Math 6510 - 1 Additional Texts

We shall basically follow the text. But much of the material is standard and widely available. I'll provide references and put copies in the math library. I will also cover additional topics that were suggested in consultation with Professors Bestvina, Bromberg and Wortman. Come to class for details and references. Here is a partial list of alternative sources that cover the material.

I have tried several different texts for this course: Bredon, Guillemin and Pollack, Singer and Thorpe, Spivak, Sternberg and Warner. Perhaps not so unexpectedly, the texts the students liked the best, Guillemin & Pollack and Singer & Thorpe are not the ones I liked the best, Bredon, Sternberg and Warner. None of the texts perfectly suits the material we teach at the University of Utah.

This course is designed to be a balance of application and theory that is optimized for the needs of students at Utah, be they interested in algebraic geometry, commutative algebra, geometric group theory, representation theory or geometric analysis. As mathematicians, it is our prerogative and, indeed duty, to understand why theorems work, so that we may modify or code them as we encounter them in the future.

The choice of topics in Math 6510 varies slightly from instructor to instructor, although the variance is far less than you might think hearing old graduate student gossip. One only needs to look at previous syllabi, or the Geometry / Topology Preliminary Exams derived from these courses over the last decade to see that they are very consistent. Guillemin & Pollack's text spins toward transversality theory. Wortman's notes designed for this course spin toward Lie Groups. A course based on John Lee's text spins towards differential geometry.

Texts Written for an Undergraduate Course in Differentiable Manifolds.

• Victor Guillemin & Alan Pollack, Differential Topology, Prentice Hall 1974.
• Isadore M. Singer & John A. Thorpe, Lecture notes on Elementary Topology and Geometry, Springer 1967; orig. Scott, Foresman & Co., 1967.
• Michael Spivak, Calculus on Manifolds, W. A. Benjamin, Inc. 1965.

Texts Suitable for a Graduate Course in Differentiable Manifolds.

• Louis Auslander & Robert MacKenzie, Introduction to Differentiable Manifolds, Dover 1977; orig. McGraw Hill 1963.
• Dennis Barden & Charles Thomas, An Introduction to Differentiable Manifolds, Imperial College Press 2003.
• Richard Bishop & Samuel Goldberg, Tensor Analysis on Manifolds, Dover 1980; orig. Macmillan Co. 1968.
• Glen Bredon, Topology and Geometry, Springer 1993.
• William Boothby, An Introduction to Differentiable manifolds and Riemannian Geometry, 2nd ed., Academic press 1986.
• Harley Flanders, Differential Forms with Applications to the Physical Sciences, Dover 1989; orig. Academic Press Inc. 1963.
• Serge Lang, Differentiable Manifolds, Addison Wesley Publ. Co. 1972.
• John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer 2013.
• John Milnor, Topology from the Differentiable Viewpoint, University Press of Virginia 1965.
• Siavash Shahshahani, An Introductory Course on Differentiable Manifolds, Dover 2016.
• Michael Spivak, Differential Geometry Vol. I, Publish or Perish, Inc. 1970.
• Shlomo Sternberg, Lectures on Differential Geometry, Chelsea Publ. Co. 1983; orig. Prentice Hall 1964.
• Loring Tu, An Introduction to Manifolds, 2nd ed., Springer 2011.
• Frank Warner, Introduction to Differentiable Manifolds and Lie Groups, Scott, Foresman & Co. 1971.
• Kevin Wortman, Math 6510 Notes, Department of Mathematics, University of Utah online notes 2017.

Specialist's Books on Specific Topics. Unsuitable for Math 6510.

• Raoul Bott & Loring Tu, Differential Forms in Algebraic Topology, Springer 1982.
• Shiing-Shen Chern, Differentiable Manifolds, Department of Mathematics, University of Chicago 1959.
• Morris Hersch, Differential Topology, Springer 1976.
• Shoshichi Kobayashi & Katsumi Nomizu, Foundations of Differential Geometry, vol I and II, Interscience Publ., 1969.

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Last updated: 9 - 4 - 18