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Questions tagged [smooth-manifolds]

For questions about smooth manifolds, a topological manifold with a maximal smooth atlas.

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How Klein Bottle in R⁴ does not have self intersection?

This is the screenshot from Feko's book on Differential Geometry. In 1.5.10 in the given hint how he deduced that in R^4 the Klein Bottle has no self intersection? Also,in 1.5.11, what does square ...
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$C^\infty$ atlas + Partitions of unity $\Rightarrow$ Second-countable

Let $X$ be a (connected) topological space with a $C^\infty$ atlas. It is a known theorem that if $X$ is second-countable and Hausdorff, then it admits partitions of unity. I'm trying to prove the "...
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Why does the injectivity of the differential implies the injectivity of the derivative(Jacobian)?

I'm reading Warner. "Foundations of Differentiable Manifolds and Lie Groups." In the proof of corollary (a) in page 24, which I will present below, the book proves that the differential $df$ of the ...
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What exactly is in the standart atlas of $R^n$?

I saw a theorem saying that any atlas on a manifold can be extended to a unique maximal atlas. Concerning the standard atlas of $\mathbb{R}^n$, we saw that this was the atlas generated by the chart $(...
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Understanding the proof that the quaternionic projective space is diffeomorphic to the 4-sphere

Let us consider the quaternionic non-commutative field $\mathbf{H} = \mathbf{R}^4$ and its projective space $\mathbf{P}_1(\mathbf{H})$ of dimension 1, which is, by definition, the set of lines in $\...
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Generalized stereographic projection

Let $(M^n,g)$ be a closed (compact, without boundary) Riemannian manifold and let $p\in M$. Let $$ \square: =c\Delta+S $$ be the conformal Laplacian of $(M,g)$. Here $c=4\frac{n-1}{n-2}$ is a ...
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Are manifolds almost diffeomorphic to $\mathbb{R}^n$?

Let $M$ be an $n$-dimensional compact connected orientable smooth manifold. Can we always find a diffeomorphism $$M\setminus S_1\cong\mathbb{R}^n\setminus S_2$$ where $S_1\subset M$ and $S_2\subset\...
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Smooth map with null differential at each point are constant on the connected component of the domain

Let $F:M\to N$ be a smooth map between smooth manifolds $M$ and $N$ (with or without boundary). I want to show that $dF_p:T_pM\to T_{F(p)}N$ is the zero map for each $p\in M$ if and only if $F$ is ...
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Well-posedness of linear ODE problem on vector bundle

Let $\gamma \colon [0,\alpha] \to \mathbb{R}^{3}$ be a smooth regular curve and let $N\gamma$ denote its normal bundle. Recall that $N\gamma$ is a smooth vector bundle, whose fiber at $\gamma(t)$ is ...
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clarification about use of immersion in defining embedded submanifolds

the definition of embedded submanifolds as given in the text of boothby is: image of a topological embedding+immersion is an embedded submanifold Suppose we have a smooth manifold $M$ and $N$ ...
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problem on embedded submanifolds [on hold]

sometime ago,this question was asked on immersion:Understanding Embedded submanifolds and immersions here,in the answer,it says that: To avoid this, you want $F$ to be an immersion, so that both ...
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Understanding Embedded submanifolds and immersions

I am trying to understand the concept of embedded submanifolds and have the following understanding: Suppose we have a smooth manifold $M$ and $N$ of dimensions $m$ and $n$ such that $m\gt n$ .Let $...
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Definition of smooth functions on arbitrary subsets of $\mathbb{R}^n$ and partial derivatives

Let $A$ be an arbitrary subset of $\mathbb{R}^n$, and let $f:A\to \mathbb{R}$ be a function. We say that $f$ is smooth if for each point $p$ in $A$ there exists an open subset $U$ of $\mathbb{R}^n$ ...
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How to show that this manifold has dimension 2?

Q. Given that $M=\{\mathbf{x} \in \mathbb{R}^4: x_1^2 + x_2^2 + x_3^2 + x_4^2 =1, x_1x_2 = x_3x_4\}$. Show that $M$ is a smooth manifold of dimension 2. I write $M = \mathbf{F}^{-1}(\{\mathbf{0}\})$,...
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Characterization of $\mathbb{R}^n$?

Let $M$ be a smooth $n$-dimensional manifold with the property that any compact subset $K \subset M$ is contained in an $n$-dimensional smooth ball $K \subset B \subset M$. If $M$ is open, does it ...
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Notation: gradient as vector field

Consider the tangent space $T_p\mathbb{R}^n$, and suppose $\{\big(\frac{\partial }{\partial x^i}\big)_p\}$ is a basis. So my textbook says that the gradient of a function $f$, $f\in C^\infty(U)$, $U\...
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Jacobi Identity of Commutator of Vector Fields

I want to proof the Jacobi identity of Lie brackets of vectorfields on smooth manifolds. Let $ X=\sum \xi^i\frac{\partial}{\partial x^i} $, $ Y=\sum \eta^i\frac{\partial}{\partial x^i} $ and $ Z=\...
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Is the condition 'connected' necessary for differentiable structure?

I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness ...
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Smooth curves and velocity

Let $M$ be a smooth manifold. Let $X:M \to TM$ be a global smooth vector field on $M$, and let $K$ be the support of $X$, i.e. $K=\overline{\{p\in M: X_p\not=0\}}$. Suppose $K$ a is compact subset of ...
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How to solve this system differential equations to determine the local flow of a smooth vector field?

I want to calculate the local flow of the following smooth vector field on $\mathbb{R}^2$, $X:\mathbb{R}^2\to T\mathbb{R}^2$, defined by $X=(x^2-x^1)\frac{\partial}{\partial x^1}-(x^1+x^2)\frac{\...
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The hypothesis $K\leq 0$ in the proof of Hadamard's Theorem

In chapter 7 from do Carmo's Riemannian Geometry, right after proving Hadamard's theorem, there is the following remark: When he says "poles can exist in non-compact manifolds which have positive ...
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The Image of sphere under the map $f: (x,y,z) \to (x^2,y^2,z^2,\sqrt{2}yz, \sqrt{2}zx, \sqrt{2}xy)$ and $\mathbb{R}P^2$

I'm going to take a course about diferentiable manifolds next semester and I'm preparing for it by solving some of the problems from the book An Introduction to Differential Manifolds by D. Barden and ...
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How to transfer the integration theory of smooth manifold to that of $C^k$ manifold

I want to consider $C^k$ manifolds because, in PDE, people want to consider $C^k$ boundaries etc, and a version of divergence theorem for $C^k$ manifold is needed. Is it necessary to repeat all the ...
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1answer
56 views

Computing the differential of a certain smooth map

Let $M \subseteq \mathbb{R}^k$ be an embedded submanifold of $\mathbb{R}^k$, with dim$M=n$. Let $v$ be in $\mathbb{S}^{k-1}$, and let $P_v:\mathbb{R}^k\to(\mathbb{R}v)^{\bot}$ defined by $P_v(x)=x-&...
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24 views

Criterion for pullback to be a manifold

Suppose we have a smooth map $F:M\rightarrow N$ and a map $G:U\rightarrow N$. We can talk about pullback $M\times_NU$ as a set given by $\{(m,u):F(m)=G(u)\}$. Suppose $F$ or $G$ is a submersion, then ...
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Smooth quotient bundle

Let $E \rightarrow M$ be a smooth vector bundle. This link gives a construction of quotient bundle for a subbundle. $E' \subseteq E$. We define an equivalence relation $\sim$ on $E$ by $v_x \sim ...
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An exact sequence of vector bundles.

Page 376, Prop 15.6.7: Let $p:E \rightarrow M$ be a vector bundle. There exists a canonical exact sequence $$ 0 \rightarrow p^*E \xrightarrow{\alpha} TE \xrightarrow{\beta} p^*TM\rightarrow 0 $$ ...
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$F:M\to N$ is surjective if $\int_M F^* \eta \ne 0$ for some $\eta \in \Omega^n(N)$

Let $M$ and $N$ be compact orientable and connected smooth $n$-manifolds and $F:M \to N$ a smooth map. Suppose $$\int_M F^* \eta \ne 0$$ for some $\eta \in \Omega^n(N)$. Then $F$ is surjective. Give ...
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1answer
49 views

Characterization of being a submersion

Let $\mathcal{M}$ be a smooth $m$-dimensional manifold and let $F\colon \mathcal{M} \longrightarrow \mathbb{R}^{n}$ be any smooth map, with m $\geq$ n . I want to prove that, if $p \in \mathcal{M}$: $$...
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show that $f(x)=a^2 + b^2 + c^2 + d^2$ is smooth on $\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})$

The space of lattices in the Euclidean plane $X=\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})$ can it have smooth functions? I'm trying to find examples of $C^\infty$ functions. Consider ...
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Submanifolds and adapted atlas

Let $M$ be a smooth manifold of dimension $n$. My notes say Theorem: A subset $S$ of $M$ could be given a structure of smooth manifold of dimension $k$ such that $S$ is an embedded submanifold ...
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some question about a problem in gtm218:the introduction to smooth manifolds

let $F:R^2\to R$ be defined by $F(x,y)=x^3+xy+y^3$.which level sets of $F$ are embedded submanifolds of $R^2$ ? For each level set,prove either that it is or that it is not an embedded manifold. My ...
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Adapted charts for smooth manifold

Let $M$ be a smooth manifold of dimension $n$ and let $S$ be an embedded submanifold of $M$, i.e. a subset of $M$ which is given a structure of smooth manifold of dimension $k\leq n$ such that the ...
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Homotopy type of smooth manifold with boundary

It seems very likely to me that a $n$-dimensional smooth manifold with boundary has the homotopy type of a $(n-1)$-dimensional CW complex. Is that true? Does the manifold need to be compact? What ...
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The nonsingular variety is a manifold and irreduciblity

For the claim that a nonsingular variety is a smooth manifold, do we need to require the nonsingular variety to be irreducible? I am thinking that each irreducible component is a smooth manifold and ...
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1answer
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Derivative of $C^k$ function $(k < \infty)$ on smooth manifolds.

If $M$ and $N$ are smooth manifolds, the jacobian matrix of the derivative of a smooth function $F: M \to N$ in a point $p \in M$, taking charts $(U,\varphi)$ around $p$ and $(V,\psi)$ around $F(p)$ ...
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Loomis and Sternberg: Tangent Space to a manifold, using equivalence classes; help justifying one step of an argument

I am currently reading through the section in Loomis and Sternberg's Advanced Calculus on Tangent Spaces, but I'm having trouble justifying one step of the argument (shown below). Here's the ...
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Properties of forms and pullbacks

Let $j: \mathbb{S}^2 \rightarrow \mathbb{R}^3\setminus\{0\}$ be the canonical injection and $\alpha$ a k-form over $\mathbb{R}^3\setminus\{0\}$. If $\alpha$ is closed or exact, is it the same for $j^*...
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Subcover of a compact Manifold

Let $M$ be a compact smooth manifold and $\{(U_i,\phi_i)\}_{i=1}^s$ a finite atlas for $M$. I want to show that exists an open cover $(V_i)_{i=1}^s$ of $M$ such that for each $1\leq i\leq s$ I ...
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Reference for cup product in deRham cohomology

Given a smooth manifold $M$, we have what are called deRham cohomology groups $H^i(M,\mathbb{R})$. deRham cohomology ring $H^*(M,\mathbb{R})$ is as a set $\bigoplus_{i=0}^{\text{dim(M)}} H^i(M,\...
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Killing homology below middle dimension with equivariant surgery

Assume a finite group acts smoothly on a manifold $M$ of dimension $n$. Suppose $a\in H_i(M)$, where $i=1,\ldots,[n/2]$. Is there a way to kill $a$ with equivariant surgery and keep the same fixed ...
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What are the Intrinsic properties of Euclidean spaces?

I am reading a book "An introduction to Manifold "by Loring W.Tu . I am not understand this line "euclidean spaces are handicap because, defined in terms of coordinates, it is often not obvious ...
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Intersection of two plaques is an open subset of both plaques.

I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from ...
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1answer
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Acyclic compact Lie groups of dimension 3

Are there any examples of compact connected Lie groups with vanishing first homology groups in dimension $3$ different from $S^3$?
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Measurable set on Manifold.

In my textbook, we have the following definition, If $M$ is a smooth manifold of dimension $n$. We say that a set $A\subset M$ is measurable if, for any chart $U$, the intersection $A\cap U$ is a ...
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Are the manifolds $N=(\mathbb{R},\text{Id})$ and $M=(\mathbb{R},x\mapsto x^{\frac{1}{3}})$ diffeomorphic?

Are the manifolds $N=(\mathbb{R},\text{Id})$ and $M=(\mathbb{R},x\mapsto x^{\frac{1}{3}})$ diffeomorphic? I have already shown that $\text{Id}: N \rightarrow M$ is a homeomorphism but not a ...
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1answer
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Showing that these vector fields commute on the image

This was a problem from my final exam I took a few days ago, and after having it graded my solution did not receive full points. The professor seems to be busy grading for other classes he is teaching ...
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1answer
39 views

Rank of a smooth map is lower semicontinuous?

Let $F:M\rightarrow N$ be a smooth map between manifolds, $p\in M$. Prove that if $\operatorname{rank}_{p}F=r$, then there exists a neighborhood $U$ of $p$, such that for $\forall q\in U$, $\...
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Example of a parallelizable smooth manifold which is not a Lie Group

All the examples I know of manifolds which are parallelizable are Lie Groups. Can anyone point out an easy example of a parallelizable smooth manifold which is not a Lie Group? Are there conditions on ...
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2answers
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$\omega$ a closed 2-form and $\bigwedge_{i=1}^n \omega \ne 0$ on a compact orientable smooth $2n$-manifold w/o boundary, $M$, then $H^2(M) \ne 0$.

Suppose $M$ is a compact orientable smooth $2n$-manifold without boundary, and let $\omega$ be a closed $2$-form such that $\bigwedge_{i=1}^n \omega_p \ne 0$ at every point $p$. Show that $H^2_{dR}(M) ...