I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.

I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.

Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes’ theorem and de Rham theorem with complete proofs.

Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?

Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.

Suggestions about important theorems and concepts to learn, and book references, will be most helpful.

**Answer**

**ADDITION:** I have compiled what I think is a definitive collection of listmanias at Amazon for a best selection of books an references, mostly in increasing order of difficulty, in almost any branch of geometry and topology. In particular the books I recommend below for differential topology and differential geometry; I hope to fill in commentaries for each title as I have the time in the future.

If you want to have an overall knowledge Physics-flavored the best books are **Nakahara**‘s *“Geometry, Topology and Physics”* and above all: **Frankel**‘s *“The Geometry of Physics”* (great book, but sometimes his notation can bug you a lot compared to standards).

If you want to learn Differential Topology study these in this order: **Milnor**‘s *“Topology from a Differentiable Viewpoint”*, **Jänich/Bröcker**‘s *“Introduction to Differential Topology”* and **Madsen**‘s *“From Calculus to Cohomology”*. Although it is always nice to have a working knowledge of general point set topology which you can quickly learn from **Jänich**‘s *“Topology”* and more rigorously with **Runde**‘s *“A Taste of Topology”*.

To start Algebraic Topology these two are of great help: **Croom**‘s *“Basic Concepts of Algebraic Topology”* and **Sato/Hudson** *“Algebraic Topology an intuitive approach”*. Graduate level standard references are **Hatcher**‘s *“Algebraic Topology”* and **Bredon**‘s *“Topology and Geometry”*, **tom Dieck**‘s *“Algebraic Topology”* along with **Bott/Tu** *“Differential Forms in Algebraic Topology.”*

To really understand the classic and intuitive motivations for modern differential geometry you should master curves and surfaces from books like **Toponogov**‘s *“Differential Geometry of Curves and Surfaces”* and make the transition with **Kühnel**‘s *“Differential Geometry – Curves, Surfaces, Manifolds”*. Other nice classic texts are **Kreyszig** *“Differential Geometry”* and **Struik**‘s *“Lectures on Classical Differential Geometry”*.

For modern differential geometry I cannot stress enough to study carefully the books of **Jeffrey M. Lee** *“Manifolds and Differential Geometry”* and **Liviu Nicolaescu**‘s *“Geometry of Manifolds”*. Both are deep, readable, thorough and cover a lot of topics with a very modern style and notation. In particular, Nicolaescu’s is my favorite. For Riemannian Geometry I would recommend **Jost**‘s *“Riemannian Geometry and Geometric Analysis”* and **Petersen**‘s *“Riemannian Geometry”*. A nice introduction for Symplectic Geometry is **Cannas da Silva** *“Lectures on Symplectic Geometry”* or **Berndt**‘s *“An Introduction to Symplectic Geometry”*. If you need some Lie groups and algebras the book by **Kirilov** *“An Introduction to Lie Groops and Lie Algebras”* is nice; for applications to geometry the best is **Helgason**‘s *“Differential Geometry – Lie Groups and Symmetric Spaces”*.

FOR TONS OF SOLVED PROBLEMS ON DIFFERENTIAL GEOMETRY the best book by far is the recent volume by **Gadea/Muñoz – “Analysis and Algebra on Differentiable Manifolds: a workbook for students and teachers”**. From manifolds to riemannian geometry and bundles, along with amazing summary appendices for theory review and tables of useful formulas.

EDIT (ADDED): However, I would argue that one of the best introductions to manifolds is the old soviet book published by MIR, **Mishchenko/Fomenko** – *“A Course of Differential Geometry and Topology”*. It develops everything up from $\mathbb{R}^n$, curves and surfaces to arrive at smooth manifolds and LOTS of examples (Lie groups, classification of surfaces, etc). It is also filled with LOTS of figures and classic drawings of every construction giving a very visual and geometric motivation. It even develops Riemannian geometry, de Rham cohomology and variational calculus on manifolds very easily and their explanations are very down to Earth. If you can get a copy of this title for a cheap price (the link above sends you to Amazon marketplace and there are cheap “like new” copies) I think it is worth it. Nevertheless, since its treatment is a bit dated, the kind of algebraic formulation is not used (forget about pullbacks and functors, like Tu or Lee mention), that is why an old fashion geometrical treatment may be very helpful to complement modern titles. In the end, we must not forget that the old masters were much more visual an intuitive than the modern abstract approaches to geometry.

**NEW!:** the book by Mishchenko/Fomenko, along with its companion of problems and solutions, has been recently typeset and reprinted by Cambridge Scientific Publishers!

If you are interested in learning Algebraic Geometry I recommend the books of my Amazon list. They are in recommended order to learn from the beginning by yourself. In particular, from that list, a quick path to understand basic Algebraic Geometry would be to read **Bertrametti** et al. *“Lectures on Curves, Surfaces and Projective Varieties”*, **Shafarevich**‘s *“Basic Algebraic Geometry”* vol. 1, 2 and **Perrin**‘s *“Algebraic Geometry an Introduction”*. But then you are entering the world of abstract algebra.

If you are interested in Complex Geometry (Kähler, Hodge…) I recommend **Moroianu**‘s *“Lectures on Kähler Geometry”*, **Ballmann**‘s *“Lectures on Kähler Manifolds”* and **Huybrechts**‘ *“Complex Geometry”*. To connect this with Analysis of Several Complex Variables I recommend trying **Fritzsche/Grauert** *“From Holomorphic Functions to Complex Manifolds”* and also **Wells**‘ *“Differential Analysis on Complex Manifolds”*. Afterwards, to connect this with algebraic geometry, try, in this order, **Miranda**‘s *“Algebraic Curves and Riemann Surfaces”*, **Mumford**‘s *“Algebraic Geometry – Complex Projective Varieties”*, **Voisin**‘s *“Hodge Theory and Complex Algebraic Geometry”* vol. 1 and 2, and **Griffiths/Harris** *“Principles of Algebraic Geometry”*.

You can see their table of contents at Amazon.

Hope this helps… good luck!

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