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Principal bundles, groupoids, and connections

100%

Kock A.

Banach Center Publications

2007

tom 76

nr 1

185-200

We clarify in which precise sense the theory of principal bundles and the theory of groupoids are equivalent; and how this equivalence of theories, in the differentiable case, reflects itself in the theory of connections. The method used is that of synthetic differential geometry.

A geometric theory of harmonic and semi-conformal maps

100%

Kock A.

Open Mathematics

2004

tom 2

nr 5

708-724

We describe for any Riemannian manifold M a certain scheme M L, lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M L; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M L.

Ordinary differential equations and their exponentials

63%

Kock A. , Reyes G.

Open Mathematics

2006

tom 4

nr 1

64-81

In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.