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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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Question about Proposition 12.9 of Tu's "Introduction to Manifolds" [closed]

I'm reading through Tu's book and having trouble understanding Proposition 12.9, where they prove the sum of two $C_{\infty}$ sections $s(p), t(p)$ is $C_{\infty}$. Specifically, how they use ...
Stephen Jiang's user avatar
1 vote
0 answers
38 views

Details in the proof of quotient manifold theorem

The theorem is stated as: Let $G$ be a Lie group acting smoothly, freely and properly on a smooth manifold $M$. Then $M/G$ is a topological manifold of dimension dim$M$ - dim$G$, and has a unique ...
saberlove lin's user avatar
1 vote
1 answer
29 views

Change of coordinates in a regular surface $S$.

I'm quite confused in the following situation: Suppose that $S \subset \mathbb{R}^3$ is a regular surface. Let $U_1$ and $U_2$ be open sets in $\mathbb{R}^2$, and consider the pair of ...
Joel Marques's user avatar
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0 answers
22 views

Non-parametric and parametric statistical manifolds: Equivalence of score functions in tangent spaces [closed]

Below is the framework to give the manifold structure to the space $M_{\mu}=\{f \in L^{1}(\mu): f>0 , \mu a.e , \int f d\mu=1\}$ Statistical Model and its Topology: The Statistical Model and ...
Andyale's user avatar
  • 181
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1 answer
83 views

Vector bundle construction theorem

Let $M$ be a k-dimensional smooth manifold, and let $\{U_\alpha\}_{\alpha\in I}$ be an open cover of $M$. Further, for any $\alpha, \beta \in I$, let there be given smooth maps $\varphi_{\alpha\beta}:...
Gao Minghao's user avatar
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0 answers
27 views

Motivating the second derivative definition given in Audin,Damian Morse Theory text.

I am reading the Morse theory and Floer Homology text written by Audin and Damian.In the first chapter they have made a remark about second order derivatives on manifolds,the remark is quoted verbatim ...
Kishalay Sarkar's user avatar
2 votes
1 answer
55 views

Sheaves of sections of vector bundles

If I have two sheaves $E$ and $G$ over a complex manifold $M$ and I want to prove something like say $\mathcal{O}(E) \otimes_{\mathcal{O}} \mathcal{O}(G) \cong \mathcal{O}(E\otimes G)$, where $\...
Tim's user avatar
  • 207
4 votes
0 answers
106 views

Is this a valid definition of tangent space?

Given a smooth manifold $M$ and a point $p\in M$, I wonder if the tangent space of $M$ at $p$, denoted $T_pM$, can be defined in the following way. First define tangent space for $\mathbb{R}^n$ in any ...
new account's user avatar
2 votes
0 answers
49 views

Sheaf $\mathscr{F}_S$ for which $\mathscr{F}_S(U)$ consists of holomorphic sections that vanish on $S \cap U$ isomorphic to $\mathscr{O}(L^*_Y)$

Let $M$ be a complex manifold and $Y \subset M$ a closed hypersurface and $L_Y$ the holomorphic line bundle associated with $Y$. How can I show that the sheaf $\mathscr{F}_Y$ for which $\mathscr{F}_Y(...
Elena's user avatar
  • 63
2 votes
1 answer
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Confusion on Notations of Partial Derivatives on Manifolds

I'm confused with the notations of partial derivative on manifolds in Tu's An Introduction to Manifolds.. Just to make clear the notations I'm using, what I've known and which part I'm confusing, ...
TheHan6edMan's user avatar
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0 answers
43 views

Total space of the frame bundle always an almost complex manifold in $dim >2$?

I seem to remember reading that the total space of the frame bundle of a smooth Riemannian manifold always admits an almost complex structure in dimensions greater than 2. However now I can't seem to ...
R. Rankin's user avatar
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1 vote
0 answers
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Lee's Smooth Manifolds Problem 10-18

Lee's Introduction to Smooth Manifolds Problem 10-18 asks us to prove the following theorem: Theorem. Let $S$ be a properly embedded codimension-$k$ submanifold of $\mathbb R^n$. Then the following ...
Joseph Kwong's user avatar
2 votes
0 answers
62 views

Map of global sections surjective under a local condition

Let $M$ be a compact complex manifold and $p \in M$. Let $L \to M$ be a line bundle on $M$ and $\mathcal{F}_{\{p\}}$ be the sheaf of holomorphic sections of $L$ that vanish at $p$. Locally around a ...
Elena's user avatar
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Smoothness of a function is well-defined for functions between smooth manifolds

Definition: A function $F: M\rightarrow N$ smooth at $p\in M$ if there is a chart $(U,\phi)$ about $p$, a chart $(V,\psi)$ about $F(p)$ such that $F(U) \subset V$ and the map $\psi \circ F \circ \phi^{...
Jeffrey Jao's user avatar
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49 views

Suppose $M/G$ is a smooth quotient manifold and $N$ is a $G$-invariant submanifold of $M$. Then is $N/G$ a submanifold of $M/G$?

Let $M$ be a smooth manifold equipped with a, not necessarily proper, smooth Lie group action $G$. Suppose $M/G$ is a smooth quotient manifold. That is, there exists a smooth structure on the quotient ...
Spencer Kraisler's user avatar
-1 votes
1 answer
52 views

Curves on a trousers space. [closed]

How does one go about defining curves on a trousers space? I want to define two curves evolving cyclically around a cylinder and then at some time let one of the curves evolve on the other cylinder. ...
manyworlds's user avatar
3 votes
1 answer
53 views

Any smooth mapping from $\mathbb{R}^n$ into $S^1$ is of the form $e^{if(x)}$?

Let $F\colon \mathbb{R}^n \to S^1$ be a smooth mapping. Then, I strongly suspect that there must be a smooth function $f\colon \mathbb{R}^n \to \mathbb{R}$ such that \begin{equation} F(x) = \exp \big( ...
Keith's user avatar
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2 votes
1 answer
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Non-proper smooth embedding

Does all smooth embedding $\iota: S\to M$ is proper if $\dim S < \dim M$? I thought then the slice criterion implies $S$ is locally closed, so closed in $M$. But this is equivalent to saying that ...
okabe rintarou's user avatar
1 vote
1 answer
65 views

Why is $\int_0^1{df(tx_1,\dots, tx_n)\over dt}=\int_0^1\sum_{i=1}^n{\partial f\over\partial x_i}(tx_1,\dots,tx_n)\cdot x_i\;dt$?

For context: Milnor's Morse theory page 6, Lemma 2.1: $V$ is a convex neighborhood of $0$ in $\mathbb R^n$. also, $f,g \in C^\infty \left( V \rightarrow \mathbb R^n\right)$ where $f(0) = 0$ and $...
Nate's user avatar
  • 894
2 votes
1 answer
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$\Lambda^n(M)$ is not isomorphic to $C^{\infty}(M)$ if M is not orientable

Let $M$ be a differentiable Manifold of dimension n. If $M$ is orientable, then there exists an $\omega \in \Omega^n(M)$ (top-degree differential form) such that $\omega(p) \neq 0$ $ \forall p \in M$. ...
Jahi02's user avatar
  • 301
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1 answer
44 views

Question about the coordinate system on the manifold

Given an $n$-dimensional manifold $M$ and a homeomorphism $\phi:U\subset M\rightarrow V\subset\mathbb{R}^{n}$ from a patch on the manifold $U\subset M$, then we can parametrize a point $p\in U$ via ...
Andyale's user avatar
  • 181
9 votes
1 answer
98 views

If the suspension of $M$ is homotopy equivalent to a smooth manifold, does $M$ bound a contractible smooth manifold?

Let $M$ be a smooth $(n-1)$-manifold. If $M$ bounds a contractible $n$-manifold $C$, then the suspension of $M$ is homotopy equivalent to $$\partial (C\times I) = C\times (\partial I)\cup (\partial C)\...
Evan Scott's user avatar
1 vote
0 answers
48 views

When do smooth equations define a manifold?

Suppose you have some $C^\infty$ functions of $N$ variables $f_k(x_1, \dots, x_N)$ for $k=1, \dots, M <N$, and you want to consider the set $$\mathcal{M} = \{ x \in \mathbb{R}^N \, : \,f_k(x_1, \...
UtilityMaximiser's user avatar
2 votes
0 answers
42 views

(Necessary and sufficient) Conditions for the Ricci tensor of an affine connection to be symmetric.

Let $\nabla$ be an affine connection on a smooth manifold $M$. It is widely known, that if $\nabla$ is torsion-free, then its Ricci tensor is symmetric iff there exists a volume form $\omega$ on $M$ ...
ProphetX's user avatar
  • 350
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1 answer
64 views

Does the definition of Lie groups rely on invariance of domain? [closed]

I have some confusion over what we can say for certainty about the dimensions of a manifold or product of manifolds, without directly applying invariance of domain/dimension. In the definition of a ...
user760's user avatar
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1 vote
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Comparing the definitions of Derivative in Guillemin Pollack and in other differential topology books.

In standard differential geometry and topology books I have seen the authors defining the derivative/differential in the following way: Let $f:M\to N$ be a smooth map between two smooth manifolds.Then ...
Kishalay Sarkar's user avatar
1 vote
1 answer
49 views

"Dividing out" a cartesian product

The specific case I'm wondering about, is the case where $M,N$ are smooth manifolds, and we have $$M\times\mathbb{R}\cong N\times\mathbb{R}$$meaning that they are diffeomorphic. Does this imply that $...
John Doe's user avatar
  • 764
1 vote
2 answers
46 views

Computations in coordinates of Hamiltonian vector fields

I just need someone to snap me out of a (hopefully small) misunderstanding: in p.574 of (the 2nd edition of) Lee's Introduction to Smooth Manifolds it is written that a Hamiltonian vector field $X_f$ ...
Sam's user avatar
  • 5,166
4 votes
1 answer
104 views

Images of structure preserving maps are structures?

Context: I have been working with (mostly concrete) categories like Sets, Groups, Rings, Vect$_k$, Ab, $R$-Mod, Top, Diff and so on. As I understand it, images of group maps are subgroups of their co-...
frelg's user avatar
  • 483
0 votes
1 answer
40 views

Chain rule to differentiate vector fields with a connection or Lie derivative

Let $X : M \rightarrow TM$ be a smooth vector field, $f : M \rightarrow \mathbb{R}$ a smooth real valued function and $\gamma : [0,T] \rightarrow M$ a smooth curve in $M$. I am interested in ...
Theo Diamantakis's user avatar
0 votes
0 answers
39 views

If a curve intersects a propperly embbeded submanifold, then it is contained in that submanifold

I saw this problem in Lee's Introduction to Smooth Manifolds: Suppose that $S\subseteq M$ is a propperly embbeded submanifold and let $V\in\frak{X}$$(M)$ such that $V$ is tangent to $S$ (i.e $V_p\in ...
Andrés Vásquez's user avatar
4 votes
0 answers
49 views

Lee Smooth Manifolds Theorem 6.24 proof

I'm currently reading the book Smooth Manifolds by John M.Lee. There is something unclear to me in the proof of theorem $6.24$, the Tubular Neighborhood Theorem. The proof goes as follows. First, if $...
Mark's user avatar
  • 41.5k
0 votes
1 answer
32 views

Why is interior multiplication by $v$ an antiderivation implying $v\lrcorner\ \omega ^n = n(v\lrcorner\ \omega)\wedge\omega^{n-1}$?

In Proposition 22.8 of Lee's Introduction to Smooth Manifolds it is written that [For $V$ a $2n$-dimensional vector space, $v$ a vector in $V$ and $\omega$ a degenerate $2$-covector of $V$] interior ...
Sam's user avatar
  • 5,166
0 votes
0 answers
34 views

A $2$-covector $\omega$ is non-degenerate if and only if $(\omega_{ij})$ is non-singular

The following is from Lee's Introduction to Smooth Manifolds: Exercise 22.1. Show that the following are equivalent for $2$-covector $\omega$ on a finite dimensional vector space $V$: $\omega$ is ...
Sam's user avatar
  • 5,166
0 votes
0 answers
25 views

Comparability Riemannian Metric in chart

Let $(M, g)$ be a Riemannian manifold. I know the following result: For each $x \in M$ there exists a chart $V$ and a constant $C \geq 1$ such that for all $y \in V$ and $\eta \in T_y \mathbb{R}^n$ $$ ...
Metalhead's user avatar
1 vote
0 answers
54 views

Is this the correct value for integrating this differential form over a sphere?

I posted this problem but made a mistake. I redid my work and got another sensible looking answer (the volume of the unit sphere). Does this look like a correct solution? This isn't an assigned ...
Nate's user avatar
  • 894
2 votes
0 answers
71 views

Smoothness of Ricci flow solution on a closed interval

In the paper "Deforming the metric on complete Riemannian manifolds" by Wan-Xiong Shi, the author proves the following theorem which I copy verbatim below: Theorem 1.1. Let $(M, g_{ij}(x)$ ...
Joseph Kwong's user avatar
1 vote
1 answer
47 views

What justifies the use of global coordinates when computing the $L^p(\mathbb{T}^n)$ norm?

Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying: $$\int_{\mathbb{T}^n}|f|^p < \infty....
CBBAM's user avatar
  • 6,277
3 votes
1 answer
61 views

Completeness of vector field generated by family of diffeomorphisms

Let $M$ be a smooth manifold. Let $I$ be an open interval containing zero. A smooth family of diffeomorphisms (parameterised by $I$) is a smooth map $\varphi:I \times M \rightarrow M$ such that $\...
Joseph Kwong's user avatar
0 votes
0 answers
35 views

Locally Compact Lie Groups and Matrix Lie Groups

Let $G$ be any Lie group with a Lie algebra $\mathfrak{g}(n, \mathbb{R})$; but assume that I only deal with a locally compact, connected (topologically) component around its $e \in G$ element, say $K \...
iliTheFallen's user avatar
1 vote
1 answer
50 views

Inverse of smooth family of diffeomorphisms [closed]

Let $M$ be a smooth manifold, and let $I$ be an open interval. We say a map $\varphi:I \rightarrow \text{Diff}(M)$ is a smooth family of diffeomorphisms if the map $I \times M \rightarrow M$ given by $...
Joseph Kwong's user avatar
2 votes
0 answers
38 views

A curve in a single orbit in the space of Riemannian metrics

Let $M$ be a smooth manifold. Let $\mathcal M$ denote the collection of all Riemannian metrics on $M$. There is a right action of the product group $\mathbb R^+ \times \text{Diff}(M)$ on $\mathcal M$ ...
Joseph Kwong's user avatar
1 vote
1 answer
40 views

Is $\int_{M_1\sqcup M_2}\omega= \int_{M_1}i_1^*\omega + \int_{M_2}i_2^*\omega$?

If I want to compute the integral of a smooth compactly supported differential form over a disconnected oriented smooth manifold $M=M_1\sqcup M_2$ with two connected components $M_1$ and $M_2$, is it ...
Uncool's user avatar
  • 962
1 vote
0 answers
16 views

Constructing an isomorphism $H^{k}(M_1-W_1)\oplus H^{k}(M_2-W_2)\rightarrow H^{k}(M_1\# M_2)$

I'm back again. Let $M_1$ and $M_2$ be compact, orientable, connected smooth manifolds. In trying to understand the first question of this question, I want to construct an isomorphism $H^{k}(M_1-W_1)\...
A Name's user avatar
  • 316
0 votes
2 answers
63 views

Relation between Christoffel Symbols and metric tensor

I am learning about Riemannian geometry, and I have a question about relation between the Christoffel symbols and the metric tensor of the manifold. Is it true that the Christoffel symbols are unique ...
PermQi's user avatar
  • 579
0 votes
0 answers
7 views

When is the leaf of a foliation on the quotient by a properly discontinuous free group action isomorphic to a leaf on the total space?

Let $\tilde M$ be a manifold and $G$ a discrete group acting freely and properly discontinuously on $\tilde M$. Let $M:= M/G$ and $p: \tilde M \rightarrow M$ is the projection; then $p$ is a covering ...
rosecabbage's user avatar
  • 1,697
0 votes
0 answers
22 views

Is it possible to 'extend' $dF_p$ for contravariant tensors as follows?

From Lee's Introduction to Smooth Manifolds page 319-20: Just like covector fields, covariant tensor fields can be pulled back by a smooth map to yield tensor fields on the domain. (This construction ...
Sam's user avatar
  • 5,166
1 vote
1 answer
57 views

Open subset of paracompact manifold is paracompact?

I assume a smooth manifold is paracompact (including Hausdorff) instead of Hausdorff and second countable. We can show that a connected topological manifold is paracompact iff it's second countable. ...
wsz_fantasy's user avatar
  • 1,688
1 vote
1 answer
31 views

Page 67 Lee smooth manifold - transition map of charts on tangent bundle clearly smooth?

Lee wrote the following on page 67 on his Introduction to Smooth Manifolds The Tangent Bundle Now suppose we are given two smooth charts $(U, \varphi)$ and $(V, \psi)$ for $M$, and let $\left(\pi^{-1}...
wsz_fantasy's user avatar
  • 1,688
0 votes
0 answers
38 views

What's the continuous $\mathbb C^0$ version for immersion? I'm trying to see if topological embeddings are 'topological immersions.'

A 'smooth embedding' $f: M \to N$ between smooth ($m$,$n$)-manifolds ($M$,$N$) is a smooth map that is both a smooth immersion and a topological embedding, which is simply defined as that the range-...
BCLC's user avatar
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