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

Proof in do Carmo's "Differential Forms and Applications"

I've trying to figure this out for a while and I am finally desperate enough to post to stack exchangeIn do Carmo's Differential Forms and Applications Proposition 2 on pg 92 (this is the proof that the Gaussian curvature is well-defined, independent of choice frame and coframe). I post the statement and proof for context: Proposition 2 Let $M^2$ be a Riemannian manifold of dimension two. For each $p\in M$, we define a number $K(p)$ by choosing a moving [orthonormal] frame $\{e_1,e_2\}$ around $p$ and setting $$d\omega_{12}(p):=-K(p)(\omega_1...Read more

exterior algebra - Divergence of a vector field and differential forms

I just started learning differential geometry, and my knowledge on exterior algebra is basically nonexistent. While reading about the Invariant Stokes theorem, I encountered this lemma: If F = (f1, f2, f3) is a vector field in ℝ3, then div(F) = ∇ · F = ∗d(∗F♭)The document where I found this does not provide a definition of *, nor of ♭. I'm having a hard time digesting all this notation... What do these symbols mean? Is * related to the wedge product? If so, how? What does this mean in terms of differential forms? Thank you for your time! (Plea...Read more

soft question - Selecting Differential Geometry Exercises

I'm self-studying differential geometry with Do Carmo's books "Differential Geometry of Curves and Surfaces" and "Riemannian Geometry" and I find those books very good, however I feel a little confused when selecting which exercises to do. What's the best way to select exercises when studying that kind of math? I know this question seems silly, it's like : "how can someone don't know which exercises to do?", but it's just the case that there's no time to work on all of them, so I feel a little confused in which to work more.Thanks in advance, a...Read more

Statistical Inference, Differential Geometry and Entropy

Context: Statistical Inference and Differential Geometry Let's consider a generic $ p(x;\theta) $ distribution with $ \theta $ Parameters Vector, it is obvious that $$ \int p(x; \theta) dx = 1 $$and by definition it should be exactly 1 for any value of $ \theta $ hence $$ \frac{\partial}{\partial \theta_{i}} \int p(x; \theta) = 0 $$Then now let's consider the negative entropy of the distribution (the expected value of the log distribution)$$ S = E_{\theta}\left [ \ln(p(x; \theta)) \right ] = \int \ln(p(x;\theta)) p(x; \theta) dx $$applying the ...Read more

differential geometry - Basic question about definition of Chern classes

Apologies if there is something I missed with a quick internet search, but why do we define Chern classes for complex vector bundles (instead of real vector bundles for example)?If we define chern classes using homotopy theory, finding a natural transformation from between ${\rm Vect}_{n}(\cdot)$ and $H^{i}(\cdot;R)$ in the case of real vector bundles reduces to computing the cohomology ring of the base space of the universal real $n$-plane bundle (which I think is the real Grassmannian) by the Yoneda embedding, just like in the complex case. I...Read more

differential geometry - Chern Weil theory-user friendly guide

I would like to learn basics about Chern Weil theory but I don't know from which place I should start. I'm after rather basic differential geometry course and have some background in algebraic topology (ordinary homology and cohomology, characteristic classes: Chern, Pontraigin, Stiefel Whitney) but I would like to see geometrical interpretation of the theory of characteristic classes. I will be grateful for any sugestions...Read more

differential geometry - How to prove Lagrange multiplier theorem in a rigorous but intuitive way?

Following some text books, the Lagrange multiplier theorem can be described as follows.Let $U \subset \mathbb{R}^n$ be an open set and let $f:U\rightarrow \mathbb{R}, g:U\rightarrow \mathbb{R}$ be $C^1$ functions. Let $\mathbf{x}_{0} \in U, c=g(\mathbf{x}_0)$, and let $S$ be the level set of $g$ with value $c$. Suppose that $\nabla g(\mathbf{x}_0) \neq 0$. If the function $f|_S$ has a local extremum at $\mathbf{x}_0$ then there is a $\lambda \in \mathbb{R}$ so that $\nabla f(\mathbf{x}_0)=\lambda g(\mathbf{x}_0)$Some textbooks give a strict pro...Read more

Who first had the idea to study surfaces via rings of functions, as in algebraic geometry?

This idea provides the foundations of algebraic geometry now; and they have certainly gone down the rabbit hole with it. As a student studying this subject, I have always found it such a great leap to think that some ring of functions could have such a strong influence on geometry. Given the idea, it seems natural. But to be the one that first had the idea, that seems a great leap. Does anyone have any information about the history of this idea? Who first thought about? Perhaps why they thought about it? I believe the answer is Riemann, when s...Read more

differential geometry - Spherical circle - Area

I am looking at the following exercise: The spherical circle of centre $p \in S^2$ and radius $R$ is the set of points of $S^2$ that are a spherical distance $R$ from $p$. If $0 \leq R \leq \frac{\pi}{2}$ a spherical circle of radius $R$ is a circle of radius $\sin R$. Show that, if $0 \leq R \leq \frac{\pi}{2}$, the area inside a spherical circle of radius $R$ is $2\pi (1 − \cos R)$. $$$$ To find the area inside a spherical circle of radius $R$ do we have to calculate the area inside a circle of radius $\sin R$ ?...Read more

differential geometry - Curvature of projection function onto smooth curve

Suppose we have a smooth curve $C$ lying in $\mathbb{R}^2$, and let us consider the orthogonal projection function $P_C(x)$ onto the curve, described by $$P_C(x) = \arg\min_{y \in C} \Vert x - y \Vert$$where $\Vert \cdot \Vert$ is a norm, it can be $\Vert \cdot \Vert_2^2$, or $\Vert \cdot \Vert_1$.My question is: is there a general relationship between the second derivative of $P_C(x)$ and the curvature of the curve $C$? For example, relationship between the norm, whether it is "positive definite", etc. If no, under what restrictions on the cur...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