# Tagged Questions

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### Existence of critical points of $f:\mathbb{C} -\{0,1\}\to \mathbb{R}$

I am trying to show that an smooth, proper map, $f:\mathbb{C} -\{0,1\}\to \mathbb{R}$ has a critical point. My attempt was to suppose there are no critical points, then the preimage of every point is ...
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### How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...
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### Integration on Manifold

I am beginning my studies on integration on manifolds and i have some theorical questions. First, in all books that I saw they says that the singular p - simplex (or p - cube) are continuous mapping ...
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### Cap-Independence

I'm just trying to figure out something regarding Cap-Independence. The problem reads $\partial S:= r(t)=(\cos t,\sin t,\sin 2t)$, $0\le t \le 2\pi;$ $\phi=z\,dzdx-6y^2dxdy$ ($\partial S$ ...
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### Spivak Calculus on Manifolds, problem 1-2

I am confused about the hint Spivak adds to problem 1-2 in his Calculus on Manifolds: When does equality hold in Theorem 1-1(3)? Hint: Re-examine the proof; the answer is not “when $x$ and $y$ ...
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### Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$.

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$. My try : By definition of derivative of a function $f:\mathbb{R^n} \to \mathbb{R^m}$ , If I know ...
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### Triangle a manifold

Let $x,y,z \in \mathbb{R}^3$ and $\Delta:=\text{conv} \{x,y,z\}$ be a triangle. My question is: Is this triangle a $C^2$ submanifold in $\mathbb{R}^3$? The reason is, that I would need this fact in ...
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### Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
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### Question on Construction in Spivak's *Calculus on Manifolds*, induced transformations

First I quote the relevant passage (page 89): If we consider now a differentiable function $f : \mathbb R^n \to \mathbb R^m$ we have a linear transformation $Df(p): \mathbb R^n \to \mathbb R^m$. ...
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### Orientability of surfaces in $\mathbb{R}^3$

I have a question: I'm currently reading a few things about orientability and understood this concept as the answer to the question: Given a surface and a unit normal vector field on it: Is there a ...
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### The unsolved extension problem on manifolds.

I have been struggeling for quite a while with this problem: Let $M \subset \mathbb{R}^n$ be a compact $C^k-$ submanifold and $\phi_i: B_i(0) \rightarrow M$ be the associated set of charts ...
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### How do 1-d compact submanifolds look like?

I was wondering whether it is true that every 1-d compact submanifold $\subset \mathbb{R}^n$ that is $C^1$ is a closed curve that is also $C^1$, cause I cannot think of more examples. Therefore, I ...
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### How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
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### Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
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### Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
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### What's the connection between derivatives and boundaries?

The (second) fundamental theorem of calculus says that $$\int_a^b f'(x) dx = f(b) - f(a)$$ which can also be stated, if one knows enough about what's coming next, as: The integral of the ...
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### Is $[0,1]$ an *oriented* manifold with boundary? (and Stokes theorem)

The definitions I am using are a manifold with boundary is something locally homeomorphic to $(0,1] \times \mathbb{R}^n$ or $\mathbb{R}^n$. an oriented manifold is one where the transition functions ...
The pushforward of maps between smooth manifolds is defined as follows: If $f: M \to N$ and $a \in C^\infty(N)$, then $Tf: TM \to TN$ takes $v \mapsto Tf(v)$ which operates on functions on $N$ as ...