Learn Riemannian Geometry: Resources for Self-Learners

[pamparana]

Can someone recommend some background texts which can build me up with the necessary pre-requisites to learn about Riemannian Geometry? I have been self studying single and multi variable calculus but lack the mathematical rigour. Some resources/textbooks that can cover the background material with worked examples and be suitable for self-learner would be greatly appreciated!

 

[Daverz]

I would encourage you to try https://www.amazon.com/s/ref=dp_byl...Do+Carmo&sort=relevancerank&tag=pfamazon01-20 Riemannian Geometry. However, it's a bit like the Feynman Lectures: everything seems so sensible and logical the way he lays it out, but you may find that you are not prepared for the problem sets. In that case, you could try the "prequel", Differential Geometry of Curves and Surfaces.

John Lee's books are popular with the PF crowd, but I'm not familiar with them:

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

 

[ibdsm]

If you lack the rigour to study Riemannian geometry comfortably, try working through Spivak's "Calculus on Manifolds". It is the single book I would recommend for studying multivariate calculus, and it paves the way to differential and Riemannian geometry.

 

[NumericalFEA]

From the practical point of view, you might be interested in the following book (it contains some unique numerical algorithms related to the differential geometry of surfaces, together with complete source codes in C/C++ and practical examples):
https://www.amazon.com/dp/0646594044/?tag=pfamazon01-20

 

[rezeew]

I highly recommend starting with a solid foundation in linear algebra and differential geometry before diving into Riemannian geometry. This will provide you with the necessary mathematical rigour and understanding of key concepts such as tensors, manifolds, and curvature.

For background texts, I suggest "Linear Algebra" by Gilbert Strang and "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo. These texts provide clear explanations and worked examples, making them suitable for self-learners.

Additionally, I recommend supplementing your learning with online resources such as Khan Academy and MIT OpenCourseWare, which offer free lectures and practice problems on linear algebra and differential geometry.

Once you have a strong foundation in these areas, you can then move on to more advanced texts on Riemannian geometry, such as "Riemannian Geometry" by Manfredo do Carmo and "Riemannian Geometry: A Beginner's Guide" by Frank Morgan.

Remember to take your time and practice regularly to fully grasp the concepts of Riemannian geometry. Best of luck on your self-learning journey!