tailieunhanh - Introduction To Differential Topology

This is a quick set of note on basic differential topogoly. It get sketchier as it goes on. The last few section are only introduce the terminology and the some of concepts. These note qre written faster than I can read and may make no sense in spots. | Introduction to Differential Topology Matthew G. Brin Department of Mathematical Sciences State University of New York at Binghamton Binghamton NY 13902-6000 Spring 1994 Contents 0. Introduction. 2 Basics . 2 2. Derivative and Chain rule in Euclidean spaces. 7 3. Three derivatives. 13 4. Higher derivatives 15 5. The full definition of differentiable manifold. 17 6. The tangent space of a manifold. 18 7. The Inverse Function Theorem . 22 8. The c category and diffeomorphisms. 30 9. Vector fields and flows . 31 10. Consequences of the Inverse Function Theorem . 37 11. Submanifolds . 40 12. Bump functions and partitions of unity. 43 13. The metric. 49 14. The tangent space over a coordinate patch. 53 15. Approximations. 54 16. Sard s theorem . 55 17. Ttansversality 57 18. Manifolds with boundary. 58 1 0. Introduction. This is a quick set of notes on basic differential topology. It gets sketchier as it goes on. The last few sections are only to introduce the terminology and some of the concepts. These notes were written faster than I can read and may make no sense in spots. Were I to do them again the first few topics would be rearranged into a different order. I am told that there are many misprints. The notes were designed to give a quick and dirty half semester introduction to differential topology to students that had finished going through almost all of Topology A first course by James R. Munkres. There are references to this book as Munkres in these notes. The notes were written so that all of the material could be presented by the students in class. This explains various exhortations to presenters that occur periodically throughout the notes. I cribbed from three main sources 1 Serge Lang Differential manifolds Addison Wesley 1972 2 Morris w. Hirsch Differential topology Springer-Verlag 1976 and 3 Michael Spivak Calculus on manifolds Benjamin 1965. The last is a particularly pretty book that unfortunately seems to be out of print. I also stole from a few pages .