Tangent spaces and tangent bundles for diffeological spaces

Abstract

We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. We prove fundamental results about these tangent spaces, compute them in many examples, and observe that while they agree for smooth manifolds and many of the examples, they do not agree in general. After this, we recall Hector's definition of the tangent bundle of a diffeological space, and show that both scalar multiplication and addition can fail to be smooth, revealing errors in several references. We then give an improved definition of the tangent bundle, using what we call the dvs diffeology, which ensures that scalar multiplication and addition are smooth. We establish basic facts about these tangent bundles, compute them in many examples, and study the question of whether the fibres of tangent bundles are fine diffeological vector spaces. Our examples include singular spaces, spaces whose natural topology is non-Hausdorff (e.g., irrational tori), infinite-dimensional vector spaces and diffeological groups, and spaces of smooth maps between smooth manifolds (including diffeomorphism groups).

v2 and v3: minor corrections and improvements to exposition; 27 pages; to appear in Cahiers de Topologie et Géométrie Différentielle

Cited by in corpus (18)

• The diffeology of Milnor's classifying space
• Higher Order Automatic Differentiation of Higher Order Functions
• A Lie-Rinehart algebra in general relativity
• Linear and multilinear algebra on diffeological vector spaces
• Integration of singular foliations via paths
• Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them
• Derived Differentiable Manifolds
• The Holonomy Groupoids of Singularly Foliated Bundles
• Differential forms on diffeological spaces and diffeological gluing, I
• Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables
• Diffeological connections on diffeological vector pseudo-bundles
• Smooth Homotopy of Infinite-Dimensional $C^{\infty }$-Manifolds
• Simplicial cochain algebras for diffeological spaces
• Diffeological Levi-Civita connections
• On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms
• Diffeological De Rham operators
• Diffeological Morita Equivalence
• Remarks on the geometry and the topology of the loop spaces $H^s(S^1,N),$ for $s\le 1/2.$