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].
1,231
questions
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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 ...
6
votes
1
answer
212
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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
votes
1
answer
279
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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 ...
1
vote
1
answer
238
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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
answer
436
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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 ...
4
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1
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442
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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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0
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337
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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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307
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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
332
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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
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1
answer
201
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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
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1
answer
172
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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 ...
1
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1
answer
320
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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
votes
1
answer
524
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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 ...
5
votes
1
answer
342
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Embedding round manifolds into low dimensional spheres
Robert Bryant's answer to Isometric embedding of SO(3) into an euclidean space mentions that there is an isometric embedding of the round tetrahedral space $ SO_3/A_4 $ into the round sphere $ S^6 $.
...
9
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0
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328
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Nash embedding for 3 manifolds
The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$ (edit: 14 is a better bound for compact 3 manifolds thanks @mme). What ...
8
votes
1
answer
567
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Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
4
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350
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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 ...
0
votes
0
answers
128
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Betti number of the boundary of a 4-manifold
Let $M$ be a compact $4-$manifold with boundary $dM$. If $M$ has the homotopy type of a wedge of $2-$spheres then is it always true that $b_1(dM)=0$? By $b_1$, I mean first Betti number.
It is known ...
10
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1
answer
618
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Algebraic atlas on smooth manifolds
A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition ...
25
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2
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Conceptual definition of the extension of a connection to 1-forms
I have a question that arose while reading Milnor's "Characteristic Classes". I will use the notation from that book.
Let $M$ be a smooth manifold and let $\zeta$ be a complex vector bundle ...
2
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0
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74
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Is the reversibility of inflation of a subset equivalent to its smoothness?
$D_r(x)$ denotes a closed ball of radius $r$ centered at $x$.
Definition. Let $M \subset \mathbb{R}^n$.
$D_r (M): = \bigcup\limits_{x \in M} D_r (x)$
$Int_r (M): = \{x ~|~ D_r(x) \subset M\}$
...
5
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0
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When are nodal lines on a sphere geodesics?
Let $(S^2, g)$ be a Riemannian sphere and let $L := \Delta_{S^2} + q$ be a Schrödinger operator on $S^2$. Suppose that $L$ has index equal to one and that $u \in C^{\infty}(S^2)$ ($u \neq 0$) lies in ...
4
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146
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Submanifolds of $4$-manifolds and their intersections
Suppose we have two (oriented) submanifolds $A,B$ of an oriented $4$-dimensional manifold $M$, that intersect transversally. Looking at the standard references for $4$-manifolds, I couldn't find a ...
2
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0
answers
74
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Which projective planes are connected manifolds?
Inspired by a nice recent MO question, I thought I would ask a similar one: which projective planes $P$ can be given a topology and structure of a smooth connected manifold where the lines form smooth ...
4
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0
answers
664
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Submersion and fiber bundle
It is known by Ehresmann's result, that proper surjective submersions are fiber bundles. The properness of a map somehow relates to the compactness of the fibers or the level sets. So my question is ...
8
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2
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533
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Presentations of exotic 4-manifolds
TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed).
Are there known presentations of $4$-manifolds $M$ with exotic structures, ...
2
votes
0
answers
68
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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 \...
4
votes
1
answer
311
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A Fréchet space characterization of smooth structures on topological spaces?
For a compact manifold $M$ the space of smooth functions $C^{\infty}(M)$ is a Fréchet space where the seminorms are the suprema of the norms of all partial derivatives. Is there some way to ...
17
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2
answers
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Are there only two smooth manifolds with field structure: real numbers and complex numbers?
Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - $\mathbb{R}$ ...
4
votes
0
answers
87
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$1$-parameter family of minimal embeddings and the maximum principle
Let $M^3$ be a closed orientable smooth manifold and let $g_t$ be a (smooth) $1$-parameter family of Riemannian metrics on $M$, $t \in \mathbb{R}$. Let $P \subset M$ a fixed closed orientable embedded ...
6
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0
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565
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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. ...
3
votes
1
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232
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do hyperfunction solutions always exist?
I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...
5
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158
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A-spaces vs generalized spaces
$\newcommand\Set{\mathit{Set}}$I've been searching about the notions of smooth generalized spaces and I come across with two definitions that seem very good ones.
John Baez and Alexander Hoffunung ...
4
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348
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Kirby's theorem for 4-manifolds
In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related ...
4
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Being a product - from homology to topology
The famous Kunneth formula expresses the homology of a product manifold as the tensor product of the two algebras.
Now suppose we know that a manifold $X$ has a decomposition $H_*(X) \simeq A \otimes ...
10
votes
1
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250
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Are there 4d state sum models, extended TQFTs or chain mail invariant that detect smooth structures?
A state sum model is a smooth invariant defined on smooth triangulated, or PL manifolds, by summing a local partition function over labels attached to the elements of the triangulation.
Typical ...
2
votes
1
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196
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Self study guide to Hamiltonian Monte Carlo
I was wondering if anybody has a suggested self-study path to understand the mathematical aspects on Hamiltonian Monte Carlo.
In this paper The geometric foundations of Hamiltonian Monte Carlo
it is ...
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0
answers
143
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Maps on Riemannian manifold agreeing with geodesics
Let $(M, g)$ be an $n$-dimensional smooth Riemannian manifold. Let
$$ \Gamma_n = \{ [0,1] \ni t \mapsto \gamma_x^y(t) = (1-t)x + ty \mid x, y \in \mathbb{R}^n \}.$$
Let us choose $m \in M$. Can we ...
4
votes
1
answer
338
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h-cobordisms between non-simply-connected 4-manifolds
Let $M_0^4$ and $M_1^4$ be two closed smooth 4-manifolds and let $M$ be an $h$-cobordism between them (i.e., a compact smooth 5-manifold with boundary the disjoint union of $M_0$ and $M_1$ and with ...
1
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1
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191
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Products of manifolds and locally ringed spaces (over $\mathbb{R}$) coincide?
Let $M$ and $N$ be smooth manifolds. The cartesian product $M\times N$ has a natural manifold structure. Moreover, $M$ and $N$ can be seen as locally ringed spaces over $\operatorname{Spec}\mathbb{R}$ ...
4
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174
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What are the "local degrees of freedom" in the space of smooth functions?
Let $C^k$ be the set of $k$th-order smooth real functions $f:\mathbb{R}\to\mathbb{R}$, and $C^\infty$ the set of smooth real functions. One can specify an $f\in C^k$ by specifying all its derivatives ...
0
votes
1
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267
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Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow
In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $...
5
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0
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131
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Specify the embedding of Lie groups (via the representation map) precisely as the embedding of two differentiable manifolds
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
4
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0
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174
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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)} \|\...
2
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0
answers
286
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Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
13
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1
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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 ...
9
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0
answers
197
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Donaldson invariants for piecewise-linear $4$-manifolds
It is well known that in dimension $4$, the notion of piecewise linear manifolds and the notion of smooth manifolds are the same [1][2]. On the other hand, the computations of Donaldson invariants ...
8
votes
2
answers
310
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Smooth rank one foliations with closed leaves
Let $F$ be a smooth rank one foliation on a manifold $M$. Suppose that all leaves of $F$ are compact (that is, circles). Then its leaf space (edit: when additional assumptions are taken) is an ...
2
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0
answers
448
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Writing a Taylor series with covariant derivatives (connections)?
A connection of a vector bundle $E$ on a manifold $M$ is a map $d_E: \Omega^0(E) \to \Omega^1(E)$ that extends uniquely to a map $d_E: \Omega^p(E) \to \Omega^{p+1}(E)$ while satisfying
$$
d_E(\omega \...
1
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1
answer
312
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How to compute singular homologies of affine hypersurface in $A^4$ [closed]
I was trying to compute singular homology in integer coefficient of the hypersurface $t^2-1=z^{n}+x(xy-1)$ contained in $A^4$. Can anyone help me computing that? Can anyone tell me some reference ...