Page 628 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 628
DIFFERENTIAL GEOMETRY AND APPLICATIONS
On PNDP-manifolds
Alexander Pigazzini, pigazzinialexander18@gmail.com
Mathematical and Phisical Science Foundation, Italy
Coauthor: Saeid Jafari
We provide a possible way of constructing new kinds of manifolds which we will call Partially
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 = Πiq=1Bi ×
Πiq=(q +1)Bi, with Πqi=(q +1)Bi an Einstein-manifold, and the fiber-manifold F is a derived-
differential-manifold (i.e., F is the form: smooth manifold (Rd)+ obstruction bundle, so it can
admit "virtual" negative dimension).
Since the dimensions of a PNDP-manifold is not related with the usual geometric concept of di-
mension (we consider they as "virtual" dimensions), from the speculative and applicative point
of view, we use special projections/desuspensions, to identify the PNDP with another type of
"object" of the same dimension, thus introducing a new type of "hidden" dimensions.
We have consider three kinds of PNDP-manifolds:
Type I) the (n, −n)-PNDP manifold that has overall, zero-dimension (dimM = dimB +
dimF = n + (−n) = 0). The speculative result may be interpreted as an "invisible" man-
ifold but made up of two manifolds with n and −n dimensions, respectively, then we try to
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 different from
d (with d a positive integer number and the fiber-manifold dimension is m = −d) such that
dim = n + (−d) > 0. The particular speculative feature of this manifold is that it appears as
another Einstein-manifold with (n + (−d))-dimension.
Type III) it is like the Type II, but dim = n + (−d) < 0. It has the speculative feature of being
considered, through our projection, like ||(n − d)||-th desuspension of a point.
These manifolds, introducing this new concept of "hidden" dimensions, could have some appli-
cations, in particular in the description of "point-like" structures, as a structure of superconduct-
ing graphene or in the MOG theory without anomalies, but recently also in the econophysical
field, in the description of financial markets influenced by ghost fields such as dark volatility.
2010 Mathematics Subject Classification: 53C25
Keywords and phrases: PNDP-manifolds; Einstein warped product manifolds; negative dimen-
sional manifolds; derived-manifold; desuspension; point-like manifold; virtual dimensions.
Vertex algebra approach to cohomology of foliations
Alexander Zuevsky, zuevsky@math.cas.cz
Czech Academy of Sciences, Prague, Czech Republic
Using the reduction cohomology theory for grading-restricted vertex algebras, we construct ver-
tex algebra bicomplexes and introduce a cohomology theory for foliations of smooth manifolds.
We then define corresponding characteristic classes.
626
On PNDP-manifolds
Alexander Pigazzini, pigazzinialexander18@gmail.com
Mathematical and Phisical Science Foundation, Italy
Coauthor: Saeid Jafari
We provide a possible way of constructing new kinds of manifolds which we will call Partially
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 = Πiq=1Bi ×
Πiq=(q +1)Bi, with Πqi=(q +1)Bi an Einstein-manifold, and the fiber-manifold F is a derived-
differential-manifold (i.e., F is the form: smooth manifold (Rd)+ obstruction bundle, so it can
admit "virtual" negative dimension).
Since the dimensions of a PNDP-manifold is not related with the usual geometric concept of di-
mension (we consider they as "virtual" dimensions), from the speculative and applicative point
of view, we use special projections/desuspensions, to identify the PNDP with another type of
"object" of the same dimension, thus introducing a new type of "hidden" dimensions.
We have consider three kinds of PNDP-manifolds:
Type I) the (n, −n)-PNDP manifold that has overall, zero-dimension (dimM = dimB +
dimF = n + (−n) = 0). The speculative result may be interpreted as an "invisible" man-
ifold but made up of two manifolds with n and −n dimensions, respectively, then we try to
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 different from
d (with d a positive integer number and the fiber-manifold dimension is m = −d) such that
dim = n + (−d) > 0. The particular speculative feature of this manifold is that it appears as
another Einstein-manifold with (n + (−d))-dimension.
Type III) it is like the Type II, but dim = n + (−d) < 0. It has the speculative feature of being
considered, through our projection, like ||(n − d)||-th desuspension of a point.
These manifolds, introducing this new concept of "hidden" dimensions, could have some appli-
cations, in particular in the description of "point-like" structures, as a structure of superconduct-
ing graphene or in the MOG theory without anomalies, but recently also in the econophysical
field, in the description of financial markets influenced by ghost fields such as dark volatility.
2010 Mathematics Subject Classification: 53C25
Keywords and phrases: PNDP-manifolds; Einstein warped product manifolds; negative dimen-
sional manifolds; derived-manifold; desuspension; point-like manifold; virtual dimensions.
Vertex algebra approach to cohomology of foliations
Alexander Zuevsky, zuevsky@math.cas.cz
Czech Academy of Sciences, Prague, Czech Republic
Using the reduction cohomology theory for grading-restricted vertex algebras, we construct ver-
tex algebra bicomplexes and introduce a cohomology theory for foliations of smooth manifolds.
We then define corresponding characteristic classes.
626
