# Tagged Questions

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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### Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for all....
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### Which manifolds are parallelizable?

Recall that a manifold $M$ of dimension $n$ is parallelizable if there are $n$ vector fields that form a basis of the tangent space $T_x M$ at every point $x \in M$. This is equivalent to the tangent ...
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### Is there an easy way to show which spheres can be Lie groups?

I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it ...
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### Is $M=\{(x,|x|): x \in (-1, 1)\}$ not a differentiable manifold?

Let $M=\{(x,|x|): x \in (-1, 1)\}$. Then there is an atlas with only one coordinate chart $(M, (x, |x|) \mapsto x)$ for $M$. We don't need any coordinate transformation maps to worry about ...
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### Determining the angle degree of an arc in ellipse?

Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ? If I have an arc length at the first quarter of an ellipse and ...
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### Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm ...
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### Why are smooth manifolds defined to be paracompact?

The way I understand things, roughly speaking, the importance of smooth manifolds is that they form the category of topological spaces on which we can do calculus. The definition of smooth manifolds ...
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### Smooth surfaces that isn't the zero-set of $f(x,y,z)$

The zero-set of any smooth function $f(x,y,z)$ with a non-vanishing gradient is a smooth surface. I was wondering if the reverse is true: is every smooth surface in $E^3$ the zero-set of some smooth ...
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### On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
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This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given $x,y\in\... 2answers 1k views ### Elementary proof of the fact that any orientable 3-manifold is parallelizable A parallelizable manifold$M$is a smooth manifold such that there exist smooth vector fields$V_1,...,V_n$where$n$is the dimension of$M$, such that at any point$p\in M$, the tangent vectors$V_1(...
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How can I prove that a connected Lie Group is generated by any neighborhood of the identity? The result is almost trivial for $R^n$ but I tried using the open subgroup generated by this neighborhood....
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### Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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### Is every Compact $n$-Manifold a Compactification of $\mathbb{R}^n$?

I read the result that every compact $n$-manifold is a compactification of $\mathbb{R}^n$. Now, for surfaces, this seems clear: we take an n-gon, whose interior (i.e., everything in the n-gon except ...
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### How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
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### An application of partitions of unity: integrating over open sets.

In Spivak's "Calculus on Manifolds", Spivak first defines integration over rectangles, then bounded Jordan-measurable sets (for functions whose discontinuities form a Lebesgue null set). He then uses ...
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### Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus): the ...
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### What is a covector and what is it used for?

From what I understand, a covector is an object that takes a vector and returns a number. So given a vector $v \in V$ and a covector $\phi \in V^*$, you can act on $v$ with $\phi$ to get a real number ...
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### Introductory texts on manifolds

I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of. I was wondering if ...
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### Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
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### Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
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### The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
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### Vector bundle transitions and Čech cohomology

I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ (...
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### Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
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### What is the practical difference between abstract index notation and “ordinary” index notation

I understand that in "normal" index notation the indexes can be thought of as coordinates of scalar values inside a tabular data structure, while in the abstract index notation they can not. However, ...
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### Properly Defining a Smooth Curve

I have seen many different definitions of what it means for a curve to be "smooth". In this question, for instance, a curve $\gamma \colon [a,b] \longrightarrow \mathbb{R^n}$ is defined to be smooth ...
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### Are all connected manifolds homogeneous

A topological space $X$ is called homogeneous, if for every two points $x,y \in X$ there exists a homeomorphism $\phi : X \rightarrow X$ s.t. $\phi(x) = y$. It is not hard to prove that all connected ...
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### Curves with constant curvature and constant torsion

Describe all curves in $\mathbb{R}^3$ which have constant curvature $κ > 0$ and constant torsion $τ$. Any ideas what we can do to describe all such curves? Do we have to use the formulas of the ...
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### Why do we need Hausdorff-ness in definition of topological manifold?

Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant. ...
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### Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
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### surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism

Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism? I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to ...
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### What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve (...
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### Curvature of planar implicit curves

I am trying to understand how the curvature equation $$\kappa = -\frac{f_{xx} f_y^2-2f_{xy} f_x f_y + f_x^2 f_{yy}}{(f_x^2+f_y^2)^{3/2}}$$ for implicit curves is derived. These curves arise from ...
### Does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?
Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$. As the question title suggests, does there exist a $1$-form $\alpha$ with $d\alpha = \omega$?