# Tagged Questions

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### Tangent bundle is orientable

I am having some trouble finishing a proof that the tangent bundle of any manifold is orientable. What I've done so far is calculate the transition function between two standard charts on the bundle. ...
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### The graph of $x\mapsto |x|$ cannot be the image of an immersion.

How can one prove that the set $\{(x,|x|)\in \mathbb{R}^2 \mid x\in \mathbb{R}\}$ cannot be the image of an immersion of a smooth manifold? This was my homework exercise in a course about ...
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### Extension of a bounded operator on manifold

I have a problem, which is quite urgent. I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. I wish to extend this operator to an ...
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### Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
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### Question about vector fields and Lie group

Notation: $\chi(G)$ is the set of smooth vector fields on Lie group $G$, which in fact forms a vector space. Given a Lie group $G$, show that there exists a smooth vector field $X\in \chi(G)$, ...
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### General Linear Group is(not) compactly generated

We know that any connected Lie Group is compactly generated. I have a feeling that $\mathbb{GL}_n\mathbb{R}$ is not compactly generated. Is it true? If it is how can I prove this? If not, what ...
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### Canonical isomorphism between complexified tangent space of submanifold fixed by antiholomorphic involution and tangent space of complex manifold

I haven't really studied complex manifolds and I am at a bit of a loss in regards how to approach this problem: Let $M$ be an $n$-dimensional complex manifold, and let $\phi:M\rightarrow M$ be an ...
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### Show that a set is a manifold.

Let $n \ge 3$. How can I show that $M:= \{(x_1,...,x_n) \in \Bbb R^n \setminus \{(0,...,0)\} | x_1^2+...+x_n^2 = x_1 \cdot...\cdot x_n \}$ is a manifold of class $C^1$? Can anyone please tell me ...
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### How to verify that this is a submanifold

Let $g: \mathbb{R}^2 \to \mathbb{R}^2$ , $g (x, y) = (x^2-y^2, y)$ be a differentiable map. Let $r$ the line passing through $(1, 0)$ parallel to the $y-$axis. Prove that $g^{-1}(r)$ is a ...
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### $V$ vector field, $\omega$ one-form, $V(\omega(V))$=?

(1-forms) Let $X$ be a manifold and $\omega \in \Omega^1(X)$ be a smooth 1-form, and $V, W \in V^{\infty}(X)$ smooth vector fields on $X$. Then, $\omega(V ), \omega(W ) \in C^{\infty}(X)$ are ...
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### $df$ vanish in a compact manifold in at least 2 points

I need to prove that if $M$ is a compact manifold and $f$ is a smooth function in $M$, then $df$ vanish in at least 2 different points of $M$. I don't know where to start. Any suggestion will be ...
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### why f(M) is sub manifold

Let map f of M into N be an injective immersion. show taht if M is compact then f(M) is submanifold of N.
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### Subset of non-units of germs of smooth functions at $x$ is an ideal

For a manifold $X$ with a point $x \in X$ define the ring of germs of the smooth functions at $x$ to be $C^{\infty}_x(X)=C^{\infty}(X)/\sim$ where $f_1 \sim f_2 \iff \exists U \in \tau_X.f_1|U=f_2|U$ ...
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### Manifold non-orientable iff. frame bundle is connected

Let $M$ be a connected smooth manifold and $L(M):=\bigcup_{x\in M}L_xM$ its frame bundle where $L_xM:=\{(v_1,\dots,v_n):\{v_1,\dots,v_n\}\text{ is a basis of }T_xM\}$. $M$ is non-orientable iff. ...
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### Prove that $g^{-1}(0)$ is a $n$-dimensional manifold.

Let $A\subset \mathbb R ^n$ be open and let $g:A\to \mathbb R ^p$ be a differentiable function such that $g'(x)$ has rank $p$ whenever $g(x)=0$. Then $g^{-1}(0)$ is an $(n-p)$-dimensional manifold. ...
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### how to calculate the curvature of an ellipse

how can I compute the curvative of an ellipse given by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ do i need to take $x=acos(t)$ and $y=bsin(t)$? please show me a way how to solve this? thank you for ...
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### Vector field on an odd sphere

Let $x^1,y^1,\ldots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$. The unit sphere $S^{2n-1}$ in $\mathbb{R}^{2n}$ is defined by the equation $\sum_{i=1}^n(x^i)^2+(y^i)^2=1$. Show that ...