1
vote
0answers
39 views

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. ...
5
votes
1answer
53 views

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 ...
0
votes
0answers
22 views

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 ...
0
votes
2answers
44 views

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?
2
votes
1answer
58 views

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 $X\in \chi(G)$, ...
0
votes
1answer
30 views

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 ...
1
vote
1answer
35 views

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 ...
0
votes
1answer
41 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
0
votes
1answer
42 views

Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$ h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
3
votes
2answers
94 views

Is this set a manifold?

For which $ ( \alpha , \beta ) \in \Bbb R^2$ set: $\{ (x_1,x_2,x_3,x_4) \in \Bbb R^4 | x_1+x_4= \alpha, x_1 x_4 - x_2x_3 = \beta \}$ is a manifold? I made a Jacobian matrix: $ \begin{bmatrix} ...
1
vote
1answer
58 views

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 ...
3
votes
1answer
47 views

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 ...
1
vote
1answer
95 views

$V$ 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 ...
1
vote
1answer
61 views

$df$ 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 ...
1
vote
0answers
23 views

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.
0
votes
1answer
31 views

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 $f_1 \sim f_2 \iff \exists U \in \tau_X.f_1|U=f_2|U$ ...
2
votes
2answers
88 views

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. ...
1
vote
1answer
52 views

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. ...
0
votes
1answer
685 views

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 ...
2
votes
0answers
40 views

problem with submersion

Given $\varphi:\mathbb{R}^{m+n}\longrightarrow \mathbb{R}^m$ is $C^{ k}$ class. If there $a\in \mathbb{R}^{m+n}$ with $\varphi^{\prime}(a)$ is surjective. Then there a mergullo $f:V\longrightarrow ...
1
vote
2answers
56 views

Please checking to find an arc-length reparametrization

Find an arc-length reparametrization of $$c(t)=\langle \cos t+t\sin t, \sin t-t\cos t\rangle$$ for $t\in [\pi, 3\pi/2]$ solution trial: $$c'(t)=\langle -\sin t+\sin t+t\cos t, \cos t-\cos t+t\sin ...
4
votes
1answer
99 views

Defining a quotient manifold with gluing

I'm trying to find conditions on the gluing map between two manifolds so that the quotient space will be a smooth manifold, and the inclusion map will be a diffeomorphism. Specifically, Suppose ...
0
votes
0answers
58 views

Showing Gr(n,k) is a manifold.

Ok so this one needs a lot of set up: Define an atlas $\mathcal{A}$ on a set $M$ to be a collection $$ \mathcal{A}=\{(U_i,\phi_i,E_i)\,:\, i\in I\} $$ Where $U_i$ is a subset of $M$, $E_i$ is an open ...
1
vote
1answer
249 views

Constant Rank Theorem and Submanifolds

I'm related to my previous question here. The problem is: I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that ...
2
votes
0answers
123 views

Is this enough to show that this map has constant rank?

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is ...
0
votes
1answer
70 views

Can anyone check my proof that $H^1(\Sigma-\{p\})=0$ for a compact and orientable surface $\Sigma$?

I have the following problem: Let $\Sigma$ be a compact and orientable surface. Show that $H^1(\Sigma-\{p\})=0$ for every $p\in \Sigma$. Can anyone check my proof and give suggestions? Sketch of ...
2
votes
0answers
140 views

Manifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence?

Let X be an n-dimensional differentiable manifold and $p \in X$ . Let $(U, h, V )$ for X around p with coordinates $(x_1 , . . . , x_n )$ in V , and let $v_i , i = 1, . . . , n$ , be the basis of ...
2
votes
0answers
58 views

Stoke's theorem application to curl theorem. I did. Please can you check it?

Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$ $dim(M)=2$ M is the subset of $\Bbb ...
0
votes
2answers
125 views

Topological manifold example

$\theta(x,x^2)=x$ $\Bbb X =${$(x,x^2)| x$ in $\Bbb R$} And V is subset of $\Bbb R$ $dim\Bbb X=1$ My instructor said that this is topological manifold. Why? Please can you explain me? This ...
2
votes
1answer
155 views

Why is the cylinder surface on $\Bbb R^3$ orientable?

Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?
1
vote
1answer
368 views

Orientation preserving diffeomorphism.

I am stuck with the question. I guess that I need to write jacobian matrix. But I could not do. Please help me thank you
1
vote
1answer
163 views

Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.

Suppose $f(x_1,...,x_{n+1})$ is a$ C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
1
vote
1answer
211 views

How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.

Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$. We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$. How to decide whether F is orientation-preserving or ...
2
votes
1answer
92 views

$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$

Let $F : \Bbb R^2 → \Bbb R^2$ be given by If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$, compute $F^{∗}(u \, du+v \, dv)$. $$F(x,y) = (x^2 +y^2,xy).$$ I am confused so much. I ...
0
votes
1answer
45 views

How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.

I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
0
votes
1answer
71 views

Find a $1$-form $ω$ on $\mathbb R^2 −\{(0,0)\}$ such that $ω(X) = 1$ and $ω(Y) = 0$.

Please ı dont know what I need to do. thus, help me to solve.
0
votes
1answer
38 views

prove that $supp(π^{∗} f) = (supp f)×N.$ Please can you check my answer? Also more explanation please.

My question is that Let $f \colon M \to R$ be a $C^{\infty}$ function on a manifold $M$. If $N$ is another manifold and $π \colon M \times N \to M$ is the projection onto the first factor, prove ...
1
vote
0answers
110 views

Deformation retract

How to prove that $r_t$ is a deformation retract $M^a=\lbrace q\in M ; f(q)\leq a\rbrace$ We have the definition : $r_t$ is a difformation retract if: $r_t$ is a continius ,onto application ...
3
votes
1answer
256 views

Question about theorem 3.2 from Morse theory by Milnor

The demonstration of the theorem 3.2 in the book Morse theory by Milnor THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index ...
2
votes
1answer
177 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
-2
votes
1answer
919 views

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 ...
1
vote
1answer
62 views

Show the regular submanifold

Please help me how sdo I show such a problem? I Will be happy to teach me. Thank you
3
votes
1answer
114 views

Smooth Structure of the Torus

Consider the torus $T^2=S^1\times S^1$(where $S^1$ is the unit circle centered at $0$ in $\mathbb C$). Define a smooth structure on $S^1$ and $T^2$. ($\checkmark$) Let $f:T^2 ...
3
votes
1answer
181 views

Lemme 2.4 in Morse theory by Milnor

This is lemma 2.4 from "Morse theory" by Milnor ,with the prove I have some questions about this prove : 1) why $\displaystyle\frac{dc}{dt}(f)=\lim_{h\rightarrow 0} \frac{fc(t+h)-fc(t)}{h}$ and ...
0
votes
4answers
74 views

Why $GL(n+1,\mathbb{C})$ is compact?

I'm trying to prove that: The set of all lines in $\mathbb{C}^{n+1}$ ($\mathbb{C}\mathbb{P}(n)$) is a complex manifold. I'm knowing that: If a compact group $G$ acts on $X$ transitively and ...
0
votes
2answers
88 views

Lie bracket in local coordinates. Find the formula $c^{k}$ in terms of $a^{i}$ and $b^{j}$

This is from T.U Loring's manifold book. I tried. But I didnt do the question. Please show me how to solve instructively and explicitly. I want to learn this topic. Thank you for help.
1
vote
2answers
92 views

Diffeomorhism of manifold

This is one of the exam questions of the previous semester. I have studied these. But I didn't do this. Please show me how to solve this question. Thank you for help
2
votes
1answer
356 views

Tangent space at the identity element of a lie group

Let G be a lie group . we know a Lie group is a group with a smooth manifold structure s.t both the multiplication map $m$ and group inversion map $i$ are smooth . Now by identifying ...
0
votes
1answer
69 views

Problem about tangent vector and the inclusion map of the unit circle.

It is so complecated for me. Please can you show how to solve. Thank you.
0
votes
1answer
73 views

The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$

Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?