arXiv subjects

Hirokazu Nishimura

Publications and source records attributed to Hirokazu Nishimura.

32 records · Page 2Linked to original sources

arXiv2011

Differential Geometry of Microlinear Frolicher Spaces III

As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity. It is also shown that the Lie derivation preserves the Frolicher-Nijenhuis bracket. Our definitions and discussions are highly geometric, while Frolicher and Nijenhuis' original definitions and discussions were highly algebraic.

math.DG↗

arXiv2010

Relative Microlinearity -Towards the general theory of fiber bundles for Frolicher spaces

In our previous papers [Far East Journal of Mathematical Sciences, 35 (2009), 211-223] and [International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have developed the theory of Weil prolongation, Weil exponentiability and microlinearity for Frolicher spaces. In this paper we will relativize it so as to obtain the theory of fiber bundles for Frölicher spaces. It is shown that any Weil functor naturally gives rise to a fiber bundle. We will see that the category of fiber bundles over a fixed Frölicher space M and their smooth mappings over M is cartesian closed. We will see also that the category of vector bundles over M and their smooth linear mappings over M is cartesian closed. It is also shown that the tangent bundle functor naturally yields a vector bundle.

math.DG↗

arXiv2010

Differential Geometry of Microlinear Frolicher Spaces II

In this paper, as the second in our series of papers on differential geometry of microlinear Frolicher spaces, we study differenital forms. The principal result is that the exterior differentiation is uniquely determined geometrically, just as grad (ient), div (ergence) and rot (ation) are uniquely determined geometrically or physically in classical vector calculus. This infinitesimal characterization of exterior differentiation has been completely missing in orthodox differential geometry.

math.DG↗

arXiv2010

Differential Geometry of Microlinear Frolicher Spaces I

The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have emancipated microlinearity from within well-adapted models to Frolicher spaces. Therein we have shown that Frolicher spaces which are microlinear as well as Weil exponentiable form a cartesian closed category. To make sure that such Frolicher spaces are the central object of infinite-dimensional differential geometry, we develop the theory of vector fields on them in this paper. The central result is that all vector fields on such a Frolicher space form a Lie algebra.

math.DG↗

arXiv2010

Beyond the Regnant Philosophy of Manifolds

Frolicher spaces and smooth mappings form a cartesian closed category. It was shown in our previous paper [Far East Journal of Mathematical Sciences, 35 (2009), 211-233] that its full subcategory of Weil exponentiable Frolicher spaces is cartesian closed. By emancipating microlinearity from within a well-adapted model of synthetic differential geometry to Frolicher spaces, we get the notion of microlinearity for Frolicher spaces. It is shown in this paper that its full subcategory of Weil exponentiable and microlinear Frolicher spaces is cartesian closed. The canonical embedding of Weil exponentiable Frolicher spaces into the Cahiers topos is shown to preserve microlinearity.

math.DG↗

arXiv2009

A much larger class of Frolicher spaces than that of convenient vector spaces may embed into the Cahier topos

It is well known that the category of Frolicher spaces and smooth mappings is Cartesian closed. The principal objective in this paper is to show that the full subcategory of Frolicher spaces that believe in fantasy that every Weil functor is really an exponentiation by the corresponding infinitesimal object is also Cartesian closed. Under the assumption that a conjecture holds, it is shown that these Frolicher spaces embed into the Cahier topos with the Cartesian closed structure being preserved.

math.DG↗

arXiv2009

The Jacobi Identity beyond Lie Algebras

Frolicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless the conceptual meaning of their discovery has been obscured by the messy techniques they exploited. The principal objective in this paper is to show that the double dualization functor in a cartesian closed category as well as synthetic differential geometry provides an adequate framework, in which their discovery's conceptual meaning appears lucid.

math.DG↗

arXiv2008

From Vector Analysis to Differential Forms

In our previous paper [International Journal of Theoretical Physics, 41 (2002), 1165-1190] we have shown, following the tradition of synthetic differential geometry, that div and rot are uniquely determined, so long as we require that the divergence theorem and the Stokes theorem should hold on the infinitesimal level. In this paper we will simplify the discussion considerably in terms of differential forms, leading to the natural derivation of exterior differentiation in the usual form.

math-ph↗

arXiv2008

The Frolicher-Nijenhuis Calculus in Synthetic Differential Geometry

Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.

math.DG↗

arXiv2008

A Functorial Approach to the Infinitesimal Theory of Groupoids

Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion of algebraic theory and Ehresmann's functorial notion of theory called sketches. We study totally intransitive Nishimura algebroids in detail. Finally we show that Nishimura algebroids naturally give rise to Lie algebroids.

math.DG↗

arXiv2007

Another coboundary operator for differential forms with values in the Lie algebra bundle of a group bundle

Kock [Bull. Austral. Math. Soc., 25 (1982), 357-386] has considered differential forms with values in a group in a context where neighborhood relations are available. By doing so, he has made it clear where the so-called Maurer-Cartan formula should come from. In this paper, while we retain the classical definition of differential form with values in the Lie algebra of a group, we propose another definition of coboundary operator for the de Rham complex in a highly general microlinear context, in which neighborhood relations are no longer in view. Using this new definition of the coboundary operator, it is to be shown that the main result of Kock's paper mentioned above still prevails in our general microlinear context. Our considerations will be carried out within the framework of groupoids.

math.DG↗

arXiv2007

The affine bundle theorem in synthetic differential geometry of jet bundles

In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geometry of higher-order total differentials, to appear in Cahiers de Topologie et Géométrie Différentielle Catégoriques] we have introduced another synthetic approach to jet bundles in terms of infinitesimal spaces D_{n}'s, and have compared it with the former approach both in the general microlinear setting and in the finite-dimensional setting. However our comparison in the finite-dimensional setting was incomplete, for our use of dimension counting techniques has tacitly assumed that jet bundles in terms of Dⁿ's and D_{n}'s are vector bundles, which is not the case unless the given bundle is already a vector bundle. The principal objective in this paper is to establish the affine bundle theorem in our second synthetic approach to jet bundles and then to compare not merely the jet bundles based on both approaches but the affine bundles constructed as a whole both in the general microlinear setting and in the finite-dimensional setting. This completes our comparsion between the two approaches in the finite-dimensional setting.

math.DG↗

arXiv2007

Curvature in Synthetic Differential Geometry of Groupoids

We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.

math.DG↗

arXiv2006

The Lie algebra of the group of bisections

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avant-garde branch of differential geometry, in which nilpotent infinitesimals are available in abundance. The principal objective in this paper is to show within our favorite framework of synthetic differential geometry that the tangent space of the group of bisections of a microlinear groupoid at its identity is naturally a Lie algebra. We give essentially distinct two proofs for its Jacobi identity.

math.DG↗