Differential geometry recommendations

In summary, the conversation is about finding a good book in differential geometry for someone who knows calculus, differential equations, linear algebra, and variational calculus. The person is looking for a book that is clear and intuitive, but also rigorous and does not take anything for granted. They prefer a book that explains the motivation behind concepts and theorems rather than just presenting formal definitions. They also mention an interest in topology and a dislike for overly formal mathematics. A recommended book is "Nakahara - Geometry, Topology and Physics", which covers topology and differential geometry in a physical context and is suitable for graduate students or advanced undergraduates.
  • #1
Terilien
140
0
what is a good book in differential geometry. I currently know calculus, a bit about differential equations, a bit of linear algebra and a bit about tensors. I also know some variational calculus.

Of course what I know won't really help. I've skimmed through some physics sources and mathematical sources and have decided exactly what I want. I currently don't have any.

I'm looking for something clear and intuitive. something that doesn't throw formal definitions in your face without explaning what motivates them. I would like something that explains what lead to certain concepts and theorems. That's what tunred me away from most mathematics books. i don't think it helps a beginner to start a book that opens with "Let a be (insert formal terms here thing)".

I would however like the book to have mathematical rigour. Things, even if somewhat obvious, should be proven and not taken for granted. NOTHING should be taken for granted. This feature turned me away from a great deal of physics texts.

so I would like something that is informal and intuitive but still rigorous. a kind of medium.

PS.: I will admit that I'm a very impatient person. I am working on it though.

Some chapters introducing topology would also be nice.

I personally think that formality these days is killing mathematics. I admit that it is important, but it is also important to see how definitions arise.
 
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  • #3
Nakahara - Geometry, Topology and Physics is an excellent book, though it is aimed at graduate students so chunks of it are perhaps a bit more advanced than you're after but it does cover topology and differential geometry (you'd never have guessed from the name :wink: ) from fairly basic things as well as put them in a physical context so you can see what applications such things have outside of "It's an interesting bit of maths". Applications like general relativity, quantum mechanics, super fluids, string theory, gauge theories, supersymmetry etc. I never got the notion of topologies until I sat down and read this book, and I'm postgrad!

I'd advise flicking through it before buying if you're a 1st or 2nd year student, but if you're a 3rd year I very much recommend it.
 

FAQ: Differential geometry recommendations

What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in space. It combines the tools of calculus and linear algebra to analyze the geometric structure of mathematical objects.

What are some applications of differential geometry?

Differential geometry has numerous applications in various fields such as physics, engineering, and computer graphics. It is used to study the shape of objects in motion, the curvature of space-time in general relativity, and the design of surfaces in computer-aided design.

What are some key concepts in differential geometry?

Some key concepts in differential geometry include manifolds, curves, surfaces, tangent spaces, and curvature. Manifolds are spaces that locally resemble Euclidean space, curves and surfaces are objects that can be described using differential equations, tangent spaces represent the local behavior of manifolds, and curvature measures the deviation from flatness.

What are the main differences between differential geometry and Euclidean geometry?

Euclidean geometry studies the properties of flat, two-dimensional space, while differential geometry deals with curved, multidimensional spaces. In Euclidean geometry, the focus is on straight lines and angles, while in differential geometry, the focus is on curves and surfaces. Additionally, Euclidean geometry is limited to a fixed number of dimensions, while differential geometry can be applied to spaces of any dimension.

What are some recommended resources for learning differential geometry?

Some recommended resources for learning differential geometry include textbooks such as "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo and "Introduction to Smooth Manifolds" by John M. Lee. Online resources such as lectures on YouTube or courses on websites like Coursera and edX are also helpful for learning differential geometry.

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