# 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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### Topological connected eccentrics, not homeomorphic to commutative Lie groups

An eccentric is a universal algebra $\ (X\ \sigma\ \lambda\ \rho)\ $ such that operations
$\ \sigma\ \lambda\ \rho\,:\,X\times X\to X\ $ satisfy:
$\quad \forall_{x\ y\,\in X}\quad \lambda(\sigma(x\ y)...

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### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...

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### Homology of topological manifolds

Let $X$ be a topological manifold of dimension $n$ (assuming perhaps that there is a countable basis of open sets). Do NOT assume that $X$ is compact, or oriented, or triangulable (so do not assume it ...

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### prove a bondle is an indefinite Hermitian manifold which is Kahler if and only if the manifold is locally flat

Let $M(J,g)$ be an indefinite Kahler manifold, then $%
TM(J^{H},g^{D})$ is an indefinite Hermitian manifold which is Kahler if and
only if $M$ is locally flat. Here $J^{H}$ denotes the horizontallift ...

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### References on integration on non-compact manifolds

I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem.
In particular I have non-compact manifold $M$ and I have an integral (in ...

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### Calculate the Jacobian of a particular diffeomorphism of parallelizable manifold onto itself

Let $M$ be $d$-dimensional parallelizable manifold. Let $e_k(x)$ for $k=1, \dots , d$ be the smooth vector fields forming orthonormal basis in tangent bundle of 𝑀, these vector fields exist because ...

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### Generators and relations for the 2-dimensional unoriented cobordism category

It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...

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### An orientable surface that cannot be embedded into $\Bbb R^3$? [duplicate]

I previously asked this question on MSE, without success.
By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$.
Now, Wikipedia states in this ...

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### Signature of a non-compact manifold

Let $v_0,\dots,v_n\in\mathbb{Z}^2$ be integer vectors which satisfy the condition $\det\begin{pmatrix}v_{k-1}&v_k\end{pmatrix}=(-1)^k$, whose relevance will become apparent in a moment. We may ...

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### Is this $a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ exists and applied for manifolds with positive curvature?

In $1969$, Margulis proved, for suitable constant $h>0$and $r$ is a positive constant that :
$a(p)=\lim_{r\to \infty} \frac{VolS(p,r)}{e^{h r}}$ with ($(S(p,r)$ is geodesic spheres), exists at ...

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### Map which is null-homotopic on compacts

This is the missing ingredient towards answering my previous question.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...

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### Is limit of null-homotopic maps null-homotopic?

The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\...

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269 views

### Non-density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...

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### Classification of involutions on $G_{2}$-homogeneous spaces

Are you aware of a systematic classification of involutions on $G_{2}$-homogeneous spaces?

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### Every _______ $d$-manifold has an $S$-structure

I am looking for some analogous nontrivial but known statements and references about statements of the form:
Every _______ $d$-manifold has an $S$-structure.
Here _______ is a placeholder for ...

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### When is a manifold boundary a deformation retract of its open neighborhood?

For this question a manifold is a locally upper-Euclidean Hausdorff space; paracompactness or second-countability is not assumed, and boundary may be present.
Let $M$ be a manifold. What are ...

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### Existence of a Euler-like formula for the continuous image of $S^1$ in an orientable surface

Let $\mathcal{M}$ be a compact 2-manifold, and let $\gamma: S^1 \rightarrow \mathcal{M}$ be a continuous map (you can assume piecewise smooth if it is convenient), with the property that the set $A = \...

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### Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves

Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the ...

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### Density of continuous functions to interior in set of all continuous functions

Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...

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### Cut out an open ball from a 2-manifold and glue the boundary

I have a possibly elementary question. Let $\mathcal{M}$ be a manifold with $\text{dim} \; \mathcal{M} = 2$. Let $U \subseteq \mathcal{M}$ be homeomorphic to $\overline{\mathcal{B}(0,1)}$, and let $\...

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### Find a circle that intersects the image of $[0,1]$ in a manifold $\mathcal{M}$ at only 1 point

Let $\gamma : [0,1] \rightarrow \mathcal{M}$ be a continuous map so that $[0,1]$ is homeomorphic to $\gamma([0,1])$, where $\mathcal{M}$ is a manifold (Hausdorff, second countable, and locally ...

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### What is a sufficient set of links in a simplicial complex to represent any PL manifold?

The link of a vertex in a $n$-dimensional simplicial complex is the $(n-1)$-dimensional simplicial complex formed by the $(n-1)$-simplices that together with the vertex span a $n$-simplex. A ...

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### Physical interpretation of the Manifold Hypothesis

Motivation:
Most dimensionality reduction algorithms assume that the input data are sampled from a manifold $\mathcal{M}$ whose intrinsic dimension $d$ is much smaller than the ambient dimension $D$. ...

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### Is open subspace of manifold with collared boundary a manifold with collared boundary?

An $n$-manifold (for this question) is a locally upper-$n$-Euclidean Hausdorff space. Hence, a manifold possibly has a boundary and possibly is not paracompact. An $n$-precell (for this question) is a ...

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### Which properties, a warped product manifold $M$, can benefit from in having a complex subspace in its tangent space?

I have a Riemannian warped product manifold $M=B \times_f F$ where $M$ is not compatible with an almost-complex structure $J$, but (for example) $B$ is compatible with an almost-complex structure $J$.
...

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### Collared boundary of a non-metrizable manifold

For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally ...

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### Does injectivity of $\pi_1(\partial U) \to \pi_1(M)$ imply injectivity of $\pi_1(U) \to \pi_1(M)$?

Let $M$ be a smooth compact manifold of dimension $n$, and let $U$ be a smooth compact manifold with boundary, of the same dimension $n$, embedded in $M$.
The embedding induces maps on $\pi_1$.
If $...

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### Immersion in $\mathbb R^3$ of a Klein bottle with Morse-Bott height function without centers

Can the Klein bottle be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers?
That is, the height function would have only Bott-type extrema and ...

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### For which locally ringed spaces is the structure sheaf given by LRS morphisms to the real line?

Let $\mathsf{LRS}_{\mathbb R}$ denote the category of locally $\mathbb R$-ringed spaces.
Given a locally ringed space $(X,\mathcal O_X)$, write $C_{(X,\mathcal O_X)}^p$ for the hom-sheaf on $X$ of ...

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### Immersion of non-orientable surface in $\mathbb R^3$ with conditions on the height function

EDIT: The answer is trivially positive; the question arose from my misunderstanding of the figure below.
Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the ...

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### Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary

I'm looking for a reference for a statement like:
Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form
$\mathcal L ...

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### Is $H$ closed in $G$?

Every smooth manifold is assumed to be Hausdorff and second-countable.
Suppose $G$ is a Lie group, $H$ is a Lie subgroup of $G$, $N$ is a closed Lie subgroup of $G$ such that $N$ is normal, $N\cap H=\...

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### Is $S$ a smooth submanifold of $M$?

Let $G$ be a Lie group and $H$ a Lie subgroup of $G$.
Let $M$ be a smooth manifold.
Let $\theta$ be a left smooth action of $G$ on $M$.
Let $S=\{p\in M| G_p=H\}$, where $G_p$ is the isotropy ...

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### flow, stable manifold and tangent

Given vector field $f: \mathbb{R}^2 \to \mathbb{R}^2$, with $f(0)=0$
ODE: $\dot{x}=f(x)$ generates a flow $\Phi^{t}$. so $\Phi^{t}(0)=0$ for all $t \in \mathbb{R}$
So time-one map $\Phi^1$ is diffeo....

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### What is a Lipschitz continuous map between Riemann surfaces in Jost's book Compact Riemann Surfaces?

This appears in the section 3.7 of the book Compact Riemann Surfaces by Jurgen Jost, right after Lemma 3.7.3. The exact words are
Now let $v:\Sigma_1\to\Sigma_2$ be a Lipschitz continuous map. ...

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### (Non-)Orientability of non-triangulable manifolds

We heard and learned from Mike Miller's answer to Not all manifolds can be triangulated: In which dimensions? that "All orientable 5-dimensional manifolds are triangulable. In dimensions at least 6, ...

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### How to check conditions for Liouville-Arnold theorem? [closed]

Arnold gives in his book "Mathematical Methods of Classical Mechanics" on p.272 the following, well known theorem:
Let $F_1, \dots, F_n$ be $n$ functions in involution on a symplectic $2n$-...

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### Existence of universal arrow from manifolds to forgetful functor of Lie groups

Let $M$ be a manifold, and $U$ be the forgetful functor from the category of Lie groups to the category of manifolds. My question is whether there is a universal arrow $(G, i)$ from $M$ to $U$? More ...

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### Line bundles trivial outside of codimension 3

Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 ...

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### Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]

Please let me denote the following
(TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold
(PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF
(PL) ...

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### Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.
Cobordism Theory for DIFF/Differentiable/smooth manifolds
However, there are ...

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### Naturality of Poincaré–Lefschetz

Let $X$ be compact and Hausdorff, $A\subseteq B\subseteq X$ both closed such that $X\setminus A$ is an open orientable $d$-manifold. Then also $X\setminus B$ is an open orientable $d$-manifold. We ...

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### How do these topological results imply the inverse function theorem?

In this MO question, Terrence Tao inquires about the everywhere differentiable inverse function theorem. This answer claims the theorem may be deduced from fairly intricate topological results of ...

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### “Smooth” Serre Fibrations (?)

Let $M,N$ be manifolds, $f:M \to N$ be a map.
In order to understand if $f$ is a serre fibration, it is enough to test it against differntiable maps $I^p \to M, I^{p+1} \to N$? What about smooth maps?...

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### How much is the theory of piecewise-smooth manifolds different from the PL and smooth ones?

There are many differences between the smooth manifolds category and piecewise linear manifolds category when it comes to classification, embedding of these manifolds inside each other or otherwise. ...

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### Notion of knotting for a compact $n$-manifold inside an $m$-manifold (with the constraint $n=m−2$)

What is the notion of knotting for a compact $n$-manifold inside an $m$-manifold?

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### Unknotting a compact manifold in the PL setting

The general position theorem asserts that any $M$ $m$-manifold unknots in $R^n$ provided $n\geq 2m+2$. The general position theorem assumes a smooth setting. Is unknotting still hold in the PL setting?...

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### Notion of linking between two general $p$ and $q$ manifolds embedded in a higher dimensional manifold

Let $M$, $N$ be compact, connected, oriented manifolds without boundary embedded in $\mathbb{R}^m$ of dimensions $p$ and $q$ respectively. I know that when $m=p+q+1$ we can define the linking between $...

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### Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere

Is there an embedding (i.e. injective continuous map)
$$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$
of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is ...

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### Closed manifold with non-vanishing homotopy groups and vanishing homology groups

Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...