: 10.5772/intechopen.92969

## Abstract

The study of the relationships between the integration invariants and the different classes of operators, as well as of functions inside the context of the integral geometry, establishes diverse homologies in the dual space of the functions. This is given in the class of cohomology of the integral operators that give solution to certain class of differential equations in field theory inside a holomorphic context. By this way, using a cohomological theory of appropriate operators that establish equivalences among cycles and cocycles of closed submanifolds, line bundles and contours can be obtained by a cohomology of general integrals, useful in the evaluation and measurement of fields, particles, and physical interactions of diverse nature that occurs in the space-time geometry and phenomena. Some of the results applied through this study are the obtaining of solutions through orbital integrals for the tensor of curvature R μν , of Einstein’s equations, and using the imbedding of cycles in a complex Riemannian manifold through the duality: line bundles with cohomological contours and closed submanifolds with cohomological functional. Concrete results also are obtained in the determination of Cauchy type integral for the reinterpretation of vector fields.

### Keywords

• complex Riemannian manifold
• cocycles
• cohomology
• cohomology of cycles
• geometrical integration
• integral curvature
• integration invariants
• integral operators
• AMS Classification: 32A19 32C26 32C30 23C15 58 K70 14 J17

## 1. Introduction

Obtaining an integral cohomology of general integral operators that determine complex analytic solutions through classes of cohomology born of the ¯cohomology is necessary to use a holomorphic language with the purpose of obtaining the holomorphic forms that involve exact forms. In fact, this methodology is a way of so many perspectives that suggest the use of complex hyperholomorphic functions in approaching functions in complex analysis, although using fibrations on some quaternionic algebra. The holomorphic forms required in this language, are good to express the integral of complex vector fields as integral of line, which have more than enough lines and hyperplanes, respectively, in CPnand Cn, visualizing these fields like holomorphic sections of complex holomorphic bundles of fibrations XM.

But the ¯cohomology exists naturally in coverings of Stein XM, like holomorphic forms. Then, the integral can be expressed on spaces Mδ, and Δz, [1, 2], that are given as lines and hyperplanes of CPnand Cn, and that as such they are integral orbital of the complex manifolds M=G/Land Δ=Γ/Σ, belonging to a ¯cohomology in holomorphic language.

The cohomologies of functionals and functions, respectively, that they can built through the complex cohomology of hyperspaces are generalizable for vector fields in the same sense of the coverings of Stein and therefore of the ¯cohomology.

The following question arises, how to establish isomorphisms of cohomological classes for functions, functional, and vector fields inside the holomorphic context possible? How to determine a cohomological theory of integral operators that establish equivalences among these objects and the geometric objects of closed submanifolds, bundles of lines, and Feynman diagrams? How everything can decrease to a single cohomology of general integrals on contours or a cohomology of generalized functionals?

Before giving an answer to the previous questions, we give some preliminary definitions that we will use to fix concepts and outlines of the wanted general theory.

Let M, be a complex Riemannian manifold and be a sheaf of germs of holomorphic sections of a vector holomorphic sheaf.

Definition 1.1.We say that a spaceHMJ, is an integral cohomology (not in the sense of the set Z, but yes in the sense of the integrals of partial differential equations) of those ¯equations, if this is a class of solutions or general integrals of these equations in M[1, 3].

Definition 1.2.An integral as generalized solution of a ¯equation is a realization of an irreducible representation of a ¯cohomology of complex closed submanifolds [2, 3, 4].

If the irreducible representations are unitary, then we have a complex L2cohomology or ¯cohomology with coefficients in L2. The corresponding integral operators to their integral cohomology are those of the complex Fourier analysis, which in the complex geometrical context (geometrical analysis) could be integrals constructed through integral transforms as the Hilbert transforms and other [3].

In the case of a real reductive Lie group, the generalized integrals come to be determined by their orbital integrals. Let G, be a real form of GC, and P, their parabolic subgroup. The generalized integrals in G, are the integrals on open orbits of the generalized flag manifold GC/P. For this way, if G=Un1, and the generalized flag manifold is then Pn, the open orbit is the group of positive lines P+, which is an Un1orbit in PnC. The integrals are of John type [3, 5, 6]:

φP=L+PnCf,E1

The general integral in this case is given by the twistor transform [7] on the corresponding homogeneous bundle of lines, that is to say:

T:H1P+On2H1P+On2,E2

Using the twistor transform like intertwining operator of induced tempered representations on a ¯cohomology, we have representations of SU22that are orbits of a fundamental unitary gKmodule of the electrodynamics [8] (see Figure 1 ). Then, it is possible to assign a vector bundle of lines with a unitary representation, where it can be classified.

The concepts of general integral and generalized integral are different because one refers to the whole class of cohomology of solutions of those ¯equations about a complex analytic manifold, and the other refers to the classes of cohomology of solutions on cycles or cocycles of the complex Riemannian manifold [1, 2].

Another example in the recovery of a space of functions mainly the space M, is the recovery of real functions in the space Rn, through values of certain integral operators. Such is the case of the formula tofx, recovered on Rn,

fx=cΓfξppξxndpωξE3

where the integral on p, is understood in terms of its regularization (role that carries out the Hilbert transform). The constant cdepends on the dimension parity of the space Rn, where it was carrying the tomography [6].

To answer the first question, we need a structure of complexes that induce isomorphisms in integral cohomology.

Definition 1.3.A covering of Stein is a set of manifolds of Stein1 Mδand Δz, of the corresponding fibers XMand XΔ, of the double fibration [1].

Let us consider the complexes given in Ref. [1], and let us consider the structure defined by a covering of Stein given by the set of open Mδ, and Δz, in the topology

τX=zξM×ΞXM×ΞMδΞz,E4

Then, a complex in Xis the space such that Ωhr, for any complex Ωr, in a corresponding long succession is given as follows:

E5

(i.e., to say, all the subcomplexes Ωhr, of the complex Ωr). Then, the integral operator cohomology HMJ, in a complex manifold M, is that whose complexes conform a holomorphic structure that induces (in the corresponding integral manifold) a generalized according structure of integral submanifolds.

The integral submanifolds represent solutions of those ¯equations in cycles of M. The integral submanifolds are the corresponding cocycles of M, under the integral operators of HMJ.

For example, if we take the complex manifold M, like a manifold of rational curves Ez, of a twistor manifold I[where should understand each manifold I, as the manifold of integral submanifolds (locally)], then its structure comes from a structure projective of their line on Ez, guided according to the vectors in TzM.

These correspond to sections of a normal bundle NEz, to the curve Ez(infinitesimal deformations to the curve), that is to say, these conform the holomorphic structure that will induce the corresponding structure (that is to say in the corresponding integral manifold). In this case, the generalized structure of integral submanifolds is the Vkconformal integrable structure given by I. The integral cohomology in this case is given by the family of rational curves.

The twistor content in this case helps and is necessary to establish the deformation of the integral curves of the vector sheaf of lines Ok. In such case, the integral cohomology is HMJ=H0IOk.

This example is interesting not only for the fact of the definition of the integral cohomology, which defines, for this way, a class of integrals for M, but also for the fact of satisfaction of the integrability condition for the equation of the tensor of Weyl Wij=0, where H0I+Ok(respectively, H0IOk) are the solutions or integral of W+=0(respectively, W=0[13, 14].

## 2. Duality: line bundles with cohomological contours and closed submanifolds with cohomological functional

We consider the following result on integral cohomology for integral geometry.

Proposition 2.1.In the integral cohomology HMJ, on complex manifolds, the following statements are equivalent:

1. The open Mδ, and Δz, are Gorbits opened up in X, and their integrals are generalized integral for M,

2. Exists an integral operator T, such that HMJTkerDequations,

3. Mδ=Mπ¯/z¯, and Δz=Δz¯/π¯, where HMJ=Hn1Uρ1OV.

Proof.The integrals on the open Gorbits satisfy the Ginvariant integration:

G/HfdΦϕr=G/HfΦ1ϕr,E6

For the theorem of Buchdall Eastwood [12], we have that the orbits generalized in X, give us a new cohomological class that is related to the previous for an integral operator T, defined for

HMJνHΞτ1Jν,E7

and such that HΞτ1JνTkerDequationsG/H, has more than enough. By the theorem II.1 [1]2, the Gorbits are Korbits in X. Then (i) == > (ii). Now for the theorem II.2 [1],3 we have that each canonical fibration of a flag manifold will give a Gorbit in Z, for some internal symmetrical Gspace M.

In particular, kerDequations, takes place the correspondence with the cycles of HMJ. In fact, kerDequations, is similar to a compact number of components on which G, acts transitively, and these belong together to the cocycles of HΞτ1Jν.

But kerDequationsonly exists as integral of those ¯equations in M(with M, integrable) if RIMj=0. This gCestablishes a generalized structure of M, which underlies in its composition (in factm+,m+m, for integrability). Then, zMand σΣiVzexist zF, such that TzF=σTwFiVw,wF. Then, M=σΣVσ, and Ξ=γΣVγ, then for ndimensional planes of a Grassmann manifold, G1,nhad that Mδ=Mπ¯/z¯, and Δz=Δz¯/π¯, which defines cycles in Hn1Uρ1OV, with UM. Then (ii) implies (iii).

However, fixed Gexists alone a finite number of flag manifolds of certain bi-holomorphism of this type. These are in bijective correspondence with the conjugated classes of parabolic subalgebras of gC, and each flag manifold admits a finite number of canonical fibrations.

Then, kerDequations, is made of a finite number of Gorbits, all which are closed and (iii) == > (ii). Then since each one of these Gorbits exists like an Korbit of the space of classes G/K, with Nijenhuis null curvature tensor, then each flag submanifold is an Korbit of the vector holomorphic Gbundle of the 2ndimensional irreducible symmetrical Riemannian manifold JM.

Its integrals are orbital, and their extensions to Mδ, and Δz, are generalized integrals (since they are integral of line along the fibers of Mδ, and Δz, respectively) for which it is continued. Then, (ii) implies (i).♦.

Proposition 2.2.The n1¯complex cohomology with coefficients in a complex holomorphic bundle of M, is a cohomology of hyperlines and hyperplanes4.

Their demonstration is a simple consequence of the digression in part II of Ref. [1], on some basic integral ¯cohomologies on ndimensional complex spaces (see Figure 2 ), of this same philosophical dissertation of integral operators published in 2007.

Proposition 2.3.The integrals of contour are generalized function in a cohomology of contours (cohomological functional).

We define the following concept.

Definition 2.1.Cohomological function of a cohomology HMSingMΩr,5 is an integral cohomology of the form HMSingMC, where M, can be understood as a twistor space corresponding to M. (see Figure 3 ).

For example, this class belongs to the Feynman integrals.

We consider p and differential q forms of the cohomologies on the complex manifolds X, and Y, respectively, αHpXS, and βHqYS.

We consider their cup product given for αβHp+qXYST, and the connecting map in the succession of Mayer-Vietoris:

:Hp+qXYSTHp+q+1XYST,E8

We consider for the inner product of α, and β, the relation is

αβ=αβ,E9

This description of the inner product has been used in a new development of the cohomology for twistor diagrams foreseen in Refs. [14, 18]. This new method is almost opposed to the procedure that we want to use in the unification of contour integrals on diagrams, in respect of the Feynman integral, although also proper to the Conway integrals, Cauchy integrals,6 and some integral transforms as the Hilbert transforms.

We want to assemble a Feynman diagram for applications of the product “cup.” The interior edges of a Feynman diagram are taken again as elements of groups H0(such extra elements have to be abandoned in a cohomology, for example, HMτ1Jν, and the interior edges form the fields (assuming that they are elementary states) in several cohomology groups H1.

Let denote M, for Π, and singM=. Likewise, if f, is one of these elements of H1, this new procedure determines an element of the cohomology HfΠ'Ωd, where ', is the union of all the subspaces defined by internal edges, always with the subspaces CP1, on whose elementary states f, are singular.

Then for Proposition 2.1 (b), the following mapping exists

HfΠ'ΩdHf+dΠ'C,E10

using the description of Dolbeault of the first group, forgetting the bi-graduation df, (d, f) and reminding only the total grade d+f. A description of Cěch of this mapping is used for the evaluation of twistor cohomology. In our case, we will only use the duality of Poincaré to know in what moment of the evaluation of an element of Hf+dΠ'C, one can need a contour in Hf+dΠ'Ωd. This can define in a more general sense the cohomological functional. Likewise, the mapping Eq. (10) is an example of the cohomological functional.

This contour “cohomologic” is easy to relate it with a traditional in HdΠC, due to that the following mapping exists

Hf+dΠ'ΩdHdΠ'C,E11

given for iteration of the constant mapping of Mayer-Vietoris (in homology) ftimes; one for each field.

For example, for diagram, product can be demonstrated that H8ΠC, and that the image of the generator of this group low two mappings of Mayer-Vietoris as is the usual in the physical contour for the product of diagrams given. This affirms that only exists a cohomological contour for the product climb (as is expected) and suggests a method for contours that verifies and observes which belong to cohomologics.

Definition 2.2.(Hyperfunction). A hyperfunction on Rn, is an element of the n1¯cohomology Hn1MJ, with M=Cn/Rn.

Proposition 2.4.The general integrals of line are functional on arches γ, in geometry of conformal generalized structure.

Proof: Consider a vector holomorphic Ginvariant sheaf and their corresponding bundle of lines associated with those r0forms on the corresponding topological vector space. Then, the integrals on the fibers of the vector holomorphic sheaf are the integrals of line on the cycles of the sections X, of the vector sheaf, given by γXδ, δΩr[where Ωris a complex defined in Eq. (5)]. Then the holomorphic structure that constitutes these complexes induces (in the corresponding integral manifold) a conformal generalized structure of integral submanifolds where the arches γ, are local parts of integral curves of the fibers of the vector sheaf of lines. In other words, γiVzexists locally an integral submanifold S, with zS, such that TzS=γ, and TwSiVw, wS. Then the integral of line can be re-written in this conformal generalized structure as

γXδ=TzSfδ,δΩr,fT,E12

where T, is the tube domain (in the local structure where the integral submanifold S, exists) T=Rn+iV, where V, is a cone, not necessarily convex (that has applicability on the fibers of the sheaf of lines). The idea is to define the expressionfδ, inside the context of the integral of line in such case that the values of f, on the arch γ, are values off, a hyperfunction represented this like a variation of holomorphic functions fzδ, in a submanifold of Stein Mδ, such that MδT.

Then, the sesquilinear coupling of the hyperfunction corresponding to f, and the function fitself, is an integral of contour, and for Proposition 2.3, a generalized functional in the cohomology H1ΠC. Indeed, let be T=Rn+iV, the tube domain where the cone V, is not necessarily convex. This cone V=γΣVγ, in the conformal generalized structure where the Vγ, are the convex maximal sub-cones in V. Considers our manifold, complex Riemannian manifold. The idea is that a holomorphic form required in this language is a good expression to write the integral of complex vector fields as an integral of line through more than enough bundles of hyperlines and hyperplanes. As for example, we have more than enough hyperlines and hyperplanes, respectively, in RPn, and Cn, visualizing these fields like holomorphic sections of complex holomorphic bundles of fibers XM. In Δ, exists qdimensional cycles such that V=δγVδ. Let be Tδ=Rn+iVδ, with covering of Stein T=δγTδ. Let us consider the vector cohomology HqTJ, using this covering. Then for proposition 2. 1, incise b), a canonical operator exists (of values frontier for f) defined for

HqTJHqCn/RnJ,E13

Then, the integral can be expressed on spaces Mδand Δz, which are affined to lines and hyperplanes CPnand Cnand that such are orbital integrals of the complex manifolds M=G/Land Δ=Γ/Σ, belonging to a ¯cohomology in holomorphic language.

In particular, if fzδdδΩTqhas regular values zRn, then

φx=γφxδdδ,xRn.E14

Then, in the integral submanifold Mδ, said integrals take the form

γXG=γφzδdδfz=γfzδ.E15

However, these integrals are integral of contour belonging to a cohomology H1ΠCof cohomological functional. Then, the integral γfzδis a functional inside the integral cohomology Hn1Cn/RnJ( Figure 4 ).

The previous Propositions 2.3 and 2.4 establish that the structure of complexes for the integral operator cohomology does suitable to induce isomorfisms in other object classes of the manifold M, doing arise the question to some procedure that exists inside the relative cohomology on J: can we induce isomorfisms of integral cohomologies?

Now, we consider a closed subset (or relatively closed) F, of a space X, and a sheaf J, on X. In a way more than enough, we choose an open covering Y, of X, with a subcovering Y', of X/F.

A relative co-chain of Cěch is a co-chain of Cěch with regard to the covering Y, subject to the condition of annulling when we restrict to the subcovering Y'. Then, it had the exact succession of relative co-chains groups:

0CFpXJCpXJCFpX/FJ0,E16

where CFpXJis the group of relative co-chains. The inherent relative co-chain to a co-opposite operator of the ordinary co-chains and the limit on fine coverings of the homology of CFXJgive the groups of relative cohomology HFpXJ.

Cgzdz=C11zz11zz2dz=02πi2πi=4πi.

This is a good example of traditional cohomological functional element of HfΠ'Ωr=C.

In this case is not necessary to take the limit since ahead of time one has the relative theorem of Leray, which establish that if HpUJ=0,p1, for each set U, in the covering Y, then this covers enough to calculate the relative cohomology. The exact long succession cohomology of the exact short succession defined in Eq. (16) determines the exact succession of relative cohomology

0HF0XJH0XJC0X/FJH1XJ,E17

where the mappings of the cohomology on X, to the given on X/F, are restrictions.

Other important result on the relative cohomology is the split theorem, which establishes in shallow terms that the relative cohomology depends only on the immediate neighborhoods of the embedding of F, in X. With more precision, giving an open subset X, such thatX/X'F=, a canonical isomorfism exists ifHFnXJ=HFnX'J.

This is the form to induce isomorfisms. In our case, the covering Y, is a covering of Stein where the integral operator cohomology HMJ, should exist such as we wish. Why? Because the natural place, where a ¯cohomology exists, is in a covering of Stein and is because we want to obtain the solutions of ¯partial differential equations.

We apply the relative cohomology to cohomologies of contours because we want generalized function as solutions of the differential equations [5, 18].

We consider the following general procedure due to Baston [8] for the exhibition of all the cohomological functional on a collection of fields given. This procedure is required for the evaluation of boxes diagram, that is to say, the obtaining of the elementary states φii=1234of the field through a local cohomology.

We consider a complex manifold given for XY, the closed subsets FX, and GY, and elements αHpXFS, and βHqYGT. Then we can use the connecting mappings in the exact successions of relative cohomology

HpXSHpXFSrHFp+1XSHp+1XS,E18

and

HqYTHqYGTrHGq+1YTHq+1YT,E19

to obtain elements rα, and rβ. Then, the cup product on relative cohomology is defined as:

:Hp+q+1XYSTrHp+q+1XYFGSTHFGp+q+2XYSTHp+q+2XYST,E20

and this demonstrates that

αβ=r1rαrβ,E21

Due to that in the diagram boxes, the interactive vector fields φ, are given as elements of groups H1, defined on different spaces, we need the vector product in relative cohomology:

×:HFp+1XSHGq+1YTHF×Gp+q+2X×YST,E22

Likewise, for diagram box of four states, we have the cohomology of the left side of Eq. (22) that can be illustrated ( Figure 5 ).

Strictly speaking ST, could be πXSπYT. As before rα×rβ, is in the image of the connecting mapping r, in:

Hp+q+1XYFGSTrHF×Gp+q+2X×YSTHp+q+2X×YST,E23

with

αβ=r1rα×rβν1Opqr,E24

The following technical question arises: how to relate contour cohomology as Hf+dΠ'C, with an integral cohomology of vector fields?

Part of the replay to this question is found when are considered the complex components Fi=PiUi, with i=1,2,3,4,,f;being Pi,P, P, and Ui, open subsets of Pi, belonging to the correct cohomology to the Penrose transform on H1UOr.

The idea is to obtain an image of the vector field as element of a cohomology on homogeneous bundles of lines in each component of the field (that is to say, determine a cohomology for each line integral of each field component). Beforehand this is foreseen that will happen with the Penrose transform, which is an integral transform on the homogeneous bundles of lines.

Let F=F1××Ff. We denote for Li, a projective line included in Fi, and let L=L1××Lf. For fvector fields, we have an element in the cohomological group H1U1Or1H1UfOrf. For relative cohomology and projective twistor diagram results [18], the inner product for the line integrals for all these fields is not lost. Then for the Künneth formula to relative cohomology, we have:

H1U1Or1H1UfOrfHF2fΠOr,E25

where r=r1××rf. Each linear continuous functional on these fields is thus an element of the compact relative cohomology group HF2fΠΠFOr. We must establish that Eq. (25) and the group HF2fΠΠFOr, are not in general dual.

Now well, considering this cohomology of vector fields, is necessary to decide how the interior of a diagram choose some of these functionals. We remember the interior of a diagram as the holomorphic nucleus hH3f,qΠOr. For example, in the scalar product (spin zero) h=DWDZWαZα2H6,qΠO22. While in the box h=DWDZDXDYWαZαWβYβZγXγYδZδ2H6,qΠ0O2222. Usually q=0. In these cases h, can be determined for integration without the interior vertices of the twistor diagram, although it is not always easy. If q0, the determination of hin none time is clear. How to do about it?

We consider the complex cohomology, and also we consider an element αHC0,fqΠΠF.Then αhHC3f,fΠΠFOr.This is an induced mapping for the inclusion

i:HC3f,fΠΠFOrHC3f,fΠΠFOr,E26

where such iαh, is a chosen functional for the interior of the diagram (that is to say h) as required. However, as this was done through α, the results are hard to view αas a contour. For it, we first note that the embedding of the constant sheaf C, in Or, induces a mapping:

HCfqΠΠFRHCfqΠΠFOr,E27

and second, the cohomology groups αH5f+qΠΠFand HCfqΠΠFRare isomorphic. Now, it is necessary to insist in that αis in the image of the mapping Eq. (27), which will produce a viewing as contour. Being αa contour, we call to iαh, the functional “associated with” the kernel h, and we remark strongly that this not exists if F, then H5f+qΠΠF=0, which is hoped. We can refer to this problem as impossible, since necessarily F, for the chosen fields in this cohomology, which are the most general possible. The idea is to wobtain an image of the vector field as an element of a cohomology on homogeneous bundles of lines in each component of the field. We note that our defined fields are generally perfect. In fact, if the vector fields are elemental states, then Fi=Li, and F, is equal to a closed submanifold Λ(of real codimension 4f, with normal orientable bundle). Using the Thom isomorphism, we have:

Hf+qΛH5f+qΠΠΛ=0,E28

which is deduced that the viewed contours are given in Hf+qΛ. If the vector fields are not elemental states along ΠΠF, then ΠΠF, is homotopic to ΠΠΛ, which establishes its generality in homology.

Likewise we have demonstrated that ifΠΠF, is homotopic to ΠΠΛ, then the functionals on H1U1Or1H1UfOrf, associated with the kernel hH3f,qΠOr, are given for elements of the homology group Hf+qΛ. Now, which of these contours are cohomological? A classes of contours are the classic or traditional contours. However, realizing extensions of these contour classes through twistor geometry, we can consider cohomological contours to all image elements of the generator of HdΠ''C, under two mappings of Mayer-Vietoris. Likewise, the box nonprojective diagram also engages three cohomological contours. Can this particular theory of contours to the spin context be understood?

The response is yes, for example, of the foreseen construction given in Figure 6b ).

If fC1ΩCΩ¯kerD±aΩ, with f=K±af¯H0ΠC, being =±a, then h=1z±aH1ΠO2. Then an integral formula in hypercomplex analysis of a vector field is an element of the integral cohomology H1ΠH. We can realize more work in this sense until we can arrive to the Penrose transform on hypercomplex numbers.

## 3. The main conjecture and some notes of integral cohomology in low dimension of a complex Riemannian manifold

Using definitions and results exposed with before can be enunciated and demonstrated the following conjectures:

Conjecture 3.1. The cohomology of closed submanifolds of co-dimensions k1,nk, and nk1, can be represented and evaluated by a function cohomology. The cohomology of contours is represented and evaluated by a complex functional cohomology. The cohomology of line bundles is represented and evaluated by a vector field cohomologies under the ¯cohomology corresponding.

There are indicium of that the differential operator class that accepts a scheme of integral cohomology (integral cohomology) like due for the Penrose transform, twistor transform, and so on is the class conformally invariant differential operators, of fact the Penrose transform generates these conformally invariant operators. Some examples of these differential operators are for the massless field equations (for flat versions and some curved versions [20]) and the conformally invariant wave operator due to the mapping:

+R/6:O1O3,E29

or also the Einstein’s operator

(A(A'B)B')+ΦA'B'AB):O1OA'B'AB)1,E30

or the conformally invariant modification of the square of the wave operator O1O4, that is to say, the wave operator that involves in its term Ricci tensors:

2:ϕbba2Rab+2/3Rgaba,E31

Then, the integration of the partial differential equations corresponding to these linear invariant differential operators is realized due to integral transforms of the Penrose type since the irreducible unitary representation scheme to these operators is unitary representations of components of the group SL4C, such as SO2n.

In fact, in the flat case, the invariant differential operator classifications were described to determine a problem of representation theory of Lie groups applied to the Lie group SL4Cand its compact subgroups. Then, own vision to these operators through SL4Cwill be as equivariant operators between homogeneous vector bundles on M, considering to SL4Cas homogeneous space or class space. The integrals in this case are realizations of these representations and are orbital integrals of the integral transform of the resolutions to these differential equations, which, in this concrete case, are the Penrose transform.

Then, the resolution problem of the partial differential equations is reduce to the use of representation theory, but for this case, no always can construct the curved analogues of conformally invariant differential operators of the flat space. This demonstrates that cannot be generated a curved analogous under an integral transform on homogeneous bundles of lines that are direct images of the operators Dn, of G, of G/L. However, yes is possible to obtain a complete list under this procedure as a mapping of unitary modules.

Also, the scheme of the Hmodules in the quaternion analysis serves to compute and determine the properties of manifold through the scheme of fibers that can be in closed complex submanifolds. In fact, this is an alternative for the determination of vector fields through line bundles, which defined these as spin bundles.

Now well, cohomologically: How similar are these two methodologies for the study in field theory? Can the direct product of Lie groups SU2T, subjacent in the structure of a complex Riemannian manifold that models the space-time to its vector field study and its integration through the isometries of the space LHbe carried? Which are the integration limitations for the integral transforms on homogenous bundles in global descriptions of the vector fields?

The first question is related to the double fibration that can be realized on some complex projective spaces and their quaternion equivalent. Since it always exists this bijection due to this double fibration with some corresponding homotopy group that is frequently given for spheres, some real and complex projective spaces that are necessarily identified with some ndimensional sphere exist. Such is the case, for example, of the projective spaces RP1S1, HP1S4, or CP7Spin26. Then can be determined isomorphic cohomological spaces via some integral transform of the mentioned for the double fibrations. Some of these integrals result be of Feynman type due to the complex projective bundles are spin bundles in some sphere that determines some state space in quantum mechanics. For example, for the complex case, is had in an infinite succession of non-trivial bundles, the infinite set of bundles S1Pk21CP,1with kZ, and k, which represents the infinite set of corresponding monopoles bundles to the case =2.

These with proper connections represent Dirac magnetic monopoles of charge k.

The constitutive integrals of these monopoles are Cauchy integrals that for diagrams of a cohomology H1ΠC, these are reduced to integrals of Feynman type on the diagrams-boxes corresponding to the state monopoles vertices. These are identified for the factors 1/ZαWα. Likewise, an integral of Cauchy type given for an integral for a ϕ4vertex representing the projective space P, or its dual P, comes given for

IFϕ4=D1ZfZαgWαhXαjYα/ZαWαXαWαXαYαZαYα,E32

which is not different to the Cauchy integral for a monopole in z=z0, and representing the space P1R.

The response to the second question also is positive since it is possible to determine a cohomology of the space–time based on light geodesics as orbits of a complex torus T, when we consider our Universe as a complex hyperbolic manifold. The corresponding integral operators on the corresponding orbits result to be ndimensional Fourier transforms Fn, that can be calculated for the relation

f,=Fn1F1Rf,E33

in a n-dimensional manifold. The operator R, is the Radon transform calculated on the corresponding cycles. It is well known that FLH, and that the integral cohomology given forFn, is the ¯cohomology of one codimensional submanifolds in M.

A response to the last question could be the limitations that are observed when it is wanted to extend the integration on the orbits of M, to a global integration of vector fields, since it is required the global integration of a vector field without the necessity of calculating previously the integrals on orbits of sections of a homogeneous bundle.

However, certain feasibility exists to obtain a methodology in this respect, generalizing, in some sense, the concept of conformal generalized structure on the manifold M.

The existing equivalences between twistor spaces, quaternion spaces, and Riemannian manifolds establish isomorphisms between different cohomology classes whose geometrical invariants are with similar invariant properties in such different cohomology classes. Likewise, we have, for example, a John integral on a complex bundle of lines F, which includes the same integration invariants with respect to the line bundle of the linear concave domains in the space Cn(respectively, CPn) for the integral of the Radon transform. The cohomology of the singularities in the description of the massless fields can be done through a twistor description of the fields using a relative cohomology of sheaves on the massless fields distributed on a real Minkowski space. Likewise, we can have other examples of equivalences for different cohomology classes.

Much results in complex analysis in C, or C2, can be generalized on a context of analytic functions more extensive, using a holomorphic language of a ¯cohomology. Example of it is the use of hyperfunctions for generalizing some contour integrals. If FR, where F, is a closed interval and the hyperfunctions of F, are given for the quotient space OCF/OC, and iff, is an analytic complex function (analytic in a real sense), the sesquilinear coupling with a hyperfunction represented by the holomorphic function φ(which can be a hypercomplex function) on CF, is given for the contour integral:

ϕf=φzfzdz,E34

where the function f, must be extended holomorphically to a little portion of F, and the contour in CF, transits around of F, sufficiently near of the definition domain off. This integral is not more different than the Cauchy integral, in fact, in certain sense, this is a generalization. Further, this is not more different than the John integrals, Conway integrals, and Penrose integrals on R. The first two are used on the circle S1, the third, for obtaining the harmonic functions of three and four variables determining the solution of the wave equation in R4.

The Penrose line integral in integral geometry has the interpretation as was mentioned in the Radon transform on lines of a flag manifold F=LLR4. All these integrals belong to a same cohomological class, which can be determined calculating the cohomology of μ1Opqr, using the spectral sequence:

Ep,q=ΓUνqRp,E35

Then it is possible to calculate the cohomology groups Hp,qU'μ1Opqr, in terms of Hp,qU'νqRp, taking the meaning of Eq. (35). These cohomology groups are sections of the sheaf with coefficients on the fibered bundle Opqr. The space μ1Opqr, represents the inverse image of said sheaf. UM, and U'=ν1UF, and μ1U'P, with μ, and ν, are the corresponding homomorphisms of the double fibration in integral geometry to relate objects in M, and P, can be a projective twistor space and F, the flag manifold.

Finally, we can say that the descriptions in Section 3 are only few examples of our theory of integrals that we want to construct, and that are examples to enforce our conjecture on the integral geometry bases obtained from the geometry and analysis.

## 4. Conclusions

The idea to obtain an integral operator cohomology is develop a theory through integral invariants, that is to say, explore the complex Riemannian manifolds though the value of its integrals along the cycles and the corresponding cocycles (submanifolds, contours, vertices, edges, complexes, and so on) of the manifold. The duality between these cycles obeys to the spectral transformation that follows much of these integrals as solution of the corresponding differential equations. For example, in some case, it is used the tomography of Riemannian manifold whose cocycles are submanifolds. However, this idea can be generalized and induced beyond the tomography, for example, the integral transforms that generate differential operators with certain property of invariance inside the manifold and establish solution classes through these properties as the case to the conformally invariant differential operators. Then, the representation of objects, such as differential operators, functions, hyperfunctions, and fields, through integrals also appears in a natural way using the cohomology groups of its cocycles as first, second, …, nth integrals for a problem of the differential or functional equations.

Likewise, much of these solutions are given through the integral transforms that search solution classes as equivalence classes in the dual problem. The inverse problems are developed in the geometrical analysis corresponding. The cohomological problem consists in developing a cohomology HMJ, the sufficiently general that means the solution to enlarge number of differential equations and that can be applied in the solution of the field equations in exploring the Universe.

The reinterpretation for physics phenomena in the case when said complex Riemannian manifold models the space-time, results interestingly, and let open the possibility of constructing an Universe theory that includes macroscopic and microscopic phenomena through a good integral theory.

## Notes

• A Stein manifold is an open orbit of a semi-simple Lie group G , in a generalized flag manifold whose nilpotent radical is opposite to the parabolic subgroup P , of G [12]. A definition of the Stein manifold that uses the Hermitian structure of a complex holomorphic manifold is:
• Theorem. The K − invariance given for the G − structure S G M , of complex holomorphic M , is induced to each closed submanifold given for the flag manifolds of the corresponding vector holomorphic G − bundle. Furthermore, the given integral cohomology on such complex submanifolds is equivalent to the integral cohomology on submanifolds of a maximum complex torus.
• Theorem. Let be M = G / K , an internal symmetric simply connected Riemannian manifold and of compact type. Then Z = { R I M j ∈ End ∧ 2 T ∗ M ⊗ R E R I M j = 0 } ,
• We can have a little digression with certain details on the complex Radon transform using submanifolds in the space C P 1 , to the ∂ ¯ − cohomology. Let be M , a complex holomorphic manifold (or complex Riemannian manifold [15]). We consider its corresponding reductive homogeneous space determined for the flag manifold F = G C / P , with P , a parabolic subgroup of G C . We consider the open orbit given for the Stein manifold F D = G / H (as was defined in the footnote 1) with H , a compact subgroup of the real form G 0 , of G C .
• Here, M , is the product of all the twistor spaces and Sing M , is the union of all the subspaces on which the Z α W α ‐ 1 , and Z α A α ‐ 1 , are singular factors. Its differential form is integrated over a contour, which can be traditional contour, for example with cohomology space H 3 υ M − Sing M C .
• For example, to this case for holomorphic functions, we have the generalized Cauchy formula:

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Francisco Bulnes (July 7th 2020). Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold, Advances in Complex Analysis and Applications, Francisco Bulnes and Olga Hachay, IntechOpen, DOI: 10.5772/intechopen.92969. Available from:

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