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Towards Riemannian diffeology

Katsuhiko Kuribayashi, Keiichi Sakai, Yusuke Shiobara

TL;DR

The article develops a framework for Riemannian diffeology by equipping diffeological spaces with a weak Riemannian metric defined on the tangent-doubled space $T_2(X)$ using the tangent functor and a plotwise positive covariant $2$-tensor. It proves that the induced pseudodistance $d$ becomes a true distance under a definiteness condition with a separating generating family, and shows that adjunction and mapping spaces admit definite weak Riemannian metrics, with applications to free loop spaces and warped products. It introduces a subdiffeology ${\mathcal D}'$ on mapping spaces to ensure metric definiteness and analyzes the behavior of concatenation and loop-space structures under this framework. The paper further develops a warped-product construction in the diffeological setting and discusses open problems, including the extension to orbifolds and the precise relation between different diffeologies on mapping spaces, outlining a broad program for Riemannian geometry in diffeology with potential ties to string topology and data analysis.

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.

Towards Riemannian diffeology

TL;DR

The article develops a framework for Riemannian diffeology by equipping diffeological spaces with a weak Riemannian metric defined on the tangent-doubled space using the tangent functor and a plotwise positive covariant -tensor. It proves that the induced pseudodistance becomes a true distance under a definiteness condition with a separating generating family, and shows that adjunction and mapping spaces admit definite weak Riemannian metrics, with applications to free loop spaces and warped products. It introduces a subdiffeology on mapping spaces to ensure metric definiteness and analyzes the behavior of concatenation and loop-space structures under this framework. The paper further develops a warped-product construction in the diffeological setting and discusses open problems, including the extension to orbifolds and the precise relation between different diffeologies on mapping spaces, outlining a broad program for Riemannian geometry in diffeology with potential ties to string topology and data analysis.

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.
Paper Structure (18 sections, 23 theorems, 56 equations)

This paper contains 18 sections, 23 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Future work and perspective
  3. Preliminaries
  4. A Riemannian diffeological space
  5. Diffeological Riemannian metrics
  6. Manifolds in Riemannian diffeology
  7. A category of weak Riemannian diffeological spaces
  8. The Riemannian pseudo-distance
  9. A diffeological adjunction space
  10. A diffeological mapping space
  11. Weak Riemannian metrics on mapping spaces
  12. A subdiffeology of the functional diffeology.
  13. An example: free loop spaces
  14. The warped product of Riemannian diffeological spaces
  15. Problems
  16. ...and 3 more sections

Key Result

Proposition 2.3

(PIZ12) An induction $i \colon (A, {\mathcal{D}}^A) \to (X, {\mathcal{D}}^X)$ gives rise to a diffeomorphism $i\colon (A, {\mathcal{D}}^A)\stackrel{\cong}{\to} (\operatorname{Im}(i), {\mathcal{D}}^{\operatorname{Im}(i)}_{\text{\em sub}})$.

Theorems & Definitions (78)

  • Example 1.1: see Example \ref{['ex:mapping_spaces_pushout']} and Proposition \ref{['prop:an_extension']} for a more general setting
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Example 2.5
  • Remark 2.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3: PIZ23
  • ...and 68 more