Starting at an introductory level, the book leads rapidly to important and often new results in synthetic differential geometry. From rudimentary analysis the book moves to such important results as: a new proof of De Rham's theorem; the synthetic view of global action, going as far as the Weil characteristic homomorphism; the systematic account of structured Lie objects, such as Riemannian, symplectic, or Poisson Lie objects; the view of global Lie algebras as Lie algebras of a Lie group in the synthetic sense; and lastly the synthetic construction of symplectic structure on the cotangent bundle in general. Thus while the book is limited to a naive point of view developing synthetic differential geometry as a theory in itself, the author nevertheless treats somewhat advanced topics, which are classic in classical differential geometry but new in the synthetic context. Audience: The book is suitable as an introduction to synthetic differential geometry for students as well as more qualified mathematicians.
Table of ContentsIntroduction. 1. Differential Calculus and Integrals. 2. Weil Algebras and Infinitesimal Linearity. 3. Tangency. 4. Differential Forms. 5. Connections. 6. Global Actions. 7. On the Algebra of the Geometry of Mechanics. 8. Note on Toposes and Models of S.D.G. Bibliography. Index.