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PreprintCC BY-NC-SA 4.0arXiv

Diffeological Riemannian orbifolds

  • David Miyamoto
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Abstract

We show that the data of a Riemannian metric on a differentiable stack presented by an orbifold groupoid is equivalent to the data of a Riemannian metric on its diffeological orbit space. As a consequence, we conclude that the classical notion of Riemannian orbifold is equivalent to that of a Riemannian diffeological orbifold. We use the framework for Riemannian diffeology introduced by Kuribayashi, Sakai, and Shiobara, and our result answers a problem they posed in the affirmative. More generally, we show that a Riemannian metric on a Lie groupoid, namely a 2-metric in the sense of del Hoyo and Fernandes, induces a Riemannian metric on its diffeological orbit space only if the Lie groupoid is regular, and that properness is a sufficient but not necessary condition.

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Cite

BibTeXRIS
BibTeX
@misc{miyamoto2026diffeological,
  title = {{Diffeological Riemannian orbifolds}},
  author = {David Miyamoto},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.08939},
  eprint = {2607.08939},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.08939},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Diffeological Riemannian orbifolds
AU  - David Miyamoto
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2607.08939
UR  - https://arxiv.org/abs/2607.08939
AB  - We show that the data of a Riemannian metric on a differentiable stack presented by an orbifold groupoid is equivalent to the data of a Riemannian metric on its diffeological orbit space. As a consequence, we conclude that the classical notion of Riemannian orbifold is equivalent to that of a Riemannian diffeological orbifold. We use the framework for Riemannian diffeology introduced by Kuribayashi, Sakai, and Shiobara, and our result answers a problem they posed in the affirmative. More generally, we show that a Riemannian metric on a Lie groupoid, namely a 2-metric in the sense of del Hoyo and Fernandes, induces a Riemannian metric on its diffeological orbit space only if the Lie groupoid is regular, and that properness is a sufficient but not necessary condition.
ER  -