Tagged Questions

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Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
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Intuition for the compactness of real projective space $\mathbb{R}\mathbb{P}^n$.

I want to have an intuition for why the $n$-dimensional real projective space defined as $$\mathbb{R}\mathbb{P}^n:=\mbox{set of 1-dimensional subspaces of }\mathbb{R}^{n+1}$$ is compact. I don't see ...
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Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
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About $C^{0}$ being topological manifold

Is that the reason why $C^{0}$ being topological manifold due to that $C^{0}=\phi$ which contains nothing? Correct me if I am wrong. I am new to differential topology.
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Smooth embeddings of the $2$-sphere

I have a past qual question here: given a smooth embedding $f \colon S^2 \to \mathbb{R}^3$, show that there must exist distinct points $p,q \in S^2$ such that the tangent planes to the embedded sphere ...
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Are the topology of a manifold and the topology induced by the metric of a manifold the same?

I am a physics student trying to understand in a rigorous way what a manifold is so please bear with me. Ok, so I am just learning what a topology is and from what I have understood up till now is ...
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Which part of differential geomety uses metrization theorems?

I learned three metrization theorems last year, which are Nagata-Smirnov,Smirnov and Bing. I thought these theorems are purely topological theorems, but i recently saw a post which says these ...
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Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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Embedding of $\mathbb{R}^2 \to \mathbb{R}^3$ with non-parallel tangent planes

I have a qual question here and I'm struggling to get a good starting point. The question asks to construct a smooth proper embedding $f\colon \mathbb{R}^2 \to \mathbb{R}^3$ such that for any distinct ...
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Is there a nice/clever way to visualize $\mathcal{S}\times \mathbb{R}^2$?

The (velocity) phase space of a double pendulum can be seen as the tangent bundle of its configuration space ($\mathcal{S}^1\times\mathcal{S}^1$), that is: ...
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Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
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Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
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Topologies on n-manifolds

In the study of n-manifolds (real and imagined), is there any reason to spend much time learning about topologies other than the usual topology?
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Alexandrov embedded disc

We say that a compact surface $\Sigma$ is Alexandrov embedded via an immersion $f:\Sigma \rightarrow \mathbb{R}^3$ if there exist, $X$ a 3 manifold and an immersion $F: X \rightarrow \mathbb{R}^3$ ...
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Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
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Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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So I understand the definition of an alternating form on $\mathbb{R}^m$, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so?
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What is the difference between a manifold and a topological manifold?

Is it the case that a topological manifold is just a topological space but we have not mentioned any specific structure on it ?
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Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
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Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$\mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx}$$ ...
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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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Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
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Integrable Manifolds

I'm trying to understand why the line of slope y passing through (x,y) is an integral manifold. My intuition tells me that there exists a point in the slope field where the distribution cannot be ...
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Link between a topological space and a manifold

A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ...
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n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
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How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
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Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
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Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...