Topics in differential geometry
Jets
Jet bundles and the jet comonad enable partial differential equations to be seen as differential-geometric objects.
Books and surveys
- Saunders, 2008: Jet manifolds and natural bundles (doi)
- Saunders, 1989: The geometry of jet bundles (doi)
- The standard reference, fairly readable
- Olver, 1995: Equivalence, invariants, and symmetry, Ch. 4: Jets and contact transformations (doi)
Papers
Category theory and differential geometry
Although category theory is not nearly as mainstream in differental geometry as it is in algebraic geometry, it is still used in certain circles.
Books
- Kolář, Michor, Slovák, 1993: Natural operations in differential geometry (doi, pdf, nLab )
- Ramanan, 2005: Global calculus (doi)
- Innocently titled book with a sheaf-theoretic view of differential geometry
- Wedhorn, 2016: Manifolds, sheaves, and cohomology (doi)
- Also sheaf-theoretic but more introductory than Ramanan’s book
Convenient categories
The category of smooth manifolds is very bad. Many attempts have been made to construct a “convenient category” with objects that generalize smooth manifolds. See generalized smooth spaces.
Tangent categories
Tangent bundle categories , or tangent categories for short, axiomatize the concept of a category equipped with a tangent bundle endofunctor.
- Cockett & Cruttwell, 2014: Differential structure, tangent structure, and SDG (doi)
- Cockett & Cruttwell, 2017: Connetions in tangent categories (pdf, arxiv)
- Cruttwell & Lucyshyn-Wright, 2018: A simplicial foundation for differential and sector forms in tangent categories (doi, arxiv)
- MacAdam, 2021: Vector bundles and differential bundles in the category of
smooth manifolds (doi, arxiv)
- Proves differential bundles in the (tangent) category of smooth manifolds are precisely vector bundles
- Cockett, Cruttwell, Lemay, 2021: Differential equations in a tangent category I: Complete vector fields, flows, and exponentials (doi, arxiv)
- Cruttwell & Lemay, 2023: Tangent categories as a bridge between differential geometry and algebraic geometry (arxiv)
A workshop on Tangent Categories and Their Applications was held in 2021:
- Cruttwell: Introduction to tangent categories (slides, video)
- MacAdam: An introduction to differential bundles (video)
- Lucyshyn-Wright: An introduction to connections in tangent categories (video)
The tangent endofunctor has also been studied as a monad:
- Jubin, 2014, PhD thesis: The tangent functor monad and foliations (arxiv)