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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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References

Baez, J. C. and Hoffnung, A. E.. Convenient categories of smooth spaces. Trans. Amer. Math. Soc. 363 (2011), 5789–5825.CrossRefGoogle Scholar
Blohmann, C., Elastic diffeological spaces, preprint, 2023. https://arxiv.org/abs/2301.02583v1.Google Scholar
Blohmann, C.. Lagrangian Field Theory, Diffeology, Variational Cohomology, Multisymplectic Geometry, preprint, 2024, https://people.mpim-bonn.mpg.de/blohmann/LFT/.Google Scholar
Chas, M. and Sullivan, D.. String topology, preprint (math.GT/0107187).Google Scholar
Christensen, J. D., Sinnamon, G. and Wu, E.. The $D$-topology for diffeological spaces. Pacific J. Math. 272 (2014), 87–110.CrossRefGoogle Scholar
Christensen, J. D. and Wu, E.. The homotopy theory of diffeological spaces. New York J. Math. 20 (2014), 1269–1303.Google Scholar
Cockett, J. R. B. and Cruttwell, G. S. H.. Differential structure, tangent structure, and SDG. Appl. Categ. Structures. 22 (2014), 331–417.CrossRefGoogle Scholar
Cockett, J. R. B. and Cruttwell, G. S. H.. The Jacobi identity for tangent categories. Cah. Topol. Geom. Différ. Catég. 56 (2015), 301–316.Google Scholar
Cockett, J. R. B. and Cruttwell, G. S. H.. Connections in tangent categories. Theory and Applications of Categories. 32 (2017), 835–888.CrossRefGoogle Scholar
Gürer, S. and Iglesias-Zemmour, P.. Differential forms on stratified spaces, Bull. Aust. Math. Soc. 98 (2018), 319–330.CrossRefGoogle Scholar
Goldammer, N., Magnot, J.-P., Welker, K.. On diffeologies from infinite dimensional geometry to PDE constrained optimization, Contemp. Math., 794, 1–48 (Providence, RI, 2024) July 18–20, 2022. (ed. Magnot, Jean-Pierre), (American Mathematical Society (AMS), Recent advances in diffeologies and their applications. AMS–EMS–SMF special session, Université de Grenoble-Alpes, Grenoble, France).Google Scholar
Goldammer, N. and Welker, K.. Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts, Appl. Math. Optim. 93, no. 1. (2026).CrossRefGoogle Scholar
Haraguchi, T.. Long exact sequences for de Rham cohomology of diffeological spaces. Kyushu J. Math. 68 (2014), 333–345.CrossRefGoogle Scholar
Hector, G., Macías-Virgós, E. and Sanmartín-Carbón, E.. De Rham cohomology of diffeological spaces and foliations. Indagationes math. 21 (2011), 212–220.CrossRefGoogle Scholar
Hicks, N. J.. Notes on Differential Geometry (Mathematics Studies). (D. Van Nostrand Company, 1965).Google Scholar
Iglesias, P.. Fibrés difféologiques et homotopie. Thèse de Doctorat d’état. (Université de Provence, Marseille, 1985).Google Scholar
Iglesias, P., Karshon, Y. and Zadka, M.. Orbifolds as diffeologies. Math. Soc. 362 (2010), 2811–2831 Trans. , Amer.CrossRefGoogle Scholar
Iglesias-Zemmour, P.. Diffeology 185 Mathematical Surveys and Monographs. (AMS, Providence, 2012).Google Scholar
Iglesias-Zemmour, P.. Variations of integrals in diffeology. Canad. J. Math. 65 (2013), 1255–1286.CrossRefGoogle Scholar
Iglesias-Zemmour, P., On Riemannian metrics in diffeology, preprint. 2023. https://math.huji.ac.il/piz/documents/DBlog-Ex-ORMID.pdf.Google Scholar
Iglesias-Zemmour, P.. Lectures on Diffeology. (Beijing World Publishing Corporation, 2025).Google Scholar
Kihara, H.. Model category of diffeological spaces. J. Homotopy Relat. Struct. 14 (2019), 51–90 10.1007/s40062-018-0209-3.CrossRefGoogle Scholar
Kihara, H.. Smooth Homotopy of infinite-Dimensional C ∞-manifolds, Memoirs of the American Mathematical Society 1436. (American Mathematical Society (AMS), Providence, RI, 2023).Google Scholar
Krepski, D., Watts, J. and Wolbert, S.. Sheaves, principal bundles, and Čech cohomology for diffeological spaces. Isr. J. Math. 264 (2024), 461–500.CrossRefGoogle Scholar
Kriegl, A. and Michor, P. W.. The Convenient Setting of Global analysis, Mathematical Surveys and Monographs. 53. (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
Kuribayashi, K.. Simplicial cochain algebras for diffeological spaces. Indagationes Mathematicae. 31 (2020), 934–967 10.1016/j.indag.2020.08.002.CrossRefGoogle Scholar
Kuribayashi, K.. Local systems in diffeology. Journal of Homotopy and Related Structures. 19 (2024), 475–523.CrossRefGoogle Scholar
Miyamoto, D.. The basic de Rham complex of a singular foliation. Int. Math. Res. Not. 8. (2023), 6364–6401.CrossRefGoogle Scholar
Miyamoto, D., Lie algebras of quotient groups, preprint, 2025. https://arxiv.org/abs/2502.10260v1.Google Scholar
Mukherjee, A.. Topics in Differential Topology, Texts and Readings in Mathematics 34. (Hindustan Book Agency, New Delhi, 2005).Google Scholar
Nezhad, A. D. and Ahmadi, A.. A novel approach to sheaves on diffeological spaces. Topology Appl. 263 (2019), 141–153.CrossRefGoogle Scholar
O’Connell, D.. Non–Hausdoff manifolds via adjunction spaces. Topology Appl. 326 (2023), 108388 10.1016/j.topol.2022.108388.CrossRefGoogle Scholar
Rosický, J.. Abstract tangent functors. Diagrammes. 12 (1984), JR1–JR11.Google Scholar
Rudin, W.. Real and Complex analysis. 3rd edition, (McGraw-Hill, 1987).Google Scholar
Schmeding, A.. An Introduction to infinite-Dimensional Differential geometry, Cambridge Studies in Advanced Mathematics 202. (Cambridge University Press, Cambridge, 2023).Google Scholar
Simakawa, K., Yoshida, K. and Haraguchi, T.. Homology and cohomology via enriched bifunctors. Kyushu Journal of Mathematics. 72 (2018), 239–252.CrossRefGoogle Scholar
Souriau, J. -M.. Groupes différentiels, Lecture Notes in Math. 91–128, (Springer, 1980) 836.Google Scholar
Stacey, A.. Comparative smootheology, Theory and Applications of Categories. 25 (2011), 64–117.CrossRefGoogle Scholar
Vincent, M.. Diffeological differential geometry, master's thesis, Univerity of Copenhagen, 2008.Google Scholar
Wolak, R. A.. Orbifolds, geometric structures and foliations. Applications to harmonic maps. Geom. Struc. on Riem. Man.-Bari. 73 (2015), 201–215Google Scholar