New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
A simpler definition of diffeological connections on vector pseudo-bundles.
problem Defining connections on diffeological vector pseudo-bundles.
method Adapting standard connection definition to diffeological context.
result Simplified definition of connections is straightforward and uses dual pseudo-bundle as tangent space.
Study pseudo-bundles of exterior algebras and their Clifford modules, addressing compatibility issues.
problem Compatibility issues in diffeological pseudo-bundles and their duals.
method Analysis of pseudo-bundles of exterior algebras, Clifford actions, and gluing conditions.
result A natural map ensuring commutativity of duals under gluing, also an isometry.
Study of pseudo-bundles and pseudo-metrics on them.
problem Developing an analog of Riemannian metrics for diffeological vector pseudo-bundles.
method Detailed study of gluing operation for pseudo-bundles, construction of pseudo-metrics, and analysis of induced pseudo-metrics.
result Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them.
We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main differenc…
We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). Aft…
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
The paper revisits a claim about a principal bundle over a contractible base and finds it non-trivial.
problem Investigating the properties of a specific quotient space construction over a smoothly contractible base.
method Revisiting a previous claim and using the concept of vector pseudo-bundles to redefine the structure as a non-trivial principal pseudo-bundle.
result The projection fails to satisfy the strict condition of local triviality, but the structure remains rich with a smooth, free, and fiber-transitive group action.
Defines Levi-Civita connections in diffeological vector pseudo-bundles.
problem Lack of standard theory for tangent bundles in diffeology.
method Uses dual of cotangent bundle and extends notions of compatibility and symmetry.
result Equivalent Levi-Civita connection exists in finite-dimensional case.
Defines De Rham operator on diffeological spaces, showing it's unique.
problem No straightforward counterpart of De Rham operator on diffeological spaces.
method Definition based on Levi-Civita connection and Clifford action.
result Only way to define De Rham operator on diffeological spaces.
The paper examines how differential forms behave under space gluing in diffeological spaces.
problem Behavior of differential forms under diffeological space gluing.
method Description of differential forms behavior under gluing of diffeological spaces.
result Describes the behavior of differential forms under diffeological space gluing.
Diffeology explores k-forms and bundles with more information than traditional differential forms.
problem Understanding k-forms and bundles in diffeological spaces. method Developed theory of diffeological vector pseudo-bundles, including limits and colimits, and various operations.
result Sections of bundles of k-forms contain more information than differential forms.