# Tagged Questions

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### $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
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### Estimate for boundary points and exterior normal vector of bounded domain of class $C^2$

Consider a bounded open set $\Omega\subset\mathbb{R}^d$, s.t. the boundary set $\partial \Omega$ is a manifold of class $C^2$. Let $x,x_0\in\partial\Omega$ be boundary points and $\nu_x$ the exterior ...
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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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### Does a chart $\varphi$ of $S^2$ exist with $S^2=\varphi(T)$?

Let $n,k\in\mathbb N$ with $1\le k\le n$. A $k$-dimensional ($C^1$-) submanifold of $\mathbb R^n$ is a non-empty set $M\subseteq\mathbb R^n$ with the property that for all $a\in M$ there exists an ...
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### Can a 1-manifold $M$ without boundary in $\mathbb{R}^2$ have a “corner”?

If $α:U→V$ is a coordinate patch on a manifold without boundary, $M$, about the point $p$, one of the conditions on $α$ is that $Dα(x)$ have rank 1 for each $x \in U$. According to Munkres in Analysis ...
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### Show $M_1\cap M_2$ submanifold iff $N_x(M_1)\cap N_x(M_2) = \{0\}$ and dimensions

Let $M_1,M_2 \subseteq \mathbb{R}^3$ two-dimendional submanifolds of $\mathbb{R}^3$ such that for every point $x\in M_1\cap M_2$ $$N_x(M_1)\cap N_x(M_2) = \{0\}.$$ Show $M_1\cap M_2$ is an ...
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### one problem on multivariable claculus

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes ...
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### Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
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### Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
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### Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
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### Horn and spindle tori

I was trying to prove that the horn torus and the spindle torus are not manifolds by definition(locally diffeomorphic to some Euclidean space.). I have no idea how to do this, but I attempted it in ...
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### Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
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### Smooth maps on a manifold lie group

 \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
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### The euclidean space $\Bbb R^n$ is orientable as a manifold.

I know that The euclidean space $\Bbb R^n$ is orientable as a manifold. I think that it is orientable because it has a nowhere vanishing $n$-form. But I am not sure. Please can you explain ...
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