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Dimension of a smooth manifold

I believe you mean "$W$ neighbourhood of $x$", in general it is not possible to find a neighbourhood $W$ of $M$ such that $W \cap M$ is diffeomorphic to an open subset of $\mathbb R^m$. In ...
Gibbs's user avatar
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Dimension of a smooth manifold

In such situations it is always a good idea to look at examples. Let $k$ be arbitrary, and consider $$M=\{(x,0,\ldots,0)\in\mathbb{R}^k\ |\ x\in\mathbb{R}\}$$ You can verify very easily that $M$ is ...
freakish's user avatar
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Is every connected surface locally symmetric?

Credit to Moishe Kohan https://math.stackexchange.com/a/3604136/724711: The Uniformization Theorem says that if $S$ is a connected Riemannian surface then it is conformally equivalent to a complete ...
Ian Gershon Teixeira's user avatar
2 votes
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Example for Partition of Unity over circle

Here's a suggestion: Think of this on the unit interval $[0,1]$ — ultimately you will identify $0$ and $1$. You have two open subsets, $V_1=(0,1)$ and $V_2=[0,1/2)\cup (1/2,1]$. Take a smooth function ...
Ted Shifrin's user avatar
3 votes

In which points is $f: \mathbb{R}P^2 \rightarrow \mathbb{R}^3$ an inmersion.

Your chart $\phi_1$ is the one you would use if working with the projective space as the quotient $\Bbb{R}^3\setminus\{0\}\to\Bbb{RP}^2$, not $S^2\to\Bbb{RP}^2$. Thus your whole calculation is wrong. ...
peek-a-boo's user avatar
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Extending Diffeomorphisms of Lie Manifolds

Not assuming $N$ is connected: Let $H \cong S^1 \times \mathbb Z_3$ which topologically is the disjoint union of $3$ circles embedded as $H$ concentrically into $M = G = \mathbb R^2$ in such a way ...
Sven-Ole Behrend's user avatar
1 vote
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Structure required for differentiability in topological context

In order to talk about differentiability you need the full structure of $\mathbb{R}^n$, i.e. its vector space structure together with the Euclidean norm. Because these are essential in defining ...
freakish's user avatar
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Reflection of the "Figure Eight" not diffeomorphic to the "Figure Eight"

A look into Boothby's book shows that he uses the concept of an imnmersed submanifold (see Definition 4.3). $N = G( \mathbb R)$ is regarded as such an object. This means that $N$ is not regarded as a ...
Paul Frost's user avatar
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Integral formula for Hopf map in local coordinates appears to vanish

This is obvious to me now that I've taken a step back to think about it! The form $\omega$ is itself a pullback, $\omega = f^*(-\cos\theta\,\mathrm{d}\phi)$, so $$ \omega\wedge\mathrm{d}\omega=f^*(-\...
xzd209's user avatar
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Immersions are local diffeomorphisms

Recall the definition of differentiability on manifolds. A map $f:M \rightarrow N$ between differentiable manifolds is called differentiable if, for all compatible charts $(U,\phi) \in A_M$, (V,$\psi)...
Lourenco Entrudo's user avatar
0 votes

Homogeneous notation for differential forms on projective space

The only reasonable interpretation is that the form is supposed to refer to $$(-1)^i \frac{1}{f(z_{0/i}, \dots, z_{n/i})} dz_{0/i} \wedge \dots \wedge \widehat{dz_{i/i}} \wedge \dots \wedge dz_{n/i}$$ ...
CJ Dowd's user avatar
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2 votes

A Weak Type of Convexity for Smooth Jordan Domains in $\mathbb{R}^2$?

Definition. A subset $A$ is locally starlike if for every $a\in A$ there exists a neighborhood $U$ of $a$ in $A$ and a point $b\in int(U)$ such that $U$ is starlike with respect to $b$. More precisely,...
Moishe Kohan's user avatar
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Finding a point on $SO(3)$ that is equivalent (up to diffeomorphism) to a point on $\mathbb{RP}^3$

It's possible to give fairly efficient proofs of all the facts that Vincent's answer highlights: The quaternions have a quaternionic conjugation $h \mapsto \bar{h}$ given by $h=a + bi+cj +dk \mapsto a ...
krm2233's user avatar
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(Lee 8-12 SM) How do you find a global extension of a vector field defined on an open set?

Let $\varphi_0 : U_0 \to \mathbb{R}^2$, $\varphi_1: U_1 \to \mathbb{R}^2$, and $\varphi_2: U_2 \to \mathbb{R}^2$ be the three standard charts on $\mathbb{RP}^2$ (page 6 of Lee). Let's call our ...
Matt Watson's user avatar
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What is a tangent vector to a submanifold (Reconciling the definition of a tangent vector as a derivation with the geometrical idea.)?

Fix a curve $\gamma$ such that $\gamma(t_0)=p$. This defines a map $\gamma':C^\infty(M)\to \mathbb{R}$ as follows $$\gamma'(t_0)f := \frac{d}{dt}f\circ \gamma(t)|_{t_0}$$ where the $\frac{d}{dt}$ is ...
Sam Kirkiles's user avatar
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Mysterious Coordinates on $S^4$ involving Quaternions

You claimed that the change of coordinates between the two stereographic projection charts take the remarkably simple form $z′=1/z$. This is false with the standard interpretation of stereographic ...
Paul Frost's user avatar
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How to define $SO(3)$ natural topology

Just a few words about rigor. As said in the comment, the topology on $SO(3)$ should be the subsapce topology induced from the inclusion $$SO(3)\subset M_{3\times 3}(\mathbb R)\simeq \mathbb R^9$$ As ...
Just a user's user avatar
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Double of a manifold and cobordism

Yes, $\partial(M \times [0,1]) = \partial M \times [0,1] \cup M \times \{0, 1\}$. Picture a standard cylinder, $D^2 \times [0,1]$, whose boundary is a cylindrical tube $S^1 \times [0,1]$ together with ...
nkm's user avatar
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1 vote

Nonexistence of a continuous injection $f:S^2 \rightarrow \mathbb{R^2}$

Here is a proof at the level of Ahlfors. Suppose continuous injective $f:\mathbb C_\infty \longrightarrow \mathbb C$ exists. Define $\gamma:[0,1]\longrightarrow \mathbb C$ given by $\gamma(t)=\exp\...
user8675309's user avatar
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making sense of calculations in the proof that the image of an embedding is an embedded submanifold

In $(7)$ "$\subseteq$" can be replaced by "$=$". Since $B_{\varepsilon}(0_m) \subset \varphi(U)$ this follows immediately from $(4)$. In fact, for $X \subseteq \varphi(U)$ we get $(...
Paul Frost's user avatar
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Relation between volume form and volume of the region created by a closed manifold

Suppose $X$ is an orientable $(n+1)$-dimensional manifold with boundary and $M=\partial X$. Let $\vec n$ be the unit outward-pointing normal to $M$, and let $\omega$ be a volume form of $X$. Then the ...
Ted Shifrin's user avatar
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How to pull back the differential form $\omega = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}$ to $S^2$

Imho the simplest but still a bit tedious approach is to use the inverse stereographic projection: \begin{align} \pmatrix{ x\\y\\z}=\frac{1}{1+X^2+Y^2}\pmatrix{2X\\2Y\\-1+X^2+Y^2} \end{align} where $x,...
Kurt G.'s user avatar
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How to pull back the differential form $\omega = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}$ to $S^2$

To answer your question in coordinates if we have $f:U \longrightarrow \mathbb{R}$, for $U \subset\mathbb{R}^n$ and $U$ open. : $$df=\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i$$
tychonovs-scholar's user avatar
1 vote
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Prove that if a smooth manifold $M$ is contractible then every vector bundle over $M$ is trivial

You might also try Hussemoller's book "Fiber Bundles", Chapter 2, Corollary 4.8. Usually one proves, in the same breath, that if $f,g : X \to Y$ are two homotopic maps and if $B$ is a vector ...
Lee Mosher's user avatar
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3 votes

What are the orientable prime compact three-manifolds that can be embedded in $\mathbb{R}^4$?

Actually, regarding connected sums, it is much more subtle than you think. On one hand, if $M_1, M_2$ are 3-dimensional manifolds each of which embeds in $\mathbb R^4$, then their connected sum also ...
Moishe Kohan's user avatar
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In complex geometry, is an holomorphic function continuous by definition?

Let $f : A \rightarrow \Bbb{C}$ be a complex function, with $A \subseteq \Bbb{C}$. Usually, to even say that $f$ is holomorphic, we require that $A$ be an open subset of $\Bbb{C}$, just like with ...
Sambo's user avatar
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5 votes

Example of smooth map that doesn't has constant rank

Trivial examples can be obtained for non-connected $M$. Take e.g. $M = (0,1) \cup (1,2), N = \mathbb R$ and $f(x) = x$ for $x \in (0,1)$, $f(x) = 0$ for $x \in (1,2)$ Then the rank of $f$ is $1$ at $x ...
Paul Frost's user avatar
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