Introduction to Smooth Manifolds
by John M. Lee · first published 2002
63% recommend38% neutral
Why HN recommends it
A standard graduate-level text on smooth manifolds, suited as a follow-up after topology and as a reference for differential geometry. It provides a solid foundation for later study.
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…(Walter Rudin) - Partial Differential Equations (Lawrence Evans) - Analysis On Manifolds (James Munkres) - Abstract Algebra (David Dummit and Richard Foote) - Algebraic Topology (Allen Hatcher) - Introduction to Smooth Manifolds (John Lee) - Foundations of Differentiable Manifolds and Lie Groups - (Frank Warner) - Galois Theory (Harold Edwards) - Linear Representations of Finite Groups (Jean-Pierre Serre and Leonhard Scott)…
on “Mathematician's anger over his unread 500-page proof” ↗* 4-Manifolds and Kirby Calculus by Andras I. Stipsicz and Robert E. Gompf * Differential Manifolds by Antoni A. Kosinski * Introduction to Smooth Manifolds by John M. Lee
on “Ask HN: Math books that made you significantly better at math?” ↗I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a…
on “What is a manifold?” ↗+1 on munkres. Also recommend Algebraic Topology by Hatcher as a followup (free on his website) and Introduction to Smooth Manifolds by Lee.
on “Elementary Applied Topology (2014)” ↗…through Baby Rudin. For topology specifically you can look into Munkres or Introduction to Topological Manifolds, J Lee. > Measure Theory T. Tao has a good book on this topic. > Differential Geometry Introduction to Smooth Manifolds by J Lee > Probability Probability Theory, Achim Klenke > Game Theory Game Theory by Maschler, Solan & Zamir
on “Ask HN: Mathematicians, what textbooks are best for learning these math topics?” ↗That's a great suggestion. I actually started with Topological Manifolds before moving on to Introduction to Smooth Manifolds and it really helped build a solid foundation. I havent read Loring Tus books before but let me look at them since I have been wanting to revisit the topic with a clearer and more relaxed approach.
on “What is a manifold?” ↗…system of notation, and while the systems are all in some sense formally isomorphic, the transformations required to get from one to another are often not at all obvious to the student. —John M. Lee, "Introduction to Smooth Manifolds"
on “Mathematical Notation Is Awful” ↗A famous quote in Lee's "Introduction to Smooth Manifolds": > differential geometry is the study of properties that are invariant under change of notation
on “Annotated equations for increased readability and understanding of papers” ↗…manifold equipped with a contact structure. A more in-depth discussion can be found in many differential topology texts, e.g. page 581 of [1]. [0] https://en.m.wikipedia.org/wiki/Contact_geometry [1] Lee, Introduction to Smooth Manifolds https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth...
on “In the 'Wild West' of Geometry, Mathematicians Redefine the Sphere” ↗