Synthetic geometry of manifolds / Anders Kock.

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

1. Calculus and linear algebra -- 2. Geometry of the neighbour relation -- 3. Combinatorial differential forms -- 4. The tangent bundle -- 5. Groupoids -- 6. Lie theory; non-abelian covariant derivative -- 7. Jets and differential operators -- 8. Metric notions.

This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighbourhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to familiarize themselves with the field.