# Questions tagged [smooth-manifolds]

Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

1,071
questions

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### Homotopy type of space of embeddings of a disk

Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ ...

2
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1
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210
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### Manifolds whose tangent spaces have a special behavior

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let
$$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$
be a local parametrization of $M$.
...

0
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0
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40
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### Does a vector over the field of meromorphic functions describe a manifold?

Assume that the variables $\mathbf x=(x_1,...,x_n)$ are coordinates on the solution manifold of a differential equation $\mathbf D(\mathbf x,\dot{\mathbf x},\ldots,\mathbf x^{(\alpha)})=\mathbf 0$ ...

6
votes

2
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381
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### The convex hull of a manifold whose cobordism class is trivial

Let $M$ be a compact orientable $n$ dimensional manifold. Assume that $M$ has trivial cobordism class.
Is there an embedding of $M$ in some Euclidean space $\mathbb{R}^m$ such that the convex ...

4
votes

1
answer

109
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### If tangent vectors are a vector space of same dimension at every point, does one has a manifold? [closed]

Let $M$ be a non-empty subset of $\mathbb R^n$, $n \geq 2$.
Recall that a vector $v$ is tangent to $M$ at the point $m \in M$ if it exists a differentiable curve $\gamma : I \to M$ such that $\gamma(0)...

5
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+100

### Is the volume functional analytic in the space of embeddings? What about locally?

Let $(M^{n+1},g)$ be an analytic Riemannian manifold and let $\Sigma^n$ be a closed analytic manifold. Denote by $\operatorname{Emb}(\Sigma, M)$ the space of all smooth (or maybe analytic) two-sided ...

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0
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203
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### Proof of Ehresmann's theorem

In Huybrechts' book Complex geometry: An introduction p.269, Proposition 6.2.2, the author gives a proof of the following theorem
(Ehresmann)
Let $\pi:\mathcal X\to B$ be a proper family of ...

7
votes

1
answer

232
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### On fixed point sets of actions of compact Lie groups

Let a compact Lie group $G$ act smoothly on a compact smooth manifold $M$. For any compact subgroup $H\subset G$ denote by $E^H$ the image in $M/G$ of the fixed point set of $H$ in $M$.
Is it true ...

3
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92
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### Geometric intuition behind definition of $\delta$-necklike points of the Ricci flow

In "The Ricci Flow: An Introduction", the authors define a $\delta$-necklike point of the Ricci flow as a point $(x, t)$ where $$\|\text{Rm} - R (\theta \otimes \theta)\| \leq \delta \|\text{...

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votes

1
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117
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### Pull back a vector field [closed]

In Voisin's book Hodge theory and complex algebraic geometry, I Section 9.1.2, p.223, the author writes:
Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi_*$ is a ...

10
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1
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382
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### Examples of 6-manifolds without an almost complex structure

Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...

11
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1
answer

248
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### Does every smooth map of rank at most d factor through a d-manifold?

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we ...

3
votes

1
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287
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### How to show that $\text{Man}(M,\mathbb{R}^n)\cong \mathbb{R}\text{-Alg}(C^\infty(\mathbb{R}^n),C^\infty(M))$?

I'm trying to show that manifolds are affine, i.e. $\text{Man}(M,N)\cong \mathbb{R}\text{-Alg}(C^\infty(N),C^\infty(M)) $. If I could show this for $N=\mathbb{R}^n$, then I know how to do the rest ...

14
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2
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691
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### Must a space that is locally injective image of $\mathbb{R}^n$ be a manifold?

Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is ...

8
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### A conjecture about homotopy $S^1\times B^3$'s

$\textbf{Conjecture}:$
Let $X^4$ be a homotopy $S^1\times B^3$ with the following properties:
Attaching a four dimensional 2-handle gives a standard $B^4$.
The $k$-fold cyclic cover is diffeomorphic ...

13
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1
answer

243
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### Mapping torus of Klein bottle

This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:
The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.
The mapping class group of ...

2
votes

1
answer

211
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### Path lifting property for $\pi:M\rightarrow M/G$ for $G$ compact Lie acting smoothly and freely

Let $M$ be a smooth manifold and let $G$ be a compact Lie group acting smoothly and freely over $M$. Let $\pi:M\rightarrow M/G$ be the canonical projection, and endow $M/G$ with the unique ...

3
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0
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63
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### Application of Santalo’s formula

Suppose that $(M,g)$ is a compact smooth Riemannian manifold with a smooth boundary and suppose that $f$ is a smooth function on $M$ with the property that
$$ \int_I f(\gamma(t))\,dt=0,$$
for any ...

3
votes

0
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47
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### The boundary of the transversal pre-image of a submanifold with boundary

A similar question on MSE without answer.
Let $M, N$ be smooth manifolds such that $\partial N=\varnothing$. Let $A$ be a smoothly embedded submanifold of $N$ such that $\partial A\neq \varnothing$. ...

1
vote

1
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88
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### Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood

A similar post on MSE without answer.
Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...

7
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1
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430
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### Intuition/meaning behind/physical content of the concept of a smooth structure

Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological ...

2
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0
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32
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### Find weak approximation by smooth unit vector fields for Sobolev fields on manifold

I am considering the Sobolev space of unit tangent vector fields on a compact manifold:
$Γ_{W^{1,2}}(M, UTM)$.
I would like to approximate those weakly with smooth vector fields ($Γ_{C^∞}(M, UTM)$).
...

1
vote

1
answer

86
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### Smoothness of the asymptotic parametrization of a ruled surface

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form
$$\begin{align}
\sigma \colon I \...

5
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0
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93
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### How is this product of tensors defined?

I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:
Here, $\Delta_{-2}$ denotes the usual Laplacian ...

2
votes

1
answer

115
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### Quantitative results for stabilizing tangent bundles of homology spheres

I'll begin with a broad question: if $M$ is a smooth manifold and $E \to M$ is a stably trivial bundle, can one determine lower bounds on the rank $k$ of the trivial bundle needed such that $E \oplus \...

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0
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70
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### Target space of Green's operator on $L^p$-differential forms on closed manifolds

Let $M$ be a closed (i.e., compact without boundary) smooth oriented Riemannian manifold endowed with a regular atlas in the sense of C. Scott [1], i.e., with a finite atlas $\mathcal{A}$ so that for ...

2
votes

1
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### Equidistant points on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold. To this Riemannian manifold, we associate a natural number $K(M,g)$ as follows:
$K(M,g)$ is the maximum of all $n\in \mathbb{N}$ such that we have at ...

7
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### A weak analogue of smooth manifolds (reformulated)

It is widely known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as ...

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0
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### Submersion function from a product space

Let $\Phi(x,y) \colon U_N \times U_M \to \mathbb{C}^n$ be a submersion, where $U_N \subset \mathbb{C}^N$ and $U_M \subset \mathbb{C}^M$.
Under which condition on $\Phi$ can I find some $s \in \...

6
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answers

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### Submersion vs fiber bundle

If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is ...

4
votes

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### Poincaré–Bendixson Theorem on a compact, connected, orientable, two-dimensional manifold

I'm currently reading the article "A Generalization of a Poincaré–Bendixson Theorem to Closed Two-Dimensional Manifolds" by Arthur Shwartz. The paper first establishes a result for minimal ...

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### Proper isotopy of proper embeddings of manifolds

We know from Theorem 2.2 in(http://www.map.mpim-bonn.mpg.de/Embeddings_of_manifolds_with_boundary:_classification#6.2) that
Any two smooth embeddings of closed oriented $n$-manifold(n>1) in $\...

3
votes

1
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179
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### Classification of "homogeneous" submanifolds of ℝⁿ

I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if:
it is a closed connected smooth submanifold of $\mathbb R^n$,
for every $p, q$ in $M$, there is a ...

3
votes

1
answer

190
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### Global choice of eigenvectors on an open surface

Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...

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0
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### Is this generalization of differentiable manifolds to mixed dimensions a known object?

Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls.
This would amount to studying the ...

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0
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### Fundamental groups and cellular walks

Suppose $M$ is a smooth manifold (compact if desired) with a cell structure or other nice stratification.
Call a path $\gamma : [0,1] \to M$ transverse to the stratification if there is a finite ...

4
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0
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### Coordinates on quotient manifold $\mathrm{SO}(3)/\Gamma$

$\DeclareMathOperator\SO{SO}$Say I have coordinates for $\SO(3,\mathbb{R})$, e.g., a parametrization by Euler angles. Is there a reasonable way to explicitly prescribe coordinates on the quotient ...

3
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### Vietoris-Begle type result for differentiable fiber bundle

In Vietoris-Begle Theorem, we consider a closed and surjective map between two paracompact and Hausdorff spaces and we get some relation involving the homologies of the fiber, total space, and the ...

5
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1
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### SRB measure and Gibbs u-state

I have been reading the famous paper of Alves, Bonatti, and Viana where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic ...

2
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1
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### In which dimensions is it true that every topological ball embedded by a smoothly embedded sphere is a smoothly embedded ball?

I asked a question on MSE with no answer. Here is my question in the generalized version.
Question 1: Suppose we are given a connected three-manifold $M$ (possibly non-compact, or non-orientable) and ...

2
votes

1
answer

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### Isometry and gluing between smooth manifolds - some references

I have a doubt that assails me.
The technique of gluing along edges between manifolds is generally considered in the topological context.
I don't know if there are other gluing techniques.
I was ...

8
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1
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363
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### Let $X$ be a manifold. Is it true that $\beta X\cong \operatorname{Specm}(C^\infty(X))$?

Let $X$ be a (smooth) manifold. It's well known that its Stone-Cech compactification $\beta X$ is homeomorphic to $\operatorname{Specm}(C(X))$, with its Zariski topology.
Is $\beta X$ also ...

2
votes

0
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138
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### Reference request for Poincaré–Lefschetz duality as an intersection pairing

I believe the following is well known after talking to some experts, but I am unable to find a reference for the case with boundary.
Fix a field $F$ and an oriented $n$-manifold $(M,\partial M)$. We ...

6
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292
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### Atiyah–Singer Index theorem for the pedestrian / layperson

So I came across the so-called Atiyah–Singer Index Theorem (ASIT) and claims of it being an extremely powerful and versatile tool.
Question. What is a truly simple application of the ASIT to obtain a ...

4
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0
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191
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### Jet Nestruev's proof that the exterior derivative $d$ on a real line is not a Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}$

The relation between the exterior derivative $d:C^\infty(\mathbb{R})\to\Omega^1\mathbb(\mathbb{R})$ and the Kähler differential $d_{C^\infty(\mathbb{R})/\mathbb{R}}:C^\infty(\mathbb{R})\to\Omega_{C^\...

1
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1
answer

278
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### Is the manifold of complex points of a quotient of compact groups just the tangent bundle?

In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...

5
votes

1
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164
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### Stable smoothing of topological manifolds relative to an embedding

Let $M$ be a topological manifold. We know that $M$ is stably smoothable if and only its tangent microbundle, up to stabilization, admits a reduction to vector bundle.
Now I wonder if there is a ...

1
vote

1
answer

96
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### Conditions for Lipschitzness of boundary normal vector, almost everywhere

Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition
(Unique CPP a.e). For almost every $x \in \mathbb R^n$, there exists a unique ...

0
votes

1
answer

125
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### Does a submanifold of nonzero codimension have measure zero under the product of non atomic measures?

Let $A$ be a non atomic measure on $\mathbb R$. Consider the product measure $\mu := A \times \dots \times A$ on $\mathbb R^n$.
Question: Let $M$ be a $n-1$ dimensional smooth submanifold of $\mathbb ...

8
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1
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392
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### What motivated Thom to relate the cobordism groups with some homotopy groups?

I would like to know what motivated or led Thom to think that the (un)oriented cobordism groups would correspond with the homotopy groups of some structure (Thom spectum), or with the coefficient ...