# 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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### Invariance of morse homology, doubt in proof in book “Morse Theory and Floer homology”

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Link to the statement of the theorem ...
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### What locales correspond to Manifolds?

I am studying the categorical equivalence between (sober) topological spaces and (spatial) locales with enough points. As the title implies, I am interested in finding localic analogues of both ...
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### Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence

Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to any self-homotopy ...
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### What are some “good” examples of Kan simplicial manifolds?

According to the definition 1.1 of the paper Kan Replacement of simplicial manifolds by Chenchang Zhu https://arxiv.org/pdf/0812.4150.pdf, A Kan simplicial manifold is a simplicial manifold $X$ such ...
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### Einstein submanifold of Einstein manifold - References

Is there any work in which the conditions under which an Einstein manifold admits an Einstein submanifold are studied? If yes, can you give me the references?
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### Existence of normal microbundles

In the same paper where Milnor introduced the concept of microbundles, he gave the following definition. $M$ has a microbundle neighborhood in $N$ if there is a neighborhood $U$ of $M$ in $N$ and a ...
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### On the dimension of the projected rank $r$ matrix space

In $d$-dimensional complex number space $\mathbb{C}^{ d}$, we can define the rank at most $r$ matrix space $$S=\{A|\ \mathrm{rank}(A)\leq r\}\subseteq \mathbb{C}^{d\times d}.$$ The dimension of ...
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### The group of isometries of a manifold is a Lie group, isn't it?

Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries ...
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### Are all monotonically normal manifolds of dimension at least two metrizable?

Alan Dow and Frank Tall recently proved the consistency of the statement Every hereditarily normal manifold of dimension at least two is metrizable. See: Dow, Alan; Tall, Franklin D., Hereditarily ...
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### A piecewise-linear or topological Fulton-MacPherson compactification

The Fulton-MacPherson compactifications of configuration spaces are smooth manifolds with corners which have the ordered configuration spaces of distinct points in a smooth manifold as their interior. ...
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### Is there a combinatorial representation of general topological manifolds similar to triangulations?

Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
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### Intersection of zero sets of continuous functions

Let the zero sets $F=\{x \in \mathbb{R}^n: f(x) = 0\}$, $G = \{x \in \mathbb{R}^n : g(x) = 0\}$, where $f$ and $g$ are $m$-dimensional real, analytic, continuous, and nonlinear vector functions. Under ...
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### Derivative $\left.\frac{\rm d}{{\rm d}t}\nu_{\partial T_t(\partial M)}\right|_{t=0}$ of outer normal field on a transformed boundary $T_t(\partial M)$

Let $d\in\mathbb N$, $v\in C_c^1(\mathbb R^d,\mathbb R^d)$, $X^x\in C^0([0,\infty),\mathbb R^d)$ denote the solution of $$T_t(x):=X^x(t)=x+\int_0^tv(X^x(s))\:{\rm d}s\;\;\;\text{for all }t\ge0\tag1$$ ...
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### minimal embedding space of a manifold in smooth and PL case

Given a manifold $M$, we can always embed it in some Euclidian space (general position theorem). Hence we can define the minimal embedding space of $M$ to be the smallest euclidean space that we can ...
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### Quotient space by discrete group and $L^2(\Gamma\backslash \mathcal{H})$

I'm reading Daniel Bump's "Automorphic forms and representations" chapter 2, and in the book they define an integral over $\Gamma\backslash\mathcal{H}$ (here, $\Gamma$ is a discontinuous ...
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### How can we calculate this tangential differential?

Let $\tau>0$, $d\in\mathbb N$ and $T_t$ be a $C^1$-diffeomorphism on $\mathbb R^d$ for $t\in[0,\tau]$ with $T_0=\operatorname{id}_{\mathbb R^d}$. Assume T_t(x)=x+\int_0^tv(s,T_s(x))\:{\rm d}s\;\;\...
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### On isospectral planar domains (and a paper by Buser, Conway, Doyle and Semmler)

I have never seen a short, elegant way (from the viewpoint of a non-topologist) which constructs isospectral planar domains from Sunada group triples, although essentially those triples live at the ...
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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 ...
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). ...
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 \$\...