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Towards Riemannian diffeology

Part of: Categories and geometry Infinite-dimensional manifolds General theory of differentiable manifolds

Published online by Cambridge University Press:  19 December 2025

Katsuhiko Kuribayashi ,
Keiichi Sakai  and
Yusuke Shiobara
Katsuhiko Kuribayashi*
Affiliation:
Department of Mathematics, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan (kurimath@shinshu-u.ac.jp)
Keiichi Sakai
Affiliation:
Department of Mathematics, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan (sakaik@shinshu-u.ac.jp)
Yusuke Shiobara
Affiliation:
Department of Science and Technology, Graduate School of Medicine, Science and Technology, Shinshu University, Matsumoto, Nagano 390-8621, Japan (24hs603d@shinshu-u.ac.jp)
*
*Corresponding author.
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Abstract

We introduce a framework for Riemannian diffeology. To this end, we use the tangent functor in the sense of Blohmann and one of the options of a metric on a diffeological space in the sense of Iglesias-Zemmour. As a consequence, the category consisting of weak Riemannian diffeological spaces and isometries is established. With a technical condition for a definite weak Riemannian metric, we show that the pseudodistance induced by the metric is indeed a distance. As examples of weak Riemannian diffeological spaces, an adjunction space of manifolds, a space of smooth maps and the mixed one are considered.

Keywords

adjunction spacediffeologymapping spacemetrictangent functor

MSC classification

Primary: 58A40: Differential spaces
Secondary: 18F15: Abstract manifolds and fiber bundles 58B20: Riemannian, Finsler and other geometric structures

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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