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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

Differential geometry is a branch of mathematics that uses techniques in calculus and linear algebra to study geometry. It is closely related to differential topology and PDEs (geometric analysis).

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I don't get the relationship between differentials, differential forms, and exterior derivatives.

I don't get the relationship between differentials, differential forms, and exterior derivatives. (Too many $d$'s getting me down!) Here are the relevant (partial) definitions from Wikipedia; essentially the same definitions/terminology/notations…
goblin GONE
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Computing the Chern-Simons invariant of $SO(3)$

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and Simon. I will recount the examples and my…
Alex Troesch
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is there any good resource for video lectures of differential geometry?

I am wondering if there is some online resource for video lectures on the topic of differential geometry. Thanks a lot
Shan
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What is a Manifold?

Now we always encounter definition of a manifold from a mathematical point of view where it is a topological space along with a family of open sets that covers it and the same old symphony. My question is from your own expertise and from what you…
PhilosophicalPhysics
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What shape does a piece of paper make when it is pushed from the edges?

When I push a piece of (A4) paper oriented landscape to me from the shorter edges, it makes a pretty shape, resembling a bell-curve. I seem to remember these sort of situations being a motivation for or concrete instance of some theorems in…
Hugh
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Geometric intuition for the Weingarten map

Parameterize a hypersurface $M$ by $r: \Omega \rightarrow \mathbb{R}^n$, and let $T_p M$ denote the tangent space at $p = r(u)$. We define the Weingarten map to be the linear map $L_p : T_p M \rightarrow T_p M$ given by $$L_p(v) = -\partial_v…
user901823
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The Taylor expansion of the metric at the origin in geodesic coordinates

It is well known that in geodesic coordinates we have $$ g_{ij}=\delta_{ij}-\frac{1}{3}\sum_{k,l}R_{ijkl}x^{k}x^{l}+O(|x|^{3}) $$ I have been trying to find a rigorous proof of it, but I cannot find a readable proof online (see this one, for…
Bombyx mori
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A smooth vector field on $S^2$ vanishing at a single point

I need to find a vector field as described in the title. I was given a couple hints, and this is what I have so far. Let $\varphi:S^2\setminus\{N\}\to\mathbb{R}^2$ be stereographic projection ($N$ is the north pole). Let $Y$ be a smooth, non-zero…
Bey
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Showing a subset of the torus is dense

Let $T^2\subset\mathbb{C}^2$ denote the (usual) torus. Let $a\in\mathbb{R}$ be an irrational number and define a map $f(t)=(e^{2\pi it}, e^{2\pi i at})$. Prove that: (a) $f$ is injective (b) $f$ is an immersion, but not an embedding (c)…
Bey
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Why is Cartan Formula just an avatar of Leibniz rule?

In this video, Arnold says that the Cartan formula $$ \mathscr L_{\mathrm X} = d i_{\mathrm X} + i_{\mathrm X} d$$ is just an avatar of $(fg)' = fg' + f'g$. Why ?
Damien L
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Isometric embedding

I am confused by the term "Isometric Embedding". To my knowledge, this refers to a distance preserving map from a space to another (a mapping $f:(E,d_1) \to (F,d_2)$ such that $d_2(f(x_1), f(x_2)) = d_1(x_1, x_2) )$. But I have the following problem…
WhitAngl
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Reading list of differential geometry **papers**

First of all, it is not a duplicate of those questions a la "best textbooks for differential geometry". What I want to ask is which papers would you recommend reading while taking (self learning) differential geometry courses? I could list (and find…
Tomas
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Calculate the Lie Derivative

In trying to get to grips with Lie derivatives I'm completely lost, just completely lost :( Is there anyone who could provide an example of calculating the Lie derivative of the most basic function you can, i.e. like in showing someone how to…
bolbteppa
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Proving that the general linear group is a differentiable manifold

We know that the the general linear group is defined as the set $\{A\in M_n(R): \det A \neq 0\}$. I have a homework on how to prove that it is a smooth manifold. So far my only idea is that we can think of each matrix, say $A$, in that group as an…
John Thompson
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What is the theory of non-linear forms (as contrasted to the theory of differential forms)?

It is often said that differential forms (sections of an exterior power of the cotangent bundle) are the things that you can integrate. But unless I'm being thoroughly dense differential forms are not the only things that you can integrate, c.f. the…
Vladimir Sotirov
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