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].
167
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Finding a 1-form adapted to a smooth flow
Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
- 109k
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Compactification theorem for differentiable manifolds ?
Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...
- 22.7k
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4
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Parallelizability of the Milnor's exotic spheres in dimension 7
Are the Milnor's seven dimensional exotic spheres parallelizable?
- 1,226
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8
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What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ to be diffeomorphic to $\mathbb{R}^n$ :
For example :
Are all open star-shaped subsets ...
- 667
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1
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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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19
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How do I make the conceptual transition from multivariable calculus to differential forms?
One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
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When is a submanifold of $\mathbf R^n$ given by global equations?
Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
- 2,501
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What is meant by smooth orbifold?
There seems to be some confusion over what the tangent space to a singular point of an orbifold is.
On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-...
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4
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Can all n-manifolds be obtained by gluing finitely many blocks?
Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
- 10.2k
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Noncommutative smooth manifolds
Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples.
Using his definition it is unclear how to separate the smooth structure from the metric.
...
- 36.3k
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can you fool SnapPea?
A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with.
What I'm looking for is a non-hyperbolizable knot ...
- 43.1k
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1
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Can the product of an exotic torus and a circle be the standard torus?
As discussed in this question from last week, if $M$ is a closed manifold such that $M\times S^1$ is homeomorphic to the torus $T^{n+1}$, then $M$ is homeomorphic to $T^n$. Is the corresponding ...
- 18.8k
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Colimits of manifolds
This question tells us that in general colimits do not exist in the category of manifolds.
However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its altas....
user2529
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5
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Which manifolds admit a diffeomorphism of order $n$?
Let $n>1$. Which smooth manifolds admit a diffeomorphism $f$ of order $n$?
For a closed orientable surface $S_g$ of genus $g$ and $n=2$ the answer is in the affirmative, since $S_g$ can be ...
- 637
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2
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The double of a smooth manifold with boundary?
$\def\mc#1{\mathcal#1}\def\seq#1{\langle#1\rangle}\def\bbR{\mathbb R}\def\gt{>}\def\dom{{\rm dom\ }}$In some instances, I have seen an appeal to the concept of "the double" of a smooth manifold ...
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Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,246
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Classification problem for non-compact manifolds
Background
It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic).
I'm also under the impression that there is ...
- 1,841
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5
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How can you tell if a space is homotopy equivalent to a manifold?
Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
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How can we detect the existence of almost-complex structures?
Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto $...
- 6,039
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Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...
- 6,139
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0
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Do there exist exotic 4-tori?
More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a ...
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Are there two non-homotopy equivalent spaces with equal homotopy groups?
Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth ...
- 2,459
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Least number of charts to describe a given manifold
Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.
E.g. a circle requires at least two charts, and ...
- 2,881
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0
answers
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Monoid structure of oriented manifolds with connect sum
Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation ...
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6
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Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
- 1,264
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Are homology spheres stably parallelisable?
A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is ...
- 18.8k
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votes
5
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Smoothness of the closest point on a submanifold
Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
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Derivation on real analytic manifolds
Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...
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Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?
Summary
Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...
- 5,535
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3
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Characterizing Hessians among symmetric bilinear tensors
I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
- 5,831
15
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1
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Are there non-smoothable homotopy/homology spheres?
A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.
A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$.
Note, by ...
- 18.8k
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2
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Slice knots and exotic $\mathbb R^4$
In the http://arxiv.org/abs/math/0606464v1 I read
"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is ...
- 4,761
14
votes
0
answers
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Which spherical space forms embed in $S^4$?
Is there any hope of getting a classification of which 3-dimensional spherical space forms are smoothly embeddable in $S^4$? I read that lens spaces cannot embed in $S^4$, but some other spherical ...
- 161
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Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-...
- 2,954
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Is it true that exotic smooth R^4 cannot be diffeomorphic to RxN, where N is a 3-manifold?
Since $\mathbb{R}$ and any 3-manifold $N$ must be non-exotic, their product $\mathbb{R}\times N$ cannot possibly be diffeomorphic to exotic $\mathbb{R}^4$, correct?
Update: Andy Putman already ...
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2
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The boundary of a domain whose interior is diffeomorphic to the ball
We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors).
My question is about a very ...
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Gluing two diffeomorphisms together
A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
$\psi(...
- 8,782
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1
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Is every closed embedded codimension-n submanifold cut out by a section of a rank-n vector bundle?
Let $M$ be a smooth manifold (over $\mathbb R$) and $N \hookrightarrow M$ a closed embedding. Locally near any point in $N$, I can find coordinates $x^1,\dots,x^{\dim M}$ on $M$ so that $N$ is the ...
- 53.1k
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1
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is there a diffeomorphism with only finite orbits but of infinite order?
I asked this in stackexchange, but got no answer, so I am trying here.
Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have the following properties:
All its orbits are ...
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2
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Relating different topologies on $C^{\infty}_c(M)$
This is somehow connected to this question.
I can think of at least four topologies to put on $C_c(M)$:
Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\...
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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 ...
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What manifolds can have a (non-piecewise) linear structure?
By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert <...
user5810
7
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2
answers
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Ehresmann fibration theorem for manifolds with boundary
All manifolds in consideration may have nonempty boundary and may be disconnected.
Let me fix a definition first. A map between smooth manifolds $M\rightarrow N$ is a fiber bundle, iff it's locally ...
- 191
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votes
1
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Is there a smooth polar decomposition for non-invertible matrices?
Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...
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What is the usual topology of $C^\infty_c(M) $
If $M$ is a smooth paracompact manifold, then what is the usual topology of $C^\infty_c(M) $, i.e., the smooth function with compact support?
- 1,361
3
votes
1
answer
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Product rule for vector bundle (Leibniz rule)
Let $\pi:E\to Y$ be a vector bundle. Write $\mathrm{T}(E/Y)\subset \mathrm TE$ for its vertical bundle. Write $\delta:\mathrm{T}(E/Y)\cong \pi^\ast E:\ell$ for the vector bundle isomorphisms over $E$ ...
- 10.3k
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votes
2
answers
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Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma
I have a general question about the motivation behind to definition the smooth morphisms
as we know it from algebraic geometry. The most common
definition of a smooth morphism $: X \to Y$ between two ...
- 6,000
2
votes
0
answers
182
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Foliation values of a manifold
Let $M$ be a smooth n dimensional manifold. The foliation values of $M$, denoted by $F(M)$, is defined as
\begin{equation} F(M)=\{ 1\leq k\leq n\mid \text{there exist an smooth $k$ dimensional ...
- 312
1
vote
1
answer
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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 ...
- 6,726
1
vote
1
answer
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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 ...
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