resources for differential geometry

Textbook

TextBook I used for Differential Geometry class at New York University, Abu Dhabi

• First Steps in Differential Geometry by Riemannian, Contact, Symplectic. Part of UTM Series
• An Introduction to Manifolds by Loring W. Tu

More Readings I found

• Introduction to Smooth Manifolds by John M. Lee
• Introduction to Matrix Analysis and Applications by Fumio Hiai , Dénes Petz
• Lie Groups, Lie Algebras, and Their Representations by V. S. Varadarajan. GTM 102
• Multilinear Algebra by Werner Greub. Part of UTX
• Riemannian Geometry by Peter Petersen
• Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo

Passages I read

Tensor

In Chinese

• "看完本文, 彻底理解张量" 系列： 看完本文, 彻底理解张量(上), 看完本文, 彻底理解张量(下), and 矢量算符是啥? 复内积怎么升降指标? 「彻底理解张量」拓展篇. I only read the first one (several time), cuz this is enough for my course. This is a really good series.

Manifold

In Chinese

• 几何学中最伟大的发明之一——流形，其背后的几何直觉与数学方法

Vector Bundle (In English)

• Vector Bundles & K-Theory pdf can be found in the link. Very good book on introducing vector bundle.

Videos

In Chinese

• 北京某高校《李群和李代数》. Still updating. This is a graduate level couse, I only watched the first three videos. But those are enough for providing a more abstract, deeper, and more concise understanding of what I am learning now.
• 最美数学系列 — 什么是微分流形. Very intuitive introduction to manifold, tangent space, etc. The tangent space part is really helpful for me.
• Ayumu 谈谈商空间、商映射、商拓扑. Very clean, detailed, easy-to-understand approach for understand Quotient space, my blog Introduction to Quotient Spaces in Topology is translated from this video.

In English, YouTube

• Tensors For Beginners. The Best video introducing tensors I've seen so far. I watched this several times on airplane. Really helpful.
• Manifolds Explained in 5 Levels of Difficulty. A general introduction to manifold.
• What is a manifold?. A general introduction to manifold. Very intuitive introduction.
• What is a Manifold?. Longer, more regious introduction.
• Real Projective Space, n = 1 and Real Projective Space, n=2. Really intuitive and clear introduction to projective space.
• Vector Bundles and Char Classes. Very good series, the book Vector Bundles & K-Theory is recommanded here.
• Manifolds. Video from The Bright Side of Mathematics, very good YouTuber on Math. This video series introduces manifold from topological perspective, while I learned from calculus and linear algebra perspective.

Course Resources

• Berkeley Math 214: Math 214: Differentiable manifolds. This couse has homework with solution, a good resource for self study.