Where do I start learning differential geometry?

In summary, the conversation discusses the need to have a strong foundation in differential geometry before delving into general relativity. The recommended resources for learning differential geometry include a book by Pressley and another book on metric spaces. The conversation also mentions the importance of topology in understanding manifolds and suggests a three-volume text by Lee for further learning. The individual is open to providing guidance and assistance in learning these topics.
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I've recently finished tackling differential equations. I want to start learning general relativity, but from what I've read, I need to have a firm footing in differential geometry first. So where do I start learning DG? I really don't want to do a half-hearted job in an attempt to quickly jump onto GR, so does anyone know some good websites where I can properly practice it? It is out of portion stuff for me so I don't have any book to refer to, but I want to do it anyways. I would also love to know the difference between topological analysis and differential geometry.
 
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The mathematical language of differential geometry consists of manifolds. But the least you can say is that manifolds are pretty abstract and confusing. In order to really grasp manifolds, you'll need to know more about curves and surfaces, because you can easily grasp and visualize that. I recommend the book by Pressley to start off with differential geometry.

https://www.amazon.com/dp/184882890X/?tag=pfamazon01-20

Then you have enough motivations to start manifolds, but you won't know enough math. Manifolds rely on topology (which basically studies the shape of a space without taking distances, angles, etc. into account). But in order to start topology, you'll need to know a bit about metric spaces. So I suggest

https://www.amazon.com/dp/1846283698/?tag=pfamazon01-20 (you can read this together with Pressley)

Then you can start the awesome 3-volume text of Lee:

https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20
https://www.amazon.com/dp/1441999817/?tag=pfamazon01-20
https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20

For questions/guidance/mentoring, feel free to PM me anytime!
 
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1. What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of geometric objects using the tools of calculus and algebra. It is concerned with the study of curves, surfaces, and higher-dimensional objects and their properties such as curvature, length, and angle.

2. What are some prerequisites for learning differential geometry?

A good understanding of calculus, linear algebra, and multivariable calculus is necessary for learning differential geometry. Familiarity with vector spaces, matrices, and basic concepts of topology would also be beneficial.

3. How can I start learning differential geometry?

A good place to start would be to study the basics of differential geometry, such as curves and surfaces, and then move on to more advanced topics such as Riemannian geometry and manifolds. Online resources, textbooks, and courses can be helpful in learning the subject.

4. What are some important concepts in differential geometry?

Some important concepts in differential geometry include curvature, geodesics, tensors, and differential forms. These concepts are used to study the properties of geometric objects and their relationships with each other.

5. What applications does differential geometry have?

Differential geometry has many applications in various fields such as physics, engineering, computer graphics, and computer vision. It is used to model and analyze curved surfaces and objects in these fields and has practical applications in areas such as robotics, medical imaging, and computer-aided design.

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