Basic Concepts of Synthetic Differential Geometry

1. Front Matter

   Pages i-xv

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2. Differential calculus and integrals

   • René Lavendhomme

   Pages 1-31

3. Weil algebras and infinitesimal linearity

   • René Lavendhomme

   Pages 33-59

4. Tangency

   • René Lavendhomme

   Pages 61-100

5. Differential forms

   • René Lavendhomme

   Pages 101-140

6. Connections

   • René Lavendhomme

   Pages 141-180

7. Global actions

   • René Lavendhomme

   Pages 181-210

8. On the algebra of the geometry of mechanics

   • René Lavendhomme

   Pages 211-267

9. Note on toposes and models of S.D.G.

   • René Lavendhomme

   Pages 269-311

10. Back Matter

    Pages 313-320

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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.

• Lie group
• algebra
• differential geometry
• proof

• Université Catholique de Louvain, Belgium

  René Lavendhomme

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