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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 to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, (x^1, …, x^n))$ is a local chart of a smooth manifold.

Let $M$ be a smooth manifold. I am getting lost in the notations. Could someone please explain me how to prove $\frac{\partial}{\partial x^r} (x^r) = 1$ where $(U, \phi = (x^1, ..., x^n))$ is a local ...
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Quotient manifold

To prove that for any $p \in M$, there exists a coordinate chart centered at $p$, John M Lee in Introduction to smooth manifold chooses the slice chart $(W, \psi)$ centered at $p$ for the orbit $G \...
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About the construction of the restriction of a Vector Bundle in John M. Lee's book

There are some details I want to understand in the construction of the Restriction of a Vector bundle given in John M. Lee's book Introduction to Smooth Manifolds. It is given in Example 10.8: ...
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1answer
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Is every compact solvmanifold the quotient of a simply connected solvable Lie group by a discrete subgroup?

Recall that a manifold is called a nilmanifold if it is a homogeneous space for a connected nilpotent Lie group. Mal'cev showed that every compact nilmanifold is diffeomorphic to the quotient of a ...
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Why is the tangential divergence the trace of the tangential gradient?

For a $C^1$function $\phi$ on a smooth surface $M$ one defined the tangential gradient $\nabla_M$ of $\phi$ by $$\nabla_M \phi(x):=(I-n \otimes n)\nabla \phi(x)$$ where $n$ is the unit normal and $\...
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Why are data points which are “close” should have a “similar” label?

In paper Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering, the basis idea is that data points which are “close” (i.e., $w_{ij}$ is large) should have a “similar” label (i.e., $...
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How to show $A=\{(x,y)\in\Bbb R^2|x^2=y^3\}$ is not an embedded submanifold of $\Bbb R^2$? [on hold]

Let $A=\{(x,y)\in\Bbb R^2|x^2=y^3\}$. How to show $A$ is not an embedded submanifold of $\Bbb R^2$? 假设$A=\{(x,y)\in\Bbb R^2|x^2=y^3\}$是$\Bbb R^2$的embedded submanifold, 根据Proposition 5.16, $(0,0)$在$...
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Finding an explicit smooth atlas for $GL_n(\mathbb{R})$

It is a standard exercise to show that $(GL_n(\mathbb{R}), \tau, D)$ is a smooth manifold, where $$\tau = \{\phi^{-1} (U) \subseteq GL_n (\mathbb{R}) | U\in \mathbb{R}^{n^2} is \quad open, \phi: ...
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Focal points in abstract sense

Guillemin and Pollack define a focal point of a hypersurface $X \subset \mathbb{R}^n$ to be a critical value of the normal bundle map $h: N(X) \to \mathbb{R}^n$ defined by $h(x,v) =x+v$. What does ...
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Darboux theorem and symplectic 2-sphere

Very rougly said, the Darboux Theorem states that all symplectic manifolds (of the same finite dimension) "look the same", i.e. their symplectic form could be brought to look like the standard one - $...
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Gradients on a hyperplane

Consider the hyperplane $H:=\lbrace x\in\mathbb{R}^{n}\ \colon\ x_{1}+\dots+x_{n}=0\rbrace$. If we treat $H$ as a manifold, and $f\colon H\to\mathbb{R}$ is smooth, then what is the gradient $\nabla f$...
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1answer
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How to find basis of a de Rham Cohomology group?

For example, we know that $H^1_{dR}(\mathbb{R}^2-\{p,q\})=\mathbb{R}^2$ where $p,q$ are two points, say $(-1,0),(1,0)$. But how can we find the basis of this cohomology group? I think I need to find ...
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1answer
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Getting acquainted with preimage orientation

I'm trying to do this problem: Let $f: S^2 \to \mathbb{R}^3$ given by $f(x,y,z)=z$. For the regular values $-1<t<1$, find the orientations of $f^{-1}(t).$ The hint is to find a positively ...
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1answer
18 views

Show that every smooth map from $S^n\to T^n$ has degree 0.

Here is my attempt: The degree of a smooth map $f:M \to N$ (where $M,N$ are manifolds of same dimension) is defined on the top form. Since the integral operator induces a natural isomorphism from the ...
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1answer
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Question about Natural Isomorphism in Definition of Tangent Spaces

I am a new learner in manifolds and have several questions about proof of the following Lemma: Lemma: $M_{m}$ is tangent space to a manifold $M$ at point $m$, and $F_{m}$ be the set of germs ...
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2answers
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How does this Lie derivative axiom follow from my definition of the Lie derivative?

$\newcommand{\L}{\mathcal{L}}$ $\newcommand{\der}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pder}[2][]{\frac{\partial#1}{\partial#2}}$ The definition of the Lie derivative which I'm starting with is $$ \...
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0answers
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Does every large $\mathbb{R}^4$ embed in $\mathbb{R}^5$? [migrated]

This question was prompted by my answer to this question. An exotic $\mathbb{R}^4$ is a smooth manifold homeomorphic to $\mathbb{R}^4$ which is not diffeomorphic to $\mathbb{R}^4$ with its standard ...
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1answer
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Equivalent Characterizations of Smoothness.

Definition. Let $M$ and $N$ be smooth manifolds, and let $F\colon M\to N$ be any map. We say that $F$ is a smooth map if for every $p\in M$, there exists smooth chart $\big(U,\varphi\big)$ containing $...
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0answers
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tangent bundle that isn't diffeomorphic to the Cartesian product [duplicate]

I am studying the notion of tangent bundle $TM$ of a smooth manifold $M$,and in my textbook I am acknowledged that for most smooth manifold of dimension $n$,it's not true that $TM$ is diffeomorphic to ...
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2answers
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Embedding of exotic manifold

While solving some problems on Differential Topology I asked myself this question: Suppose a smooth manifold having atleast two exotic smooth structures can be embedded in some Euclidean space under ...
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2answers
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Question about distribution on a smooth manifold

Let $N$ be a smooth manifold. Let $\Delta$ be a $C^{\infty}$ distribution on $N$. Suppose we have for all $q \in U$, an open set, $$ \Delta_q = Span ( X'_1(q), ..., X'_r(q) ) $$ for $X'_j \in X(N)$ ($...
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1answer
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Definition of smooth functions on a Manifold

Definition. Let $M$ a smooth manifold of dimension $n$. A funciton $f\colon M\to \mathbb{R}$ is said to be smooth at a point $p$ in $M$ if there is a chart $(U,\varphi)$ about $p$ in $M$ such that $f\...
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2answers
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$SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$

Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$. I want to prove that $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$. An idea is to use the ...
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0answers
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Degree of a map globally constant

I'm having trouble with the following proof: Denote by $deg_pf$ the deegree of a proper differentiable map $f:X\rightarrow Y$ between manifolds, where $p$ is a regular value of $f$ and the degree is ...
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0answers
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Lie algebra of $G \times H$.

We have to prove that the Lie algebra of $G \times H$ is $\mathfrak{g}\oplus \mathfrak{h}$. We know that $\mathfrak{g}=T_eG$ and $\mathfrak{h}=T_eH$. But is it true that $T_e(G\times H)=T_eG\oplus ...
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1answer
42 views

Prove that the graph of a function is a manifold

I know how to show this if $X$ and $Y$ are euclidean spaces using IFT but wanted to confirm proofs about the abstract case. Q) a) $X$, $Y$ are smooth manifolds and $f:X\rightarrow Y$ is smooth. Show ...
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2answers
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How to prove that $f(x)=\frac{x}{\sqrt{1+|x|^2}}$ is smooth?

I have a function $f(x)=\frac{x}{\sqrt{1+|x|^2}}$ on the open ball $B_1(0)$ in $\mathbb{R}^n$. I have to show that this function is a diffeomorphism between $\mathbb{R}^n$ and $B_1(0)$. I have already ...
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1answer
40 views

Derivative of products of exponential maps

Let $G$ be a (finite-dimensional) Lie group with Lie algebra $\mathfrak g$. Then for $f\in C^\infty(G)$, $X,Y\in\mathfrak g$ and $g\in G$ one can define $f(ge^{tX})\in C^\infty(\mathbb R)$ which ...
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Defining charts on the hypersphere as a smooth manifold

I have a small conceptual difficulty regarding coordinate charts of (smooth) manifolds. In my terminology a chart (of dimension n) on a topological space X is a pair $(U, \phi)$ with $U \subseteq \...
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2answers
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The meaning of “metric preserving” connection.

Let $g$ be a metric on a smooth manifold $M$. In local coordinates the metric takes the form $$g=g_{ij}(dx^i\otimes dx^j+dx^j\otimes dx^i).$$ From here, a connection $\nabla$ on $M$ is said to be ...
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0answers
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Can someone explain why we take Second Countability and Hausdorffness conditions in Manifold definition? [duplicate]

The definition of Differential Manifold or Smooth Manifold include $\text{Second countability}$ and $\text{Hausdorffness condition}$. My question is why we include Second countability and ...
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2answers
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How can a velocity vector be determined literally as a derivative in the calculus sense?

I am following Tu's book on manifolds. On pages 178-179 he proves that the Lie algebra of $\mathrm{SL}(n,\mathbb{R})$ is the set of traceless real $n\times n$ matrices. Specifically he writes: ...
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1answer
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Smooth manifold $F$ which is a vector space of finite dimensional $k$, and diffeomorphism with $\mathbb{R}^k$

I have a question about the manifold, especially when the manifold is as well a vector space of finite dimensional $k$. Actually, let $(v_1, \dots, v_k)$ be a basis of F as a vector space. I would ...
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1answer
149 views

A set as a graph of a function

The unit sphere $n$ dimensional is the set $$\mathbb{S}^n=\bigg\{(x_1,x_2,\dots, x_{n+1})\in\mathbb{R}^{n+1}\;|\;\big(x_1^2+x_2^2+\cdots+x_{n+1}^2\big)^{1/2}=1\bigg\}.$$ For all $i=1,\dots, n+1$ ...
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2answers
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Must any shortest line between two surfaces coincide with normals to both surfaces?

In differential geometry I know of a result which says something along the lines of ...
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1answer
34 views

Open Submanifolds

Let $M$ be a smooth $n$-manifold and let $U\subseteq M$ be any open subset. Define an atlas on $U$ $$\mathcal{A}_{U}=\big\{\text{smooth charts}\;(V,\varphi)\;\text{for}\; M\;\text{such that}\;V\...
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1answer
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Non-surjective map $F$ implies $\int_M F^* \eta=0$ [duplicate]

This is a problem from Lee 17.12: Suppose $M$ and $N$ are compact, oriented, smooth n-manifolds, and $F:M\rightarrow N$ is a smooth map. Prove that if $\int_M F^*\eta \neq 0$ for some $\eta \in \...
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1answer
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Existence of degree of smooth map between manifold and sphere

I came across this statement and couldn't figure out why this is true, please help: Let $M$ be an n-dimensional compact, connected, orientable smooth manifold without boundary. Prove that there ...
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1answer
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Rational Euler number of oriented Seifert manifolds

I'm studying Michele Audin's book - Torus Actions on Symplectic Manifolds and stumbled across an exercise I can't prove. Exercise I.13 Prove that the Euler class of the Seifert manifold with ...
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0answers
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Integration of 1-form $\omega = f(xdx + ydy)$.

I am self-learning integration on manifolds, and I'm trying to find an answer to the following question. For the manifold $M=\{(x,y) \in \mathbb{R} : (x,y) \neq (0,0) \}$, let $f: M \rightarrow \...
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1answer
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Does the interior semidirect product of Lie groups $G = N \rtimes H$ respect the projection $G \to H$?

Suppose we have some matrix Lie groups $N$ and $H$ both subsets of the $n \times n$ matrices, and a matrix Lie group $G$ for which we know $G = N H$, $N$ is a normal subgroup of $G$, and $N \cap H = \...
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1answer
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Vertical bundle of a principal bundle. Isn't it always trivial?

I may be totally wrong here, but after a bit of studying I've reach the conclusion of the title. My reasoning is as follows: For a principal bundle $P(M,G),$ and $VP$ is vertical bundle, let $\sigma :...
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2answers
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Why doesn't ever smooth vector bundle admits a line bundle?

Let $E \to M$ be a smooth vector bundle. Consider $G = \sqcup_{p \in M} F_p$ where $F_p$ is just a 1 dimensional subspace of each fiber $E_p$. The trivialization is just coming from the restriction of ...
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0answers
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Horizontal lifts of velocity fields

Let $X\overset{f}{\to}Y$ be a submersion equipped with an Ehresmann connection. Let $\mathbb R\overset{\gamma}{\to}Y$ a curve. Following wiki I want to show the velocity field of $\gamma$ uniquely ...
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A relationship between the values of an invariant vector field at the identity and another element

Let $G(R^n)$ be the general linear group, and $g(R^n)$ the set of all linear transformations on $R^n$, and $\mathscr g$ the Lie algebra of $G(R^n)$. We define a function $J:\mathscr g \to g(R^n)$ by $&...
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0answers
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Why can't we define any $C^{\infty}$ structure on the single point $0 \in \Bbb R^n$?

I'm studying the book "Differentiable Manifolds" by Brickell &Clark . On page 94, in the example 6.3.6, it was written that we can't define any $C^{\infty}$ structure on the single point $0$. ...
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1answer
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Smooth images of manifolds are immersed?

in various papers in symplectic geometry, I have encountered the following argument. Statement: Suppose $f: M \rightarrow N$ is a smooth map of constant rank. Then its image $f(M)$ can be equipped ...
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Is this function an open mapping? $(X,Y)\mapsto Yh(p)\cdot X(p) - Xh(p) Y(p). $

Let $M$ be a compact smooth $3$-manifold, and $h: M\to \mathbb{R}$ a function such that $\{0\}$ is a regular value of $h$, and define $\Sigma = h^{-1}(0).$ Moreover, we will denote $\mathfrak{X}^r(M)$ ...
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1answer
58 views

Chern classes, classification of bundles, and Bockstein morphism

I'm doing a work on Chern classes and I have the following doubts, I do not know if anyone could support me with the doubts, or with bibliography / references Given a $ n \in \mathbb{Z} $, is it ...
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0answers
33 views

Differential of the inverse map gives rise to the bracket of vector fields being zero

Suppose that $J:G\to G$ is the inverse map on a Lie group $G$ And suppose that $X,Y$ are vector fields. We know that if $\phi:G\to G$ is an arbitrary diffeomorphism then $d\phi[X,Y]=[d\phi X, d\phi Y]$...