For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

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Quotient by a discrete subgroup of a Lie group

I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let $H$ be a Lie group and $T$ be a discrete subgroup of its center $Z(H)$. Then there exists ...
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30 views

Understanding Hempel's proof of uniqueness of cube with handles

In Hempel's 3-Manifolds book, Theorem 2.2 says that if $P$ and $Q$ are two cubes, both with $n$ handles, and both are orientable, then they are homeomorphic. He defines a cube with handles as a ...
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1answer
25 views

Is every convex cone a manifold?

Let $C \subseteq \mathbb{R}^n\setminus \{0 \}$ be a connected convex cone*. Question: Is $C$ always a topological manifold (perhaps with boundary)? A smooth one? Does anything change if we do not ...
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1answer
32 views

Vector bundle vs Total Space

On page 59 in Lee's "An Introduction to Smooth Manifolds" the author writes, "Let $E$ be a smooth vector bundle over a smooth manifold $M$, with projection $\pi:E\to M$." I thought the vector bundle ...
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1answer
54 views

How to define CW-complex structure on cubic surface in $CP^3$?

I have read roughly this blog and I have following question. I changed my original question to following. How to define CW-complex structure on cubic hypersurface $M$ in $\mathbb CP^3$ defined by ...
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30 views

Does anything obstruct Mostow-Prasad rigidity for orbifolds?

If we phrase the Mostow-Prasad rigidity theorem algebraically, it goes like this (let $\mathcal{H}^n$ be a model for hyperbolic $n$-space). For $n>2$: if ...
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2answers
30 views

Choice of chart and altas

Manifold is second countable space. Should charts in altas also be countable? I don't think it has to be. But some how the second countable condition enforces me to think like that..
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57 views

The classifying space of open covers of a manifold

Let $M$ be a manifold of dimension $d$ and let $\mathsf{Disk}_{/M}$ be the category of open subsets of $M$ that are diffeomorphic to $\mathbb{R}^d$ with morphisms given by inclusions. Let $\mathrm{B} ...
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1answer
25 views

Tensor field acting on vector fields and covector fields gives a function?

If I have a $(p,q)$ tensor field that acts on $p$ covector fields and $q$ vector fields then does $T\left(X_1,\dots,X_p,Y_1,\dots,Y_q\right)$ return a function $f$ defined on the manifold by ...
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34 views

Is $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$?

I was reading a little about how to imagine the projective plane and I have some weird intuition that says $T^2 \# \mathbb RP^2\cong \mathbb RP^2\# \mathbb RP^2\# \mathbb RP^2$. Is this true, and if ...
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Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
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37 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...
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1answer
23 views

Calculating the second fundamental form of surfaces

I am asked to prove that for a surface in $\mathbb{R}^3$ with local coordinates in a chart, u,v, the coefficients of the second fundamental form can be calculated as follows: eg.: (first entry in II) ...
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20 views

How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
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1answer
24 views

Integrate 2-Form over surface

Problem: Calculate $\int_S dx \wedge dy + dy \wedge dz$, where $S$ is the surface given by $S = \{(x,y,z) : x = z^2 +y^2 -1, x < 0\}$. Wikipedia says: Let $$ \omega=f_{z}\, \mathrm dx \wedge ...
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9 views

Sum two nearest function of two class are the nearest function of the sum class

Suppose $x,\mu:[0,1]\rightarrow \mathbb{R^2}$ two smooth function and $\Gamma = \{\gamma : [0, 1] \rightarrow [0, 1]| \gamma (0) = 0, \gamma (1) = 1, \gamma$ is a diffeomorphism $\}$. Here $\Gamma$ ...
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1answer
31 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
33 views

Why does $(X_a,Y_b)(f\circ \mu(a,b)) = X_a(f\circ R_b(a)) + Y_b(f\circ L_a(b))$?

On the first page of these notes: if $\mu\colon G\times G\to G$ is the multiplication map on a Lie group $G$, then given a point $(a,b)\in G\times G$ and letting $R_b$ and $L_a$ denote ...
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1answer
97 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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1answer
203 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
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26 views

$M=M_1\cup M_2$ is not necessarily a manifold, when $M_1\cap M_2=\emptyset$, $M_i$ a manifold

While $M_1\cap M_2=\emptyset$, $M_i$ a manifold, show $M=M_1\cup M_2$ is not necessarily a manifold. Another question, prior to this one, was to show the union is a manifold, where the conditions ...
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39 views

calculate the surface of the manifold in $\Bbb{R}^4$

How to calculate the surface area of the following manifold : $$ x_1^2 + x_2^2 = x_3^2 + x_4^2, 0 \le x_1^2+x_2^2 \le a^2$$ I know I should first describe this manifold as a map or a graph of a ...
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29 views

Linearity in quotient space

Let $\mathcal{C}$ be the space of all parametric curves $x:[0,1]\rightarrow \mathbb{R}^2$. Also let $\mathcal{C}$ is a linear manifold in the sense that $x_1,x_2\in \mathcal{C}$ implies that ...
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5 views

Vector bundle morphisms $T(I\times I)\longrightarrow A$?

Let $I:=[0, 1]$ be the unit interval in $\mathbb R$ and $\pi:A\longrightarrow M$ a vector bundle. Is there a nice characterization of the vector bundle morphisms $T(I\times I)\longrightarrow A$? ...
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34 views

Diffeomorphism group of product manifold

For a given differentiable manifold $M$, the diffeomorphism group $\mathrm{Diff}\left( M \right)$ of $M$ is the group of all $C^\infty$ diffeomorphisms of $M$ to itself. Consider a product manifold of ...
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1answer
29 views

Topological boundary as a submanifold

Let $U$ be an open subset of a smooth $n$-manifold. Consider $\partial U$ the topological boundary of $U$. Is the following true ? : If $\partial U$ is a smooth $n-1$ submanifold without boundary, ...
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1answer
25 views

Whats the surface area of the surface $0 \leq z, (x-1)^2 + y^2 \leq 1$?

Whats the surface area of the manifold $0 \leq z, (x-1)^2 + y^2 \leq 1$? The surface is the intersection of the sphere $x^2 + y^2 + z^2 = 4$ and a cylinder centered at $(1,0,0)$. I'm just not sure how ...
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Is there a locally compact, locally connected, Hausdorff and second countable space that is “nowhere locally Euclidean”?

When I study topological manifold, I think some property of manifolds are so important that they can "almost characterize" manifolds. But I know a topological manifold is not easily to be ...
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Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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Orientability of projective space

Q: Show that $\mathbb {RP}^n$ is not orientable for $n$ even. First I looked at the definition for orientability for manifolds of higher degree than 2, because for surfaces I know the definition with ...
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32 views

Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set

Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set. Given a $k$ dimensional manifold, $M$, and a mapping $r: U \rightarrow M$, and a null-set $E \subset M$, prove that ...
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Differential of a map including a manifold

Let $f\in C^{k}(M,\mathbb R)$ with $M$ is a $m$-Manifold and $d_xf:T_xM\to\mathbb R$ is the surjectiv differential. Let $m\lt l$ and $L:\mathbb R^l\to\mathbb R^{m-1}$ be a linear map and ...
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1answer
40 views

Why are are integrals of functions in coordinates on manifolds not invariant under coordinate transformations?

I'm reading the book Introduction to Smooth Manifolds. And there is a question that confuse me on page 202. Can anyone tell me why it would change under coordinate transformations graphically? Thank ...
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What are the “technical troubles” with using a metric space rather than a topological space when defining an abstract manifold? (As in Spivak)

One thing I think is interesting about Spivak's book A Comprehensive Introduction to Differential Geometry is that Spivak uses metric spaces instead of topological spaces when defining an abstract ...
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4 views

Finding this transition function for homeomorphisms over a cylinder

Let $C = \left\{ \ (x,y,z) \in \mathbb{R}^{3} \ | \ x^{2} + y^{2} = 1,\ 0 \leq z \leq 1\ \right\}$ I've got the functions $f:(0,1) \times (0,1) \to C$ and $g:(0,1) \times (0,\tfrac{1}{3}) \to C$ ...
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40 views

Embedded submanifolds

Set $L= \{ (x,y)\in \mathbb{R} : x^3=y^5\} $. Consider the parametrization of the curve $ t \to (t^5,t^3) $. Then the derivate of this curve at 0 is 0. Hence $L$ is not an embedded submanifold? ...
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It can happen that $M_1 \cap M_2 =\emptyset $ but $M_1 \cup M_2$ is not a $k$-dimensional manifold. Give a counter example.

Let $M_1,M_2 \in \mathbb{R}^n$ be $k$-dimensional manifolds, $ M=M_1 \cup M_2$ It can happen that $M_1 \cap M_2 =\emptyset $ but $M$ is not a $k$-dimensional manifold. Give a counter example. ...
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2answers
273 views

About Sectional Curvature

In a paper by Yann Ollivier: Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
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37 views

What do we need to guarantee that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$?

I am trying to figure out the conditions such that $[X, Y]_p$ is linearly independent with $X_p$ and $Y_p$ for some vector fields $X, Y$ and some $p$ in a three-dimensional manifold. I have that ...
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M is a k-manifold if and only if $\phi(M)$ is a k-manifold

Let $\phi: \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a diffeomorphism and $M\subset \mathbb{R}^n$ M is a k-manifold if and only if $\phi(M)$ is a k-manifold. Prove it. So what I did was try to ...
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How to define an atlas on this manifold with boundary?

Consider the set $\mathcal{M} = \{\ \mathbf{x} \in \mathbb{R}^{3}\ | \ 1 \leq ||\mathbf{x}|| \leq 2 \ \}$. This is a $3$-submanifold with boundary. Obviously, we have $\partial \mathcal{M} = \{\ ...
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Coefficients Relative to a Smooth Frame

An exercise from Loring Tu's textbook asks the following question: Let $\pi:E\to M$ be a $C^\infty$ vector bundle and $s_1,\ldots,s_r$ a $C^\infty$ frame for $E$ over an open set $U$ in $M$. Then ...
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1answer
62 views

Is $\mathbb{R}\times\{0,1\}$ a manifold?

The definition of a $k$-manifold we are given is a set $M\subset\mathbb{R}^n$ such that the following equivalent conditions hold for each $x\in M$: There exists a mapping ...
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35 views

Is the restriction of a smooth vector field to a regular submanifold also smooth?

Let $S$ be a regular submanifold of a manifold $M$, meaning a subset of $M$ such that for all $p\in S$ there is a coordinate neighborhood $(U,\phi) = (U,x^{1},...,x^{n})$ of $p$ in the maximal atlas ...
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32 views

What's the surface measure (volume) of the manifold $x_1^2 + x_2^2 = x_3^2 + x_4^2$, $0 \leq x_1^2 + x_2^2 \leq a^2$?

What's the surface measure (volume) of the manifold $x_1^2 + x_2^2 = x_3^2 + x_4^2$, $0 \leq x_1^2 + x_2^2 \leq a^2$? I'm trying to figure this out in terms of an integral of a manifold but can't ...
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33 views

An exercise about immersion map

The map \begin{align*} F \colon \mathbb{R} \times \mathbb{C} &\to \mathbb{C}^2 \\ (t,z) & \mapsto (z^2,tz) \end{align*} restricts to an immersion $f \colon S^2 \to \mathbb{C}^2$, where $ S^2 ...
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1answer
12 views

Extending covering projection of the boundary

Let $M$ and $E$ be (topological) manifolds with boundaries $\partial M$ and $\partial E$ respectively and assume we have a finite-sheeted covering $\rho: \partial E\to \partial M$. Is it possible to ...
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1answer
18 views

Jacobian of a diffeomorphism

Let $U,V\subseteq \mathbb{R}^{n}$ be open. Let $\alpha:U \to V$ be a smooth homeomorphism. Furthermore, assume that $\mathcal{J}_{\alpha}(\mathbf{x})$ (the Jacobian matrix) has rank $n$ for all ...
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1answer
31 views

Orient Manifold

$\mathbf{Problem \,2.}$ Consider the $2$-manifold in $\Bbb R^3$ given by $$x^2+y^2+z^2=1,\qquad z\ge 0.$$ Orient $M$ such that $\alpha$ in the Equation $(2)$ belongs to the orientation, and give ...
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61 views

Smooth vs topological orientation

I am confused about the different notions for a closed smooth manifold to be oriented. In my mind there are several equivalent ones: 1) Coherent pointwise orientation of the tangent spaces. 2) ...