### Physics and Differential Geometry

Currently I have asked to me if physics is needed to study differential geometry. I know that in the theory, we can study differential geometry without any concept of physics; however, so many ideas in differential geometry has arose from physics, which lead me to think that, if I study differential geometry but I don't know anything about physics (for instance, relativity) then I will repeat theorems like a parrot, because I don't have a motivation of a problem.I would like to know that it is not the case. PD. (I'm sorry for my english, it is ...Read more

### reference request - Non-English-language graduate-level textbooks on differential geometry

I'm looking for modern graduate-level non-English-language differential geometry textbooks. I'm interested in original works by non-English-language speakers in their native languages, not translations into non-English languages. I'm also not interested in non-English language textbooks which are available in English translations, such as Manfredo Perdigão do Carmo's "Riemannian geometry".I have already made a list of about 45 English-language textbooks, and I have already acquired the following non-English-language DG books.Francisco Gómez Rui...Read more

### Exercises to help in the understanding of differential forms?

I've been trying to grasp the concept of differential forms, which I have been encountering while studying the text "Geometric Measure Theory" by Frank Morgan. Unfortunately the explanation is very sparse and while the internet contains many definitions, I have a hard time getting the bigger picture from just reading them. Are there any lists of exercises that I could do to assist my understanding by just working with them?Thank you....Read more

### differential geometry - Repeated application of the gradient on a Riemannian manifold

While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here:1) What kind of object id $\nabla ^k f$? Is it a $k$-vector? How is it defined?2) How is the metric $g$ extended to the space of these objects?I suspect a strong similarity to differential forms and their norms as defined in Hodge theory, yet there must also be some differences, since forms are a purely differential object, whereas the gradient is a Riemannian one....Read more

### differential geometry - Showing a manifold is smooth by constructing a diffeomorphism

Let $M$ be a smooth manifold, and $N$ be a manifold that may or may not be smooth. Let $f:M \rightarrow N$ be a homeomorphism. If I can show that both $f$ and it's inverse are smooth, i.e. $f$ is a diffeomorphism, does that mean $N$ is necessarily a smooth manifold?I raise this question because from every source I read, whenever one has a diffeomorphism between two manifolds, they are assumed to be smooth. One stackexchange post said that it makes no sense to construct a diffeomorphism between non-smooth manifolds, just like it makes no sense t...Read more