# Questions tagged [manifolds]

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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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)...

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### Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...

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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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### Converse to Hopf degree theorem

Below, I mean smooth oriented closed connected manifolds and smooth maps (but am happy to hear about the topological category, or unoriented manifolds, etc instead).
Say that $X^n$ has the Hopf ...

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### composition of two tangent vector fields and connection [closed]

Let $X, Y$ be two tangent vector fields on a manifold.
Consider their composition:
$(X \circ Y)(f) = X(Y(f)) = X^i\frac{\partial Y(f)}{\partial u^i} = X^i\frac{\partial Y^j}{\partial u^i}\frac{\...

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### Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest .
Consider a ...

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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 ...

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### (Homotopy) colimit and manifold

Suppose that I have an arbitrary regular CW complex. By associating a topological space to each vertex of the CW complex, I can have a diagram of topological spaces, denoted by $D$, over the CW ...

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### Universal cover with one end

Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$.
Is $M $ necessarily homeomorphic to the total space of some ...

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### Persistence of planar trajectory converging to a node / focus

I consider a planar system $\dot u =F(u,p)$ where p is a scalar parameter. Suppose that the flow $\phi^t(u_0; 0)$ from $u_0$ converges to a stable node / focus $x^{eq}_0$ for the parameter value $p=0$....

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### Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold?

After reading some recent questions on mathoverflow about universal coverings, I am curious about the following:
Is it possible to construct a closed $6$-manifold $M$, with universal cover ...

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### Universal covers of non-prime 3-manifolds

Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...

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### Is there a closed manifold whose universal cover is $\mathbb{R}^n\setminus\{x_1, \dots, x_k\}$ for some $k > 1$?

There are many closed manifolds with universal cover homotopy equivalent to $\mathbb{R}^n$, they are precisely the closed aspherical manifolds. There are also many closed smooth manifolds with ...

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### Which quantity could hold its convergence under Gromov-Hausdorff convergence?

Recently I've been reading T.H.Colding's paper of Ricci curvature and volume convergence. A proof of the continuity of volume functions was given under the lower Ric bounded condition.
Having searched ...

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### Sobolev embedding theorems on manifolds

I had asked the following question on math.stackexchange but did not get any response:
I'm looking for a reference which states the Sobolev embedding theorems on Riemann manifolds for fractional ...

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### When does the diagonal cohomology class of a non-compact oriented manifold vanish?

Let $M$ be a non-compact, connected and oriented topological $d$-manifold without boundary. My understanding is that there are two (equivalent) ways of defining the diagonal class $\delta_M \in H^{d}(...

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### Can someone explain this proof on aspherical manifolds?

I am trying to understand this proof that the fundamental group of an aspherical manifold is torsion free. The proof is lemma 4.1 from Aspherical manifolds at the Manifold Atlas Project. The proof is:
...

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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 $\...

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### Mathematics Roadmap [closed]

I immediately apologize for my English, Google translator is my assistant. I couldn't find the information in my own language.
My question is addressed to people who understand mathematics. I hope for ...

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### Is there a version of the Poincaré–Hopf theorem for manifold with corners?

As we know, the square $S=[0,1]\times[0,1]$ is not a manifold with boundary. Instead, it's a manifold with corners. For a tangent vector field on a compact manifold with boundary, we have the Poincaré–...

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### On eigenfunctions of the Laplace Beltrami operator [closed]

How can we generate the eigenspace for the Laplace Beltrami operator on SU(2)?

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### Is every (not necessarily PL-) triangulation of a manifold pure, non-branching and strongly-connected?

A triangulation of a topological manifold $\mathcal{M}$ possibly with boundary is an abstract simplicial complex $\Delta$ together with a homeomorphism $\varphi:\vert\Delta\vert\to\mathcal{M}$, where $...

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### Analogue of Kolmogorov/Arnold superposition for general manifolds?

Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...

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### Decomposition of manifolds with toroidal boundary

Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following this article, it is stated that $\mathcal{M}$ can be written as
...

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### Classification of 3-dimensional manifolds with boundary

It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as
$$\mathcal{M}=P_{1}\#\dots\# P_{n}$$
where $P_{i}$ are prime manifolds, i.e. ...

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### Obstruction of smooth structure

The first 24 lectures of Jacob Lurie on Geometric Topology [1]
gave a concise introduction to the comparison of smooth manifolds
and piecewise-linear manifold. In the first five lectures, it is
shown ...

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### Equivariant embeddings

According to the article I got from this question, closed Riemann manifolds together with their isometry groups (which is always a compact Lie group) can be smoothly and equivariently embedded into ...

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### Relation between cohomological dimensions of manifolds

$\DeclareMathOperator\Ch{Ch}$Let $M$ be a connected manifold of finite type. We denote $\Ch_{\mathbb{Q}}(M),$ $\Ch_{\mathbb{Z}}(M)$ and $\Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of ...

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### CW-complexes that cannot be homotopically compressed

Definition. A CW-complex $A$ can be elementary compressed to a CW-complex $B$ if there is a deformation retraction $A \to B$ or a quotient map by a contractible subcopmlex $A \to B$ (meaning according ...

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### Domain of definition of a certain mapping

Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold.
I am studying the mapping
$$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...

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### Set of proper homotopy classes of arcs in a manifold

Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that $...

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### Approximating maps by embedding

A continuous map $f$ between two metric spaces is said to be a $r$-map if preimage of each point under $f$ has diameter atmost $r$. Suppose $D^n=\{x\in \mathbb{R}^n\mid ||x||\leq 1\}\subset \mathbb{R}^...

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### Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary?

Does there exist a manifold with finitely generated homology groups that is not homotopy equivalent to a compact manifold with boundary?
I am also interested in several variations of this question. ...

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### A Riemann surface is automatically paracompact

[A question I remember from many years ago.]
Definition
A Riemann surface is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is ...

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### non-smooth manifold with circle action (with fixed points)

I am interested to know if there a non-smooth manifold (i.e. a closed topological manifold admitting no smooth structure) $M$, having a continuous action $M \times S^1 \rightarrow M$, and the number ...

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### Open manifolds which have stable $\pi_1$ at infinity but are not inward tame

Let $M$ be a $1$-ended open manifold. An important result of Siebenmann states that (in dimension $\geq6$) if $M$ is $(i)$ inward tame, i.e. for every closed neighborhood of infinity there exists a ...

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### Can we embed a closed manifold into a homotopy equivalent CW complex?

Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (...

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### Classifying singularities of the Ricci flow

Context:
A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and:
$$
\sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...

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### Are there examples of Einstein manifolds with unbounded curvature?

Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$
I'm looking for these examples because they might provide a counter-example to a ...

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### Roadmap for L-Theory

Background: I spent sometime reading about algebraic K-theory and started reading research papers on the subject with relative facility at least I do understand constructions, statements of the ...

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### Homotopy equivalent cartesian product of closed manifold

I'm little bit lost with the following question:
I have four connected closed orientable manifolds $M,N,S,S'$ such that $S$ and $S'$ are homotopy equivalent and $M\times S$ is homotopy equivalent to $...

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### Do most manifolds have symmetries? or not?

Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere ...

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### Contractible set in a manifold

Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F(...

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### Is $\operatorname{Spin}(8)$ a direct product of $\operatorname{Spin}(7)$ and $S^7$?

Is $\textrm{Spin}(8)$ a direct product of $\textrm{Spin}(7)$ and $S^7$?
I met this statement in the literature, but without a reference. If it is true, where is it explicitly written?

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### Hypothesis on $W$ to achieve $d(x,F(W))<\sum\epsilon_n$

Writing a paper, I'm trying to formulate the following technical result:
Let $X$ be a manifold, $W\subset\subset X$.
Let $f_k:U_k\to X$ be continuous, where $U_k\subset X$ is an open neighborhood of $...

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### Do colimits of manifolds coincide with underlying colimits as topological spaces?

Categories of manifolds (possibly with extra structure) tend not to have all colimits.
Other questions have addressed when colimits of manifolds exist.
I'd like to know what we can say in general ...

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### When is a triangulation of sphere two-colorable?

Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors.
I ...

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### Birth of chaos due to nonautonomous perturbation

Let $\sigma, b>0$. I want to study the dynamics of the map
$$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that
$$T_{\sigma,b}(n,\theta,y) = (\...

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### Embedded submanifold in a cylinder

Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $...

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### Suspension of a topological space

Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?