This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

Chapter 1. Examples of Manifolds
Chapter 2. Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings
Chapter 3. The Degree of a Mapping. The Intersection Index of Submanifolds. Applications
Chapter 4. Orientability of Manifolds. The Fundamental Group. Covering Spaces(Fibre Bundles with Discrete Fibre)
Chapter 5. Homotopy Groups
Chapter 6. Smooth Fibre Bundles
Chapter 7. Some Examples of Dynamical Systems and Foliations on Manifolds
Chapter 8. The Global Structure of Solutions of Higher-Dimensional Variational Problems
Bibliography

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