aa r X i v : . [ m a t h . DG ] M a r On PNDP-manifold
A. Pigazzini, C. ¨ O zel, P. Linker and S. JafariDedicated to Professor Maximilian Ganster on the occasion his retirementand Renata Albizetti 08/11/1925 - 06/01/2020 Abstract
We provide a possible way of constructing new kinds of manifolds which we will callPartially Negative Dimensional Product manifold (PNDP-manifold for short).In particular a PNDP-manifold is an Einstein warped product manifold of special kind,where the base-manifold B is a Remannian (or pseudo-Riemannian) product-manifold B = Π q ′ i =1 B i × Π e qi =( q ′ +1) B i , with Π e qi =( q ′ +1) B i an Einstein-manifold, and the fiber-manifold F is a derived-differential-manifold (i.e., F is the form: smooth manifold ( R d )+ obstruc-tion bundle, so it can admit negative dimension).Since the dimension of a PNDP-manifold is not related with the usual geometric conceptof dimension, from the speculative and applicative point of view, we try to define thisrelation using the concept of desuspension to identify the PNDP with another kind of”object”, introducing a new kind of hidden dimensions. Keywords : PNDP-manifolds; Einstein warped product manifolds; negative dimensionalmanifolds; derived-manifold; desuspension; point-like manifold, virtual dimension. : 53C25
1. Introduction and Preliminaries
The concept of negative dimensional space is already used in linguistic statistics [1].Also in supersymmetric theories in Quantum Field Theory, negative dimensional spacesare used [2].Let E ∼ = M × F be a fiber bundle with base space M and its fiber F . We will discussnow a case, where the fiber has negative dimension. Note that the total dimension of the fiber bundle is given by the relation dim E = dim M + dim F . We will consider the case,where the base manifold has greater positive dimension than the negative dimension ofthe fiber is dim M > − dim F . In this case, the dimension of the total fiber bundleis still positive. Since the base manifold is obtained by projection of the fiber bundlealong the fiber by projection operator π F , we have π F E = M i.e. the projection ofthe lower-dimensional fiber-bundle along the fiber yields the higher dimensional basemanifold space. Therefore, the projection operator π F along the negative-dimensionalfiber, is a suspension operator that raises the dimension of topological spaces.Let dim F = − d, d >
0. Then, the operator π F will be a d -fold suspension (a singlesuspension extends a topological space by dimension one). By a slight redefinition ofthe notion of a fiber bundle to the new mathematical structure, the inverse fiber bundle,we are enable to turn fiber bundles with negative-dimensional fibers into inverse fiberbundles with positive-dimensional fibers. More precisely, we define the inverse fiberbundle as follows: - An inverse fiber bundle E ∗ is locally isomorphic to the space M ∗ × ∗ F ∗ with some base manifold M ∗ and a fiber F ∗ .- However, we have a suspension along the fiber Σ F ∗ such that Σ F ∗ E ∗ = M ∗ instead ofa projection along a fiber in an inverse fiber bundle. - Because of the use of suspensioninstead of projections, the inverse cartesian product × ∗ is not the ordinary cartesianproduct. It can be defined functorially as follows: in the category of fiber bundle topology F ib , the ordinary cartesian product is a morphism between some topological spaces (theobjects in this category). By defining an inversion functor F that maps the category F ib to the category of inverse fiber bundles
InvF ib by mapping projection operators(are also morphisms) to suspensions F ( π F ) = Σ F ∗ . Objects remain the same, but on thefiber, the sign of the dimension is changed (negative fibers in F ib change to positive onesin
InvF ib ). Thus, it holds F ( × ) = × ∗ . We define this product to be compatible withsuspensions instead of projections. The functor also preserves commutative diagramsthat are frequently used in the definition of fiber bundles. It is also easy to generalizethe above discussion to the case of negative fibers with arbitrary negative dimension. Wecan treat a fiber bundle with negative-dimensional fiber as an inverse fiber bundle, wherethe fibers will be positive-dimensional. Cartesian products, however, must be changedto inverse cartesian products. The base manifold in inverse fiber bundles has the highestdimension, since it is obtained by a suspension. Imposing local isomorphisms betweenan inverse fiber bundle to the inverse cartesian product of base manifold with fiber, wecan see that the inverse cartesian product reduces dimensions by some amount. It holds:dim E ∗ = dim( E ∗ × ∗ F ∗ ) = dim M ∗ − dim F ∗ . Definition 1:
We say that dimension of X is − X , Σ X is diffeomorphicto a point. By induction and suspension operation we can define all negative dimensions.Since if X and Y are difeomorphic then their suspensions Σ X and Σ Y are diffeomorphic.Also a metric on X is compatible with Σ X . It is also well-known that differentialsand suspensions are compatible on manifolds. This means that the differential on thesuspension space Σ X is induced from the differential on X .The operator × ∗ acts similar as the topological quotient, but in such a way that it iscompatible with respect to above defined functor. The inverse cartesian product we canalso view as a quotient operator obtained by using the F -functor. We can alternativelywrite: × × ∗ = / F . Definition 2:
In general, given an n-dimensional space X , the suspension Σ X hasdimension n + 1. Thus, the operation of suspension creates a way of moving up indimension. The inverse operation Σ − , is called desuspension. Therefore, given an n -dimensional space X , the desuspension Σ − X has dimension n −
1, (see [14]).
Einstein warped product manifolds are a mathematical object that is considered incurrent research. We will treat an Einstein warped product manifold in this section ofthe paper, where we will introduce also the concept of negative dimensions. As ”math-ematical tool”, for the fiber-manifold, we will use derived geometry and about this werecall that if A → M and B → M are two transversal submanifolds of codimension a and b respectively, then their intersection C is again a submanifold, of codimension a + b . Derived geometry explains how to remove the transversality condition and makesense out of a nontransversal intersection C as a derived smooth manifold of codimension a + b . In particular, dimC = dimM − a − b , and the latter number can be negative. Sothe obtained dimensions are not related to the usual geometry concept of ”dimension”,but they are ”virtual dimensions”, and therefore from a speculative/applicative pointof view we will try to relate the dimensions obtained from the PNDP-manifolds withdesuspensions (special projections), interpreting this, as the negative dimensions of thefiber-manifold ”hide” the positive dimensions of the base- manifold, making the PNDP-manifold appears as if desuspensions had been performed. Definition 3:
A warped product manifold ( M, ¯ g ) = ( B, g ) × f ( F, ¨ g ) (where ( B, g )is the base-manifold, ( F, ¨ g ) is the fiber-manifold), with ¯ g = g + f ¨ g , is Einstein if only if:(1) ¯ Ric = λ ¯ g ⇐⇒ Ric − df ∇ f = λg ¨ Ric = µ ¨ gf ∆ f + ( d − |∇ f | + λf = µ where λ and µ are constants, d is the dimension of F , ∇ f , ∆ f and ∇ f are,respectively, the Hessian, the Laplacian and the gradient of f for g , with f : ( B ) → R + a smooth positive function.Contracting first equation of (1) we get:(2) R B f − f ∆ f d = nf λ where n and R B is the dimension and the scalar curvature of B respectively, and fromthird equation, considering d = 0 and d = 1, we have:(3) f ∆ f d + d ( d − |∇ f | + λf d = µd Now from (2) and (3) we obtain:(4) |∇ f | + [ λ ( d − n )+ R B d ( d − ] f = µ ( d − . Definition 4:
We called PNDP-manifold a warped product manifold( M, ¯ g ) = ( B, g ) × f ( F, ¨ g ) that satisfies (1), where the base-manifold ( B, g ) is a Rie-mannian (or pseudo-Riemannian) product-manifold B = Π qi =1 B i = B × B × ... with g = Σ g i , which we can write as B = B ′ × e B where e B is an Einstein manifold Π e qi =( q ′ +1) B i (i.e., e Ric = λ e g where λ is the same for (1) and e g is the metric for e B ), with dim e B = e n and B ′ = Π q ′ i =1 B i with dimB ′ = n ′ , so dimB = n = n ′ + e n . The warping function f : B → R + is f ( x, y ) = f ′ ( x ) + e f ( y ) (where each is a function on its individual man-ifold, i.e., f ′ : B ′ → R + and e f : e B → R + ) and can also be a constant function. Thefiber-manifold ( F, ¨ g ) is a derived differential Riemann-flat manifold with negative integerdimensions m , where with derived differential manifold we consider the derived mani-folds as smooth Riemannian flat manifolds by adding a vector bundle of obstructions . Inparticular we consider, for F , only the spaceforms R d , with orthogonal Cartesian coor-dinates such that g ij = − δ ij , by adding a vector bundle of obstructions, E → R d , (so F := R d + E ), with dimension m = d − rank ( E ), where rank ( E ) > d (for issues related to derived-geometry or obstruction bundle see [9]), and more specifically we consider F such that its dimension is m = − d . Since F = R d + E , and on E (obstruction bundle)the Riemannian geometry does not work, we consider and define that every operationof Riemannian geometry done and defined on the underlying R d , is considered done anddefined on F (i.e., for example, if we calculate the Ricci curvature of R d , which is ob-viously zero, then we will say that the Ricci curvature of F is zero, this because it willbe the definition of Ricci curvature for F ). Moreover if we want the quantity n − d > n ′ = d = − m , and in the special case where n − d > B ′ anEinstein-manifold with the same Einstein- λ , then B ′ i = e B i , or B ′ = e B . Important Note 1:
Since the usual Riemannian geometry works for the underlying Rie-mannian manifold as for ”ordinary” manifolds, we can, therefore, consider ourselves tobe able to work with the (pseudo-)Riemannian geometry on our derived fibers-manifold F , defining all Riemannian geometry operations on R d as Riemannian geometry oper-ations made on F , paying attention only to the dimension; In particular, as mentionedabove, we work on F by actually working on the underlying R d , (in the sense that,since F := R d + E , the calculation results obtained on R d are considered to be ob-tained and defined on F ), but taking into account that F has negative dimensions, infact on the obstruction bundle we cannot put the metric, then we do not work on theobstruction bundle with the Riemannian geometry, but it gives negative dimensions on F . Obviously for what has been said, the tangent space and the vector fields are thoseof R d . The scalar product with two arbitrary vector fields F g h V, W i is define on F as: g ij v i w j = − δ ij v i w j = − ( v i w i ).We want to underline that the analysis does not differ from the usual Einstein warpedproduct manifold analysis, and that the Riemannian curvature tensor and the Ricci cur-vature tensor of the product Riemannian manifold can be written respectively as the sumof the Riemannian curvature tensor and the Ricci curvature tensor of each Riemannianmanifold (see [15]). This could be also a special kind of Einstein Sequential warpedproduct manifold ( M × h M ) × ¯ h M , (see [16]), where h = 1, M is an Einstein-manifoldand M is a derived-smooth-manifold with negative dimensions, that said: Proposition:
If we write the B-product as B = B ′ × e B , where:i) Ric ′ is the Ricci tensor of B ′ ,ii) e Ric is the Ricci tensor of e B , iii) g ′ is the metric tensor referred to B ′ ,iv) e g is the metric tensor referred to e B ,v) f = f ′ + e f , is the smooth warping function f : B → R + , where f ′ : B ′ → R + and e f : e B → R + ,vi) ∇ f = τ ′∗ ∇ ′ f ′ + e τ ∗ e ∇ e f is the Hessian referred on its individual metric, but since e B is Einstein, we have e τ ∗ e ∇ e f = 0, (where τ ′∗ is the pullback),vii) ∇ f is the gradient (then |∇ f | = |∇ ′ f ′ | + | e ∇ e f | ), andviii) ∆ f = ∆ ′ f ′ + e ∆ e f is the Laplacian, (since e τ ∗ e ∇ e f = 0, then e ∆ e f = 0), so the Riccicurvature tensor will be:(1****) ¯ Ric (Σ q ′ i =1 X i , Σ q ′ i =1 Y i ) = Ric ′ (Σ q ′ i =1 X i , Σ q ′ i =1 Y i ) − df ∇ ′ f ′ (Σ q ′ i =1 X i , Σ q ′ i =1 Y i )¯ Ric (Σ e qi =( q ′ +1) X i , Σ e qi =( q ′ +1) Y i ) = e Ric (Σ e qi =( q ′ +1) X i , Σ e qi =( q ′ +1) Y i )¯ Ric ( U, V ) = ¨
Ric ( U, V ) − f ¨ g ( U, V ) f ∗ ¯ Ric (Σ q ′ i =1 X i , Σ e qi =( q ′ +1) X i ) = 0¯ Ric (Σ qi =1 X i , U ) = 0 , where f ∗ = ∆ ′ f ′ f + ( d − |∇ f | f , Σ q ′ i =1 X i = X ′ , Σ q ′ i =1 Y i = Y ′ , Σ e qi =( q ′ +1) X i = e X andΣ e qi =( q ′ +1) Y i = e Y . Theorem:
A warped product manifold with derived differential fiber-manifold F := R d + E , and dimF a negative integer, is a PNDP-manifold, as defined in Definition 4 , if and only if:(1***) ¯
Ric = λ ¯ g ⇐⇒ Ric ′ − df τ ′∗ ∇ ′ f ′ = λg ′ e τ ∗ e ∇ e f = 0 e Ric = λ e g ¨ Ric = 0 f ∆ ′ f ′ + ( d − |∇ f | + λf = 0 , (since Ric is the Ricci curvature of B , then Ric = Ric ′ + e Ric = λ ( g ′ + e g ) + df τ ′∗ ∇ ′ f ′ ).Therefore (2) and (3), for n − d = 0 and n − d <
0, become:(1**) ¯ R = λ ¯ n ⇐⇒ R ′ f − ∆ ′ f ′ d = n ′ f λ e ∆ e f = 0 e R = λ e n ¨ Ric = 0 f ∆ ′ f ′ + ( d − |∇ f | + λf = 0 . where n ′ and R ′ are the dimension and the scalar curvature of B ′ respectively, while for n − d >
0, we must set d = n ′ . We have(1*) ¯ R = λ ¯ n ⇐⇒ R ′ f − ∆ ′ f ′ n ′ = n ′ f λ e ∆ e f = 0 e R = λ e n ¨ Ric = 0 f ∆ ′ f ′ + ( n ′ − |∇ f | + λf = 0 . Proof.
We applied the condition that the warped product manifold of system (1****) isEinstein. (cid:3)
Recapitulating, we used the Derived-geometry to define the fiber-manifold, which ex-plains how to remove the transversality condition and make sense of a non-transversalintersection and therefore admit the presence of negative dimensions. Since we considerthe fiber-manifold F as R d adding a vector bundle of obstructions (then we work withthe Riemannian geometry on F working only on the underlying R d , such that all theRiemannian geometry operations made on R d are considered as defined on F ), then theclassical construction for the warped product manifold (see [10]) is the same; we con-sider the vertical vector fields U , V , as lift of vector fields of F the development of theformulas remains the same; X, Y are lift from B , they are horizontal and so constanton fibers, then for example V [ X, Y ] = 0, and we continue to consider the inner productbetween a vector field on B with one on F as zero (i.e., h X, V i = 0). We obtain anEinstein warped product manifold ( M, ¯ g ) = ( B, g ) × f ( R d , ¨ g ), because from the geomet-ric point of view the fiber-manifold is R d , but taking into account that the manifoldobtained has the fiber-manifold with negative dimensions, then the outcome manifold is( n + ( d − rank ( E )))-dimensional manifold.Another important observation is that we consider R d because, from a speculative pointof view, negative dimensional space is used only to make the PNDP-manifold dimensionsmaller than the base-manifold dimension. Therefore for this purpose we consider it inthe simplest possible form. PNDP-metric:
Referring to a PNDP-manifolds, with negative dimensional fiber, andfor not confusing its metric with the metrics of a ”classic” Einstein warped productmanifold, we denote the Riemannian or pseudo-Riemannian metric of the fiber-manifoldwith the following notation to indicate that F has negative dimensions:¯ g = g − f (Σ ni =1 ( dψ i ) ) ( m ) , where g is the metric of the base-manifold B , and m is the negative dimension of F . Then the general metric form of a PNDP-manifold is:¯ g = g − f (Σ ni =1 ( dψ i ) ) ( m ) = ( g ′ + e g ) + ( f ′ + e f ) (Σ ni =1 ( dψ i ) ) ( m ) . Example 1 - A type of flat PNDP-manifold:
The manifold ( R × R × R × R ) × [( R + E )] (with rank ( E ) = 4) is a (4 − f = 1 ( f ′ and e f both constants),we have n + m = n − d = 4 − >
0. Thus we have to consider dimB ′ = dimF . Inthis special case the B-product is composed of only Einstein-manifolds where B ′ = e B ,that is R , and its metric is: ds = dt + dx + dy + dz − ( du + dv ) ( − . Example 2 - A type of PNDP-point-like manifold:
For this purpose, we consider fiber-manifold F = ( R + E ) (with rank ( E ) = 6), 1-dimensional B ′ -manifold and λ = 0. Likein the previous case, if for example we choose e B as R , then our PNDP-manifold will be: M = ( R × R ) f ( R + E ), and with the following metric: ds = dx + dy + dz + h ( y ) ( du + dv + dw ) ( − . Example 3 - A type of PNDP-manifold with negative ”virtual” dimension:
Followingwhat was done in the previous examples, keeping the same setting, and considering, forexample, dim ( F ) = −
4, we obtain that space ”will emerge” as a desuspension of a point,i.e., negative dimension.
2. Possible speculative interpretation of the types of PNDP manifolds
We assume that negative quantities exist in nature, here we consider the possible ex-istence of negative spatial dimensions, but the fact remains that the PNDP-manifoldscould be used to describe anything in nature that has to do with negative quantities.From the speculative point of view, in this session we consider 2 types of PNDP-manifoldsand try to relate their ”virtual dimensions” to the usual concept of dimensions in Rie-mannian geometry, and this interpreting that, through the product, the negative virtualdimensions of the fiber-manifold ”hide” the positive dimensions of the base-manifold,making the PNDP-manifold appears as if desuspensions had been performed.In practice the derived-manifold F has negative dimensions, which are far from the usualgeometric concept of ”dimension”. Being that with the ”warped product” we must con-sider both the dimension of F (virtual) and the dimension of B, (which is a Riemannianor pseudo-Riemannian manifold and therefore its dimension is in perfect relationship with the usual concept of ”dimension”), the obtained virtual dimension of our PNDP-manifold is equivalent to the dimension of the base-manifold B with the dimension of theunderlying manifolds R d subtracted, and we get that the result will be smaller than thedimension of B . From the speculative point of view, we want to consider the dimensionsobtained for the PNDP-manifold interpreting that when the dimensions of B come into”contact” with the dimensions of F , these latter interact by ”hiding” the dimensions of B . This means that the PNPD-manifold, will ”visually appears” as if a kind of desus-pensions had been performed on B , then with extra hidden dimensions which we cannot see.The same thing can be assumed for F ; it ”will appear” only as desuspensions of thepoint, with extra hidden dimensions which we can not see. Below we have consider twokinds of PNDP-manifolds: Type I) the ( n, − n )-PNDP manifold that has overall, zero-dimension ( dimM = dimB + dimF = n + ( − n ) = 0). Then the result may be interpreted as an ”invisible” manifoldbut made up of two manifolds with n and − n dimensions, respectively. Then we tryto consider it as a kind of ”point-like manifold” (zero-dimension), but with ”hidden”dimensions, and Type II) the ( n, − d )-PNDP manifold, where n (the base-manifold dimension) is differ-ent from d such that dim = n − d >
0. The particular feature of this manifold is that itappears as another Einstein-manifold with ( n − d ) dimension.In (Type I) we try to consider dim = 0 referring to a point-like manifold. It wouldbe thought that the points of B and of the underlying R d of F appear as if they were”projected” on the PNDP-manifold, which in this case degenerates into a point (dim= 0), and we observe the manifold as a point-like; in effect this perception is given byour interpretation that negative dimensions ”hide” positive dimensions and show us themanifold as if it were a point, i.e. the PNDP-manifold is ”hidden” except for a point.Also in (Type II) case, since the dimension of the PNDP-manifold is calculated by thedimension of the base-manifold with the dimension of the fiber-manifold subtracted, wetry to interpret it as if the negative dimensions ”hide” the positive dimensions of thebase-manifold. Since the total dimension will be smaller than the dimension of B, butstill positive,we consider that the PNDP-manifold appears as a projection of B with dim = n − d . Therefore our projection interpret this operation by considering that each negative di-mension can act equally on each positive dimension, so that the final dimension is thedifference between positive and negative dimensions.In our speculative interpretation we consider ( n, − d )-PNDP manifold like an Einstein-manifold projected from the product-manifold B , which has hidden dimensions, but wewill see this part better in the next section.Concluding this section, we can say that our interpretation makes Type I and II ”ap-pear” as point or classic Einstein manifold, different from what they actually are.
3. Desuspension as Projection
This section helps to better define how the desuspensions work.As we have mentioned in the previous paragraph, we consider the relation betweenPNDP-manifold dimension and the usual geometric concept of ”dimension”, as desus-pensions interpreted as projections, and for this, since each negative dimension can actequally on each positive dimension, in order not to be faced with the choice on whichprojection of B the result can be, we must ”orient” the result of the projections to whatreflects certain characteristics and to do this we must consider B as a Riemannian base-manifold expressible as a Riemannian product-manifold B = B ′ × e B with g B = g B ′ + g e B ,such that e B is an Einstein manifold with the same dimension and the same constant λ of the PNDP-manifold, the latter will be the result of the projection we are going toconsider, so: B = Π q ′ i =1 B i × Π e qi =( q ′ +1) B i and F = R d + E , then PNDP= B × f F = [Π i ∈ I B i × f F ], with dimB − dim R d = dim PNDP= dim ( π ( n − d ) : PNDP) = ( n − d ).- If ( n − d ) > π ( n − d ) :PNDP → (Π e qi =( q ′ +1) B i ) = e B ,- if ( n − d ) = 0, (i.e., system solutions (1**)), we have the projection: π :PNDP → point-like manifold, and- if ( n − d ) <
0, (i.e., system solutions (1**)), we have the projection: π ( n − d ) :PNDP → Σ n − d ( p ), with Σ n − d ( p ), we mean the ( n − d )-th desuspension of point. Example 4:
Let ( B ′ × B ′ × B ′ × e B × e B × e B ) × f F be a (6 − with f non-constant, and since n + m = n − d = 6 − >
0, then dimB ′ = dimF .So our PNDP-manifold is such that: e B × e B × e B will be an Einstein-manifold, i.e., Ric ( e B × e B × e B ) = λ ( g e B + g e B + g e B ), and since n − d = 3 we have the desuspen-sion/projection: π :(6 − → (Π i =4 B i ), by identifying the (6 − e B × e B × e B . Example 5:
Let ( B ′ × B ′ × e B × e B ) × f F be a (8 − f non-constant, and also in this case, since n + m = n − d = 6 − > dimB ′ = dimF .Then our PNDP-manifold is such that: e B × e B will be Einstein, i.e., Ric ( e B × e B ) = λ ( g e B + g e B ), then we will have: π : (8 − → ( e B × e B ). Hencewe identify the (8 − e B × e B . Example 6:
If we consider the manifold of the
Example 1 , where we have the special case B ′ = e B , that is R , the desuspension/projection will be: π : ( R × R × R × R ) × ( R + E ) → R , i.e., we identify the (4 − R . Example 7:
If we consider the manifold of the
Example 2 , the desuspension/projectionwill be: π : ( R × R ) × h ( s ) ( R + E ) → point-like manifold (zero-dimension). Example 8:
If we consider the manifold of the
Example 3 , the desuspension/projectionwill be: π − : ( R × R ) × h ( s ) ( R + E ) → Σ − ( p ), i.e., a desuspension of a point. Important Note 2:
In summary, from a speculative point of view, we hypothesize thatin nature there are hidden dimensions that can be interpreted as the interaction of nega-tive dimensions, (which we have introduced with manifolds generated by non-transversalintersections, derived-geometry), with positive dimensions, which by closing together,are not perceived, therefore hidden. To show this effect, we used special projectionssuch as desuspensions, which show only a part of the manifold, its virtual dimension,while the remaining part is hidden. In this speculative context the PNDP is identifiedby ( π ( n − d ) , λ, ( n, m ) , g ).
4. Possible speculative application about the ( n, − n ) -PNDP-manifoldsand ( n, − q ) -PNDP-manifolds In this section we present possible applications in light of the interpretations made above.Since m = d − rankE , where rankE > d (such that m = − d ), here we consider d = n , then m = − n , where n is the dimension of the base-manifold. Referring to amanifold with n negative dimensions, we denote the pseudo-Riemannian metric of thefiber-manifold with the following notation to indicate that F has negative dimensions:¨ g = − (Σ ni =1 ( dψ i ) ) ( − n ) , where ( − n ) is the dimension of the fiber-manifold F . In this waywe don’t confuse it with the metrics of a ”classic” product manifold.Then the ( n, − n )-PNDP metric has the form: ¯ g = g − f (Σ ni =1 ( dψ i ) ) ( − n ) , where g is themetric of the base-manifold B .First of all we recall and highlight that the purpose of the PNDP-manifolds is preciselyto present the point-like manifolds from a mathematical point of view, and introduce atype of manifold with a new kind of hidden dimensions.In [12], Capozziello et al. introduced the concept of the ”point-like manifold” buildingsuperconductors with graphene. In particular they argued that superconductor graphenecan be produced by molecules organized in point-like structures where sheets are consti-tuted by ( N + 1)-dimensional manifold. Particles like electrons, photons and “effectivegravitons” are string modes moving on this manifold. In fact, according to string theory,bosonic and fermionic fields like electrons, photons and gravitons are particular “states”or “modes” of strings. In their important work, they show that at the beginning, thereare point-like polygonal manifolds (with zero spatial dimension) in space which stringsare attached to them, where all interactions between strings on one manifold are thesame and are concentrated on one point which the manifold is located on it. They alsoshow that by joining these manifolds, 1-dimensional polygonal manifolds are emerged onwhich gauge fields and gravitons live and so, these manifolds glued to each other buildhigher dimensional polygonal manifolds with various orders of gauge fields and curva-tures.As in [12], the authors in [13], propose a version of Moffat’s Modified Gravity (MOG)without anomaly, where they showed the same ”configuration” for space-time where atfirst, there are only point-like manifolds with scalars attached to them.In these contexts, the ( n, − n )-PNDP manifolds can play an important role.In fact ( n, − n ) -POLJ appears as a point (point-like), because in general from our inter-pretation, it is a point (positive and negative dimensions hide each other out and andthe total dimension equals zero), but in special it is composed by two manifolds, B and F with nonzero dimensions. Let’s begin with the
MOG without anomaly case , and try to consider a tPNDP-manifoldto describe the point-like version of the spacetime where to induce a Morris-Thornewormhole or a Schwarzchild black hole.Our aim is to try to demonstrate that with this manifold and the Time, we are able topresent a new concept of flat spacetime, that it is point-like version: a PNDP-manifoldwith dim = n + m = 0 plus Time.From 5th equation of (1*), i.e., f ∆ ′ f ′ + ( n ′ − |∇ f | + λf = 0, since R ′ = 0 and λ = 0,we obtain |∇ f | = 0, then f =constant.For this situation, we consider a negative real line I = − e r ) t (where t is the Timeand Φ( r ) is an adjustable function of r ) and choose n = 3, so a (3,-3)-PNDP manifold T = B × F where B = R × R × R , but for simplicity we will write as B = R andthe same for F that we will write as F ( − = R + E (bundle of obstruction), such that dimF = −
3. Then:(5) L = R × F ( − × I , with ds = − e r ) dt + dx + dy + dz − ( dψ + dϕ + dσ ) ( − .In [11], the authors, show that a (2 + 1)-dimensional wormhole can be induced from(3 + 1)-dimensional flat space-time.They considered the (3 + 1)-dimensional Minkowski space-time in the cylindrical coordi-nates. Then, following the authors’ analysis, we can write the metric of B in cylindricalcoordinates:(6) ds = − e r ) dt + dr + dz + r dϑ − ( dψ + dϕ + dσ ) ( − .If from now we work only temporarily on B plus Time and set z = ξ ( r ), we find:(7) ds = − e r ) dt + (1 + ξ ′ ( r ) ) dr + r dϑ ,(and if we consider Φ( r ) = 0 and ξ ′ ( r ) = 0, we have (2 + 1)-Minkowski spacetime), sothe form of energy density becomes: ̺ = ξ ′ ξ ′′ r (1+ ξ ′ ) .The relation (7) is comparable with the Morris-Thorne static wormhole:(8) ds = − e r ) dt + ( − b ( r ) r ) dr + r dϑ , in our specific case we have: b ( r ) = r ( ξ ′ ( r ) ξ ′ ( r ) ).It is well-known that the topological structure of a wormhole includes a throat thatconnects two asymptotically flat spaces and in order to have and maintain this struc-ture, the geometric flare-out condition is necessary, i.e. the minimum dimension to thethroat. In the case of Morris-Thorne type wormhole, this condition is given by the hugesurface tension compared to the energy density times the square of the speed of light and this implies that, if r = r is the location of the throat, then (*) b ( r ) = r and(**) for r > r , b ′ ( r ) < b ( r ) r . In the new setting, (*) implies that at the throat ξ ′ = ±∞ and (**) states that ξ ′ ξ ′′ < r > r . Besides these conditions at the throat, we have z = ξ ( r ) = 0.By the same token, we can construct a Schwarzchild Black Hole, by setting: ξ ( r ) = M ( Mr − M ) / , and Φ( r ) = (1 / ln ( Mr − −
1) + 1) − spatial dimension Morris-Thorne wormhole or (( −
1) + 1) − spatialdimension Schwarzchild black hole.Our ( n, − n )-PNDP manifold model consists of two manifolds with nonzero dimensions(one with n -dimension and one with − n -dimension, where these two manifolds can bethought as a result of intersection between other manifolds). Then we can consider thesetwo manifolds as contained in a ” p -dimensional BULK”, but their product (which gen-erates the ( n, − n )-PNDP) will create the point-like manifolds.Returning to the Graphene wormhole case , we can carry out the same analysis doneabove considering R ′ = e R = 0 and f = const. , and setting a ( n, − q )-PNDP-manifoldswith n = 4 and m = − q = −
3. In this way we obtain the following metric:(9) ds = − e r ) dt + dx + dy + dz + dw − ( dψ + dϕ + dσ ) ( − ,that as we saw at the beginning of the section, it is a 1-dimensional manifold plus Time,which in this case we consider to appear as R .Therefore proceeding with the analysis made for the previous case, but with an extradimension to the base-manifold B , we can obtain:(10) ds = − e r ) dt + ( − b ( r ) r ) dr + r ( dϑ + sin ϑdζ ) − ( dψ + dϕ + dσ ) ( − ,which is [(3 + ( − n, − n )-PNDP manifoldcould be considered as possible mathematical interpretation of point-like manifolds.The PNDP-manifold is a new type of Einstein warped-product manifold that uses thederived-geometry to introduce a concept of virtual dimension, which, from the specula-tive point of view, is the dimension with which we perceive our PNDP-manifold, the restis hidden, invisible.
5. Conclusions and Remarks We have introduced this new kind of Einstein warped product manifold:(Π q ′ i =1 B i × Π e qi =( q ′ +1) B i ) × f F , where the base-manifold contains an Einstein-manifoldΠ e qi =( q ′ +1) B i (or e B ), and where e B has the same dimension and the same Einstein- λ -constant of the PNDP-manifold.For this new kind of manifold we have shown a Ricci-flat example with f constant, andwe have shown the existence of a non-Ricci-flat case with non-constant warping func-tion f . While from the speculative point of view we have relating its dimension to theusual geometric concept of dimension, by means of desuspensions (special projections) π ( n − d ) : PNDP, where in the case n − d > e Π e q ( i = q ′ +1) B i ) with equal dimension and where the respective scalar curvatures have thesame constant λ , in the case n − d = 0 it is identified with a point, and in the case n − d < n − d )-th desuspensions of a point.To conclude, in addition to the possible application seen as point-like versions of thegraphene-wormhole and spacetime, the PNDP-manifold could describe a new nature oftime. In fact the negative dimension of the fiber-manifold could represent the ”past”time, the positive dimension of the base-manifold could represent the time ”future” (forexample 1-dimensional time ”future”), while the ”present” time which is instantaneous,could be represented by a point (dimension zero dim(M)=0). It is well-known thatEinstein considered the five dimensional Kaluza-Klein theory in his investigation for aunified field theory, i.e., a unification of gravity and electromagnetism, during the years1938-1943. But by finding a non-singular particle solution, he discarded this theory asan impossible model for a unified field theory. Since then some other theories have beenformulated which are based on Kalazu-Klein’s theory [[3],[4], [5]]. In this paper we gavesome ideas concerning a supersymmetric Kaluza-Klein theory in which tPNDP-manifoldcan be considered as hidden extra 1-time-like dimensions in a theory with more than 10dimensions. To get rid of massless ghosts we have a 4-dimensional spacetime, a sponta-neous compactification of ground state is necessary by assuming that the cosmologicalconstant is zero. Indeed, in Einstein theory the vierbein field of the 4-dimensional space-time has 10 degrees of freedom. But 5-dimensional Kaluza-Klein theory the metric fieldhas 5 more degrees of freedom.Suppose that ground state is Σ × S with a radius of Planck length. Moreover, letus assume that the metric field is independent of the fifth coordinate. The providedaction creates a theory of Σ which together with the Brans-Dicke scalar field has 15 degrees of freedom. It is well-known that 11-dimensional supergravity is an importanttheory among other supergravity theories since eleven is the maximal space-time dimen-sion which does not allow particles with helicity greater than two (see: [6]). Elevendimensional supergravity has been discarded as it does not provide a realistic model in 4dimensions. But it is considered as description of low-energy dynamics of M -theory (see:[5]). It is true that the eleven-dimensional super gravity constructed by Cremmer et al.[7], did not consider the chiral phenomenology since it is eleven-dimensional and eleven isodd. But as Wang Mian [8] proposed, one can construct a consistent supergravity theorywith an extra hidden time-like dimensions. We believe that it is possible to constructa consistent theory of supergravity with hidden extra time-like dimensions, and thesehidden extra time-like dimensions are identified with our PNDP-manifold with metric ds = dt + dt − ( dt + dt ) − , which we denote by M P NDP .Our M P NDP is a hidden extra time-like manifold. In fact the ”role” of the negativedimension of the fiber-manifold is to hide the dimension of the base-manifold mak-ing our M P NDP as a point-manifold (recall, in general, that dim (( n, − n ) − P N DP ) is dimB + dimF = n − n = 0).Now it is possible to have the topology of vaccum as Σ = Σ × S × S × S × H \ Γ,where Σ is the 4-spacetime with the condition that the cosmological constant is zero.Moreover H \ Γ is a hyperbolic manifold and Γ is a discrete isometry which is not actingfreely. This implies that there is no Killing vectors in this quotient space.This suggests that we have no gauge field which is associated with the hidden extratime-like dimensions, and our gravitational model can be written as: Σ × M P NDP . Inthis way the Time has 5 dimensions, the usual temporal dimension of space-time and4 extral ”hidden” dimensions. We believe that the theory can successfully describe asuper-Yang-Mills field theory coupled with gravitation after spontaneous compactifica-tion.
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