Questions tagged [smooth-manifolds]

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

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Constructing cut-off function on geodesic ball

Let $M$ be a complete non compact Riemannian manifold, $p \in M$ be a fixed point, $r > 0$ be an arbitrary (as large as one may want it to be) constant and $B_r(p) = \exp(B_r(0))$ the geodesic ball ...
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Can we know that a map is a submersion with using only its level sets

Let $M$ be a manifold of dimension $2n$, $f:M\rightarrow\mathbb{R}^n$ a smooth map. We know that if $f$ is a submersion, then for any $c\in f(M)$, $f^{-1}(c)$ is a $n$-submanifold of $M$. But what ...
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Do unique horizontal lifts of path homotopic paths through a point are always path homotopic?

Let $\pi:E \rightarrow M$ be a principal $G$-bundle for a Lie group $G$. Let $\omega$ be a connection on the principal bundle. It is a well known fact, that if a path $\gamma:[0,1] \rightarrow M$ is ...
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Does the Differential Topology/Geometry frameworks being able to model solutions to diff. eqs. that are Non-Smooth?

I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to ...
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Definition of uniqueness of tubular neighbourhoods

The tubular neighbourhood theorem states that if $M \subset N$ is an embedding of smooth manifolds without boundary and $\nu: E \to M$ is the normal bundle of $M$ in $N$, then there is a smooth ...
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Any orbit of $X$ passing through a point of $M$ is entirely contained in a surface

I am trying to prove this result: Given a field $X$ in $\mathbb R^n$ of class $C^1$ and let us consider a surface $M\subset\mathbb R^n$ of class $C^2$ such that $p\in M$ implies $X(p)\in T_pM$. Show ...
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Local cross section in smooth manifolds

I have managed to prove that for every field $X$ of class $C^1$ in $\mathbb R^n$ there is a local cross section at a regular point of $X$. I would like to prove that this fact is also true for ...
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Almost closed forms

Suppose we have a closed (compact, without boundary) manifold $M$. Let's assume that it is orientable, although it might play no role in the question. Now, De Rham cohomology measures how far a closed ...
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Why the poincaré theorem is not possible in $\mathbb R^n$?

Hello I am reading about the Poincaré-Bendixson theorem in the plane and then in compact two-dimensional manifolds. But I have some doubts that I would like you to help me clarify: I know the example ...
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Help proving statement in Lee's Introduction to Riemannian Manifolds about smooth curves into manifolds with nonempty boundary.

The statement appears on page 33 of the second edition of Professor Lee's Introduction to Riemannian Manifolds. It is in the section on Lengths and Distances in Riemannian manifolds, but I think the ...
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Books for learning Hyperbolic Dynamical Systems and differentiable manifolds

I am looking for some books/lectures that cover Hyperbolic Dynamical systems and supplemental materials that cover the very basics of differentiable manifolds, enough to understand everything relevant ...
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How to prove that a pullback map is linear

The following question was left as an exercise in my assignment of Manifolds and I am not able to prove this. Question: Define the map $T^{*} : L^{k}(W) \to L^{k} (V)$ , where $\alpha \in L^{k}(W)$ ...
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Gauß-Bonnet for non-compact manifolds

Is there a reasonable approach to extend the Gauß-Bonnet theorem to non-compact manifolds? and use this to distinguish between i) the 2-sphere and ii) the punctured 2-sphere?
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f-related and smooth map [duplicate]

This question was left as an exercise in my class of manifolds and I am not able to prove this. Question: Let $f : M \to N $ be a smooth map. Suppose that $X_1 , X_2 \in X(M)$ are f-related to $Y_1 , ...
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Convexity in "usual Partition of Unity arguments"

I stumbled upon Problem 13-2 on p.344 in John Lee's Introduction to Smooth Manifolds (2nd Edition) where Lee explains that the proof for the existence of a Riemannian Metric on a manifold is done by a ...
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Examples of hyperbolic sets of dynamical systems

I am studying the definition of hyperbolic set: a compact invariant set $\Lambda$ of a diffeomorphism $f$ such that the tangent space in each point of the set $\Lambda$ can be split in two invariant ...
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How to see that "two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps"

This question is about the diffeomorphism of $\mathbb{C}P^1$ and $S^2$. At the end of youler's answer, we read "the general fact that two manifolds are diffeomorphic when you can give them each ...
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Image of annulus under flow of vector field and differential equations

I'm studying Lee's introduction to smooth manifolds, in chapter 9 he introduces integral curves and vector flows, I'm using the following definitions. Definition: Let M be a manifold a flow domain for ...
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Lee Smooth Manifolds, why does the Whitney Approximation Theorem fail when the co-domain has non-empty boundary?

I am trying to study chapter 6 of Lee's Introduction to Smooth Manifolds. In a remark after the Whitney Approximation Theorem, Lee stated that this theorem do not hold because it might not be possible ...
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+50

Regarding the winding number

My main question is about part B, but I would also be grateful if you can tell me what you think about part A. Define a smooth vector field $X$ on $S^1$ as follows: $X(x,y)=(-y,x)$. For a smooth map $...
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Utility of the coordinate free definition of the derivative on manifolds.

Preface: I am not an expert on the topic of smooth manifolds, nor do I have the perspective gained from knowing many theorems proven on smooth manifolds. Please try to look at the problem from the ...
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Does defining cotangent space as quotient space work on $C^k$ manifold

I saw three different definition of tangent space on wikipedia#tangent-space: via equivlent class of tangent curves: It works for both $C^k$ manifolds and smooth manifolds via derivations: It only ...
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Lee Smooth Manifolds Theorem 6.23's Jacobian Matrix

I am trying to study the tubular neighborhoods and the normal space/bundle section of Lee's Introduction to Smooth Manifolds. I have a minor question about the proof of Theorem 6.23 which states: If $...
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Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
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Help understanding smooth covector field on a real closed interval in standard coordinates

In page 138 in John Lee's Introduction to Smooth Manifolds, he stated the following: "We begin with the simplest case: an interval in the real line. Suppose $[a,b] \subset \mathbb{R}$ is a ...
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$T\setminus\{p\}$ and $S^1\wedge S^1$ are smoothly homotopically equivalent: rigorous proof

Let $T=S^1\times S^1$. I want to prove that $M:= T\setminus\{p\}$ and $N:=S^1\wedge S^1$ are smoothly homotopically equivalent. I Just know the following facts: M and N are $C^0$-homotopically ...
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Lee Smooth Manifolds - Lemma (6.14) for Whitney Embedding Theorem

I am studying Lee's Introduction to Smooth Manifolds but I am stuck by a lemma for Whitney's Embedding Theorem. This lemma is trying to prove that: If a smooth n-manifold admits a smooth embedding $\...
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does the definition of a $C^k$ differential form rely on a smooth structure?

I have learnt some basic content of smooth manifolds from Tu's Introduction to Manifolds, in which a smooth 1-form on a smooth manifold $M$ is defined as a smooth section of the cotangent bundle $T^*M$...
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Resources for finding the dimension of the orbits

I am working on a problem concerning a Lie group acting on a smooth manifold or more generally, groups acting on topological manifolds or topological spaces. I am wanting to become more familiar with ...
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Equivalent definitions of Sobolev space on manifold and references

It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$: D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm. D2. All functions ...
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Can germs be defined as a quotient of vector spaces?

Summary: Let $M$ be a smooth manifold and $p\in M$. I know of two notions of germs of functions at $p$, the more restrictive of which can be written as a quotient vector space. I am curious whether ...
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Linear independence and nondegeneracy of a submanifold.

Consider a $d$-dimensional submanifold $M = \{f(x) : x \in U \}$ of $\mathbb R^n$ where $U$ is an open subset of $\mathbb R^d$ and $f = (f_1,\dots,f_n)$ is a $C^m$ imbedding of $U$ into $\mathbb R^n$. ...
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3 votes
2 answers
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Relating the Lie derivative to inner product of 2-tensors

Let $(M,g)$ be a Riemannian manifold. Let $f \in C^\infty(M)$, $X$ be a vector field and $h$ be a (symmetric) covariant $2$-tensor. Denote by $\langle \cdot, \cdot \rangle$ the inner product induced ...
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4 votes
2 answers
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Submersion, diffeomorphism, and embedding

Let $\pi:M \rightarrow N$ a submersion and $f:S \rightarrow M$ a smooth map such that $\pi \circ f : S \rightarrow N$ is a diffeomorphism. Show that $f:S \rightarrow M $ is an embedding. I'm having ...
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If $U$ is an open subset of $U_a$ for some $(U_a,\phi_a)$ in the atlas, then $U$ is smooth compatible with the charts in the atlas.

I'm trying to prove the statement: Given a smooth atlas $\{(U_{\alpha},\phi_{\alpha})\}$, if $U$ is an open subset of $U_a$ for some $(U_a,\phi_a)$ in the atlas, then $U$ is smooth compatible with ...
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Flow has no singularities

Context: Let $M$ be a compact, connected, edgeless bidimensional differentiable manifold and let $f:\mathbb R\times M\to M$ be a flow of class $C^2$ in $M$. What does it mean that a flow has no ...
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Covering transformation and orientation ( in Synge Theorem )

Case 1: $M_1$ is a compact orientable manifold. And $$ \pi_1:\tilde M_1\rightarrow M_1 $$ is the universal covering of $M_1$. Introduce on $\tilde M_1$ the orientation and metric such that $\pi_1$ ...
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1 vote
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1 -form on Compact Manifold

This question was asked in my quiz on smooth manifolds and I couldn't solve it during the exam. I tried this problem again but still couldn't solve it. I am not really good in solving problems related ...
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-1 votes
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$S^2$ is an orientable manifold

This question was left as an exercise in my class of orientable manifolds and I am having a hard time solving this. Question: (a) Prove that $S^2$ is an orientable manifold. (b) Let M and N be two ...
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Every $w\in \Omega^2 (V)$ is decomposable if $\operatorname{dim}(V) =3$

This question was asked in my assignment on tensors and I am stuck on this question. Question: Let $V$ be a vector space. An element $ w\in A^k (V)$ is called decomposable if $w = \phi_1 \wedge \...
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-1 votes
0 answers
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Wedge product, coordinate system and

This question was asked in my mid term term on smooth manifolds but I was not able to solve it. I tried this question again at home and I am not able to make significant progress. Question: Write ...
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Smooth functions and pullback map

This question was asked in my assignment on manifolds and I am struck on it. Question: For a 1-form w= f(x) dx on $\mathbb{R}$ , define $\int_{a}^{b} w = \int_{a}^{b} f(x) dx $ for $a, b \in \...
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0 votes
1 answer
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1 - form and smooth vector fields

This question was asked in my mid term of smooth manifolds and I couldn't solve it in exam time. I tried it again at home and I think I need help. Question: Let w be a 1-form on $\mathbb{R}^n$. If ...
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  • 1,596
1 vote
1 answer
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Smooth traversal of 𝑆𝑂(𝑛)

I am trying to constrain the space of matrices used for the layers of a neural network to those in 𝑆𝑂(𝑛). It is proven that 𝑆𝑂(𝑛) is a manifold. I'm trying to find a way to smoothly traverse ...
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2 votes
3 answers
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Hyperbolic local replacement in the neighborhood of a curve's vertex

The proof of the Rotation Index Theorem in Introduction to Riemannian Manifolds by John M. Lee, claims (without further elaboration) that, given a vertex $o$ on an otherwise smooth parametric curve ...
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1 vote
1 answer
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What is the constant of hyperbolicity?

I am studying dynamical systems of discrete time, and I am having some trouble in understanding what is the constant of hyperbolicity for a closed hyperbolic set $\Lambda \in M$ of a diffeomorphism $f:...
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  • 139
6 votes
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Regarding linking number of oriented knots

Let $K,L\subset R^3$ be oriented knots. Assume that $S\subset R^3$ is a two-dimensional compact connected oriented sub-manifold, $\partial S=K$, and the orientation given on $K$ merges with the ...
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  • 702
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Wedge product and linear independence

The following question was from my assignment on smooth manifolds and I am struck on this problem Let V be a n-dimensional vector space and $w_1,...,w_k \in A^{1} (V)$. Show that {$w_1,...,w_k$} are ...
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1 vote
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homotopy groups of Wu manifold

I'm interested in the homotopy groups of the Wu manifold $ W=SU_3/SO_3 $. From LES homotopy we have $$ \pi_1(W)=\pi_0(SO_3)=0 $$ since $ SU_3 $ simply connected $$ \pi_2(W)=\pi_1(SO_3)=C_2 $$ since $ \...
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0 votes
1 answer
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non-degenerate 2 forms, dimension and wedge product

Consider the problem from my assignment on tensor forms: Problem: We say that a 2-form n on a vector space V is non-degenerate of for $u\in V$ , n(u,v)=0 for all v$\in V$ implies u=0. Let w be a non-...
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