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Diffeological vector pseudo-bundles
No full textWe 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 difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topological bundle is even trivial). Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions
themselves), we consider the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that we indicate, provides when applicable a substitute for said local trivializations. We quickly discuss the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then consider the notion of a pseudo-metric on a diffeological vector pseudo-bundle, showing in particular that it does not always exist
Differential 1-forms on diffeological spaces and diffeological gluing
Full text linkThis paper aims to describe the behavior of diffeological differential 1-forms under the operation of gluing of diffeological spaces along a smooth map (the results obtained actually apply to all k-forms with k>0). In the diffeological context, two constructions regarding diffeological forms are available, that of the vector space Omega^1(X) of all 1-forms, and that of the (pseudo-)bundle Lambda^1(X) of values of 1-forms. We describe the behavior of the former under an arbitrary gluing of two diffeological spaces, while for the latter, we limit ourselves to the case of gluing along a diffeomorphism
Diffeological connections on diffeological vector pseudo-bundles
Full text linkWe consider one possible definition of a diffeological connection on a diffeological vector pseudo-bundle. It is different from the one proposed in [7] and is in fact simpler, since it is obtained by a straightforward adaption of the standard definition of a connection as an operator on the space of all smooth sections. One aspect prominent in the diffeological context has to do with the choice of an appropriate substitute for tangent vectors and smooth vector fields, since there are not yet standard counterparts for these notions. In this respect we opt for the simplest possibility; since there is an established notion of the (pseudo-)bundle of differential forms on a diffeological space, we take the corresponding dual pseudo-bundle to play the role of the tangent bundle. Smooth vector fields are then smooth sections of this dual pseudo-bundle; this is one reason why we devote a particular attention to the space of smooth sections of an arbitrary diffeological vector pseudo-bundle (one curiosity is that it might easily turn out to be infinite-dimensional, even when the pseudo-bundle itself has a trivial finite-dimensional vector bundle as the underlying map). We concentrate a lot on how this space interacts with the gluing construction for diffeological vector pseudo-bundles (described in [10]). We then deal with the same question for the proposed notion of a diffeological connection
Diffeological Levi-Civita connections
Full text linkA diffeological connection on a diffeological vector pseudo-bundle is defined just the usual one on a smooth vector bundle; this is possible to do, because there is a standard diffeological counterpart of the cotangent bundle. On the other hand, there is not yet a standard theory of tangent bundles, although there are many suggested and promising versions, such as that of the internal tangent bundle, so the abstract notion of a connection on a diffeological vector pseudo-bundle does not automatically provide a counterpart notion for Levi-Civita connections. In this paper we consider the dual of the just-mentioned counterpart of the cotangent bundle in place of the tangent bundle (without making any claim about its geometrical meaning). To it, the notions of compatibility with a pseudo-metric and symmetricity can be easily extended, and therefore the notion of a Levi-Civita connection makes sense as well. In the case when , the counterpart of the cotangent bundle, is finite-dimensional, there is an equivalent Levi-Civita connection on it as well
Morse functions on simple polyhedra and a related way of presenting special spines of manifolds
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Generalized Mom-structures and ideal triangulations of 3-manifolds with non-spherical boundary
No full textDiffeological Dirac operators and diffeological gluing
Full text linkThis manuscript attempts to present a way in which the classical construction of the Dirac operator can be carried over to the setting of diffeology. A more specific aim is to describe a procedure for gluing together two usual Dirac operators and to explain in what sense the result is again a Dirac operator. Since versions of cut-and-paste (surgery) operations have already appeared in the context of Atiyah-Singer theory, we specify that our gluing procedure is designed to lead to spaces that are not smooth manifolds in any ordinary sense, and since much attention has been paid in recent years to Dirac operators on spaces with singularities, we also specify that our approach is more of a piecewise-linear nature (although, hopefully, singular spaces in a more analytic sense will enter the picture sooner or later; but this work is not yet about them). Most of it is devoted to the diffeological versions of the components that go into the standard definition of a Dirac operator as the composition of a Clifford connection with Clifford action by sections of the cotangent bundle; a diffeological Dirac operator is then standardly defined
On the notion of scalar product for finite-dimensional diffeological vector spaces
No full textIt is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to dispense with this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenerate symmetric bilinear form on a given space. We apply this notion to make some observation on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in particular, any pseudo-metric on the initial space induces, in the obvious way, a smooth scalar product on the dual
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