# Tagged Questions

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### A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
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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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### A question about the existence of a smooth function [duplicate]

Does there exists a smooth function $f: R^2 \rightarrow R$, such that $f(x,y)\ge0$, for any $(x,y) \in R^2$, and $f$ has exactly two critical points $(x_1,y_1), (x_2, y_2) \in R^2$ with ...
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### Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
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### Prove no existing a smooth function satisfying … related to Morse Theory

i) Show that there does not exist a smooth function $f:\mathbb{R} \rightarrow \mathbb{R}$, s.t. $f(x) \geq 0$, $\forall x \in \mathbb{R}$, $f$ has exactly two critical points, $x_1,x_2\in\mathbb{R}$ ...
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### Reference requestion: Existence/construction of bump functions

I'm not much of an analyst myself, but I've time and time again come across proofs which require knowledge of the existence of bump functions. However, I've never studied them, so I'm missing ...
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### One example of open map which is not submersion. [closed]

It is hard to find one open map which is not submersion? Give me one example.
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### English edition of Vol 9 of Dieudonné's Foundations of Modern Analysis?

I have found the first 8 volumes of Dieudonné's Foundations of Modern Analysis in English translation, but I'm having difficulty locating volume 9. I have searched the catalogues of numerous libraries ...
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### compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
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### Crosscap function in $\mathbb{R}^4$ - and how to show it is proper?

I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$ My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue): Q1: Is ...
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### Is the classical derivative the weak derivative on any domain?

I was looking at the following definition of the weak derivative: Let $\Omega$ be a domain (ie an open connected subset of $\mathbb{R}^n$). Suppose $u,v \in L_{1,loc}(\Omega)$ and ...
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### Is it possible to learn differential topology before analysis?

Currently I'm self studying for my own enjoyment topology and algebra (munkres and herstein). Since I start at the university next year everything I'm learning now is for my own enjoyment and I will ...
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### What is basis of $\mathbb{R}$

I think it is just 1; but I am also under the impression that it is just any open interval on $\mathbb{R}$. Furthermore, I am trying to figure out how a compact interval $X = [0,1]$ inherite standard ...
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### Prove that a compact cone is not diffeomorphic to the 2-sphere

In Tapp's "Matrix Groups for Undergraduates" he briefly states (p.103) that a compact cone (he just shows a picture of a manifold with a ''cone point'') is not diffeomorphic to a 2-sphere. I would ...
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### Is [0,1] closed?

I thought it was closed, under the usual topology $\mathbb{R}$, since its compliment $(-\infty, 0) \cup (1,\infty)$ is open. However, then then intersection number would not agree mod 2, since it can ...
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### On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
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### Prove that a standard torus is diffeomorphic to $\mathbb S^1\times \mathbb S^1$

I was asked to prove that a standard torus(which means we don't consider those pathological cases where it intersects with itself, e.g horn torus) is diffeomorphic to $\mathbb S^1\times \mathbb S^1$. ...
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### Show that the tangent space of the diagonal is the diagonal of the product of tangent space

I'm stuck on this question for quite a few days and still haven't got a clue what to do. The question is as follows: If $\Delta$ is the diagonal of $X\times X$ where $X$ is a manifold, show that its ...
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### Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k

In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
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### Parametrization of $n$-spheres

This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$). I ...
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### A surjective map which is not a submersion

Is there an example of a smooth map between smooth manifolds which is surjective, but not a submersion? I feel there can't be one, but don't know of a proof. Nor do I know of a counter-example. ...
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### Why is $\partial\partial M=\varnothing$?

Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
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### Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$

Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
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### How does degree theory imply that this mapping $f$ is locally onto?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth vector field ($\mathcal{C}^1$ mapping). Let $0$ be a critical point of $f$, i.e. $H f(0) = 0$. Assume that the index of $f$ at $0$ is ...
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### Topological degree of a map with finite energy

Suppose that $\phi:\mathbb{R}^3 \to S^2$ is of class $\mathscr{C}^1(\mathbb{R}^3\setminus \left\{a\right\}) \cap \mathscr{C}^0(\mathbb{R}^3\setminus \left\{a\right\})$, that is $\phi$ might have a ...
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### Extending a $C^2$-function from a $C^{1,1}$-curve to some neighbourhood

Suppose I have a simple, compact $C^{1,1}$-curve $L$ in $\mathbb{R}^3$ and a $C^2$-function $f$ on it ($C^2$ meaning with two continuous arclength derivatives). Can it be extended to a $C^2$-function ...
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### Smooth maps between Euclidean spaces

There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
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### Approximate continuous mapping by smooth mappings on manifold(Bott, Tu book)

So I was looking at the proof given in Bott, Tu "Differential Forms in Algebraic Topology" of how to approximate continuous mapping by smooth mappings between manifolds. It is Proposition 17.8 on Page ...
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### Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
I've tried a few methods but I can't seem to work this one out. Consider the charts $$f(s) = (\cos s, \sin s) \in \mathbb{R}^2$$ for $-\pi < s < \pi$ and g(t)=(\frac{2t}{t^2 + 1}, \frac{t^2 ...
It's know that if $M$ is a compact, smooth manifold of dimension $n$ and the map $f: M \to \mathbb{R^m}$ is injective, smooth, $n \le m$ and $Jf(a)$, the Jacobian, has rank $n$ for every $a \in M$, ...