Differential geometry, what book is good for a first timer?

In summary, the conversation is about the topic of "Manifolds" and the speaker's curiosity about it. They ask about the prerequisites for reading a book on this topic and express their interest in a rigorous book that proves theorems. Another person recommends "Calculus on Manifolds" by Spivak or "Differential Topology" by Guillemin and Pollack, assuming the speaker has knowledge of metric spaces and basic topology.
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I've been wanting to see what this topic is all about for awhile now. I see the word "Manifold" and other terminology floating around on the forum. It got me really curious.

I wonder what the prerequisites are to reading a book like this?

Hypothetically I have the prerequisites, what would you recommend for a first time read?

I care about how rigorous the book is in proving theorems. Something unfounded wouldn't be worth reading.
 
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1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces using techniques from calculus and linear algebra. It also has applications in physics, engineering, and computer graphics.

2. How is differential geometry different from other branches of geometry?

Differential geometry differs from other branches of geometry by focusing on the study of smooth curves and surfaces rather than discrete or piecewise-defined objects. It also utilizes tools from calculus and linear algebra to analyze these objects.

3. What are some practical applications of differential geometry?

Differential geometry has many practical applications, including computer graphics and animation, robotics, physics and engineering, and even in the study of biological shapes and patterns.

4. What are some good books for beginners in differential geometry?

Some good books for beginners in differential geometry include "Introduction to Smooth Manifolds" by John M. Lee, "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo, and "Differential Geometry" by Wolfgang Kühnel.

5. Do I need a strong background in math to understand differential geometry?

While a strong background in math, including calculus and linear algebra, can be helpful in understanding differential geometry, it is not necessary. Many introductory books and courses provide the necessary background for beginners to understand the concepts and applications of differential geometry.

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