INTRODUCTION
TO
DIFFERENTIABLE MANIFOLDS

Fall 1999 - Math536

Instructor - Krzysztof Galicki

Time and Place: MW 4:00 - 5:15 p.m.; HUM422

COURSE DESCRIPTION

The two main textbooks for this course are Differentiable Manifolds. A First Course by Lawrence Conlon, Birkhäuser Advanced Texts, Basler Lehrebücher, Birkhäuser (1993) and Introduction to differential topology by Th. Brocker and K. Jänich, Cambridge University Press (1982). To see the Birkhäuser's catalogue entry for Conlon's book (with its table of contents cilck on it). Brocker and Jänich is out-of-print now but I will try to arrange that you get a copy. The lectures will cover the following topics:

• Topological manifolds

• Manifolds and differentiable structures

• Tangent bundle

• Vector bundles and multilinear algebra

• Submanifolds, immersions, and embeddings

• Vector fields, flows, and foliations

• Lie groups, principal bundles, and homogeneous spaces

• Covectors and smooth p-forms

• Integration and cohomology

• Forms and foliations

• Symplectic manifolds

• Riemannian manifolds

Recommended reading during and after this course

• Fundamentals of Differential Geometry by Serge Lang Graduate Texts in Mathematics, Volume 191, Springer-Verlag (1998). To see the detailed Springer catalogue web page description of the new edition of Lang's book click on it. I have ordered this book as a recommended text so you will find it in the UNM bookstore soon.

• Differential Geometry. Cartan's Generalization of Klein's Erlangen Program by R. W. Sharpe, Graduate Texts in Mathematics, Volume 166, Springer-Verlag (1997). To see the detailed Springer catalogue web page description of Sharpe's book click on it. I have ordered this book as a recommended text so you will find it in the UNM bookstore soon.

• Manifolds, Tensor Analysis, and Applications by R. Abraham, J. E. Marsden, and T. Ratiu, Applied Mathematical Sciences, Volume 75, Springer-Verlag (1988). To see the Springer catalogue web page entry on this book click on it. This book is a very useful source of examples and various interesting applications. I highly recommend it.

• Geometry I. Basic Ideas and Concepts of Differential Geometry R.V. Gamkrelidze (Ed.), Encyclopedia of Mathematical Sciences, Volume 28, Springer-Verlag (1988). To see the Springer catalogue web page entry on this volume click on it. The book was written by D. V. Alekseevsky, A. M. Vinogradov, and V. V. Lychagin very much in the spirit of the whole series. It is not a textbook to learn all the details and proofs but it provides a superb overview of what differential geometry is all about. Great reading and a "must have" if you plan to use differential geometry beyond this course.

• Differential Geometry: Manifolds, Curves, and Surfaces by M. Berger and B. Gostiaux, Graduate Texts in Mathematics, Volume 115, Springer-Verlag (1988). To see the Springer catalogue web page entry on this volume click on it. Excellent textbook.

• Foundations of Differentiable Manifolds and Lie Groups by Frank W. Warner, Graduate Texts in Mathematics, Volume 94, Springer-Verlag (1983). To see the Springer catalogue web page entry on this volume click on it. This is a very well written textbook with emphasis on the theory of Lie groups and homogeneous spaces.

• Differential Topology by Morris W. Hirsch, Graduate Texts in Mathematics, Volume 33, Springer-Verlag (1976). To see the Springer catalogue web page entry on this volume click on it.

• Differentiable Manifolds by Antoni A. Kosinski, Pure and Applied Mathematics, Volume 138, Academic Press (1993). Excellent book but perhaps a bit more advanced for this course.

• Comprehensive Introduction to Differential Geometry, Volume I by Michael Spivak, Publish or Perish, Inc. (1979). Famous five volume lectures of Michael Spivak have educated generations of differential geometers. Volume one contains many topics that will be discussed in class. I highly recommend it.

• Differentiable manifolds by Georges de Rham, Grundlehren der mathematischen Wissenschaft, Volume 266, Springer-Verlag (1984). To see the Springer catalogue web page entry on this classic click on it. This is a beautiful book certainly worth having but a bit expensive. It is a more advanced text and I will not use it during the semester.