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
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Almost isometric manifolds are diffeomorphic
I am looking for a reference to the following statement.
(It should be known --- I saw it before, don't remember where; search by keywords did not help.)
Let $f\colon M\to N$ be a homeomorphism ...
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0
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80
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What 'large' surfaces are there?
I answered this question on "is there a longest geodesic" by a kind of a joke, which I couldn't resist: the long line! Simply going by the name it had to be the 'longest geodesic'! I didn't ...
2
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1
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Approximating continuous functions via diffeomorphisms on compact manifolds
Let $M$ be a compact and connected manifold without boundary.
My question is how to prove the following fact which I believe is true:
If $f : M \to \mathbb{R}$ is a continuous function that attains ...
2
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0
answers
188
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Intuition behind Nakano positivity
I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don't understand what is the geometric meaning of it. Let me briefly ...
4
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1
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483
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Path integral presentation of solutions of Dirac equation
It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold).
Is there a way to present solutions of the Dirac equation using path ...
12
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0
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252
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Smooth dual cell structure
Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric ...
4
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0
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639
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Equality of Hausdorff measure and Lebesgue measure on manifolds (reference)
Let $\mathcal{M} \subset \mathbb{R}^N$ be an $n$-dimensional $C^1$ submanifold (connected). We have two metric functions on $\mathcal{M}$:
The Euclidean distance inherited from $\mathbb{R}^N$.
The ...
3
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0
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123
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Restriction of complete 1-forms to closed submanifolds (Sharpe's book on Cartan geometries)
In his Book Differential Geometry: Cartan's generalization of Klein's Erlangen Program, Sharpe gives the following definition of a complete 1-form:
Soon thereafter he gives the following example:
I ...
4
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1
answer
434
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The maximum number of vertical independent vector fields on the tangent bundle
Let $M$ be a differentiable manifold.
Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for ...
1
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1
answer
123
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Realizing a set as the image of a smooth map
Consider the following subset of $\mathbb{R}^2$:
$S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$
(See here for a plot on Wolfram Alpha.)
The origin $(0, 0)$ is a kind of singular point of $S$....
4
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0
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139
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Push forward of Chern character and index theorem
I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998).
I expose here the setup for my ...
0
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0
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236
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Homology of a closed $3$-manifold with balls removed
This question has been posted on MSE with no answers.
Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
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1
answer
171
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A question regarding the action of a Lie subgroup
Suppose $H$ is a closed subgroup of a Lie group $G$. Then in Lee's book Introduction to Smooth Manifolds (Ch. 9) he showed that the action $H\times G\to G$ $(h,g)\mapsto gh$ is a smooth, free, proper ...
1
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0
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281
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Number of smooth structures on a compact manifolds
I'm new to this field: is there a compact smooth manifold of dimension $n\geq 5$ with uncountably many smooth structures no two of which are diffeomorphic. This is motivated by Milnor's exotic sphere $...
0
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1
answer
131
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Local triviality condition in vector bundles [closed]
Let $E$ and $M$ be smooth manifolds (of finite dimension, Hausdorff and second countable). Let $\pi:E\longrightarrow M$ be a surjective submersion such that:
$E_p:=\pi^{-1}(p)$ is a real vector space ...
4
votes
0
answers
201
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Fréchet subdifferentiation on riemannian manifolds
Context. I'm looking for a "natural" definition of subdifferentials on riemannian manifolds.
Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \...
2
votes
0
answers
73
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Is this family of minimal tori compact?
Let $\Sigma$ be a smooth $2$-sphere and let $M = \Sigma \times \mathbb{S}^1$. Fix an integer $n \geq 0$. Is there generic set $\mathcal{S}$ of Riemannian metrics on $\Sigma$ such that the following ...
6
votes
1
answer
173
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Decomposition of real algebraic varieties into manifolds
I apologize in advance if this question is too elementary for MO. I am new to the field of algebraic geometry.
I am dealing with a (real) algebraic variety $V$ of (Krull) dimension $n$. I keep reading ...
2
votes
1
answer
210
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Embedding of the adjoint group into $\mathrm{GL}(\mathfrak{g})$
Given a connected Lie group $G$ with corresponding Lie algebra $\mathfrak{g}$, the adjoint representation/action $\mathrm{Ad} : G \to \mathrm{GL}(\mathfrak{g})$ induces a Lie group homomorphism. It's ...
1
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0
answers
84
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Curvature of a superconnection
Let $E\rightarrow X$ be a $\mathbb{Z}_2$-vector bundle (or superbundle for connoisseurs) and consider the superconnection
$$A=\nabla + B$$
where $\nabla$ is a connection on $E$ and $B\in\Gamma(End(E))^...
1
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0
answers
65
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Higher order Leibniz rule for higher order tangent space
Let $M$ be a smooth manifold (finite dimensional, Hausdorff and second-countable) and $p\in M$ a point.
The higher cotangent space at $p$ is defined to be quotient:
$$ {T^*_p}^rM:= \eta_p/\eta_p^{r+1} ...
5
votes
2
answers
846
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What are the sufficient and necessary conditions for surjective submersions to be locally trivial
Let $E$, $M$ be smooth finite dimensional, Hausdorff and second-countable manifolds. Let $\pi:E \longrightarrow M$ be a surjective submersion.
$\pi$ is locally trivial if $\forall p\in M$, $\exists U \...
4
votes
0
answers
229
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Infinitely many simple closed geodesics in any compact orientable surface but the sphere
My question is the following: if $(\Sigma, g)$ is any compact orientable Riemannian surface of genus $g \geq 1$, is it true that there are infinitely many closed, simple and geometrically distinct ...
6
votes
1
answer
272
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Geometric intuition for $R[x,y]/ (x^2,y^2)$, kinematic second tangent bundle, and Wraith axiom
This is a sort of continuation of this question.
In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that ...
13
votes
2
answers
710
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Can every element of a homotopy group of a smooth manifold be represented by an immersion?
I originally posted this on MSE but didn't get much of a response, so I'll attempt to post it here. Let me know if this is not appropriate.
Let $M$ be a smooth manifold of dimension $n$. Let $\alpha \...
5
votes
1
answer
493
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When is a diffeomorphism a bundle map?
Let $F\rightarrow E_0 \rightarrow B$ and $F\rightarrow E_1\rightarrow B$ be two smooth fiber bundles. Suppose $E_0$ and $E_1$ are diffeomorphic.
What are the obstructions for $E_0$ and $E_1$ to be ...
3
votes
2
answers
2k
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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 ...
3
votes
1
answer
217
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What is the name of this geometric structure, where we identify each sphere of vision with the sphere at infinity?
If you consider hyperbolic $n$-space $H^n$, modeled by the open unit ball $B^n \subset \mathbb{R}^n$, then given any two distinct points $x_1$, $x_2$ in $H^n$, there is a natural way of identifying ...
3
votes
1
answer
516
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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$ ...
2
votes
0
answers
325
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Differences between induced vector fields on a smooth manifold and on a principal bundle
In the context of the connections on fibre bundle, I have found some difficulties trying to understand the fundamental vector field (my reference is Nakahara, but I'm having some problems with the ...
9
votes
1
answer
549
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Local behavior of smooth triangulations
If $M$ is a smooth $n$- manifold, a smooth triangulation is defined to be a homeomorphism from a simplicial complex $K$ to $M$ whose restriction to each simplex is a smooth embedding. It's a well-...
17
votes
2
answers
983
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Homotopy groups of Diff(X) and Homeo(X)
For a compact closed smooth manifold $X$, the group Diff(X) has a natural homomorphism $\Phi$ to the homeomorphism group Homeo(X). If $X$ has dimension at least $5$, I'm looking for some general ...
3
votes
2
answers
274
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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 ...
5
votes
1
answer
585
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What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?
(I asked it first in MathStackExchange but I haven't get an answer yet)
Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms.
For unramified ...
0
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0
answers
57
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Every disk in $(S^2 \times S^1) \setminus B$ whose boundary lies in $\partial B$ separates
Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. Is it true to assert that every embedded $2$-disk $D \subset M$ such that $D \cap \partial M = \partial D$ ...
1
vote
0
answers
71
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Conditions ensuring that the outward-pointing unit normal field on the boundary of a manifold has a Lipschitz continuous extension
Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $\alpha\in\mathbb N\cup\{\infty\}$, $M$ be a $k$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary, $\nu_{\partial M}$ denote the ...
2
votes
0
answers
211
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Show that the manifold interior is invariant under this flow
Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ be continuous in the first argument with $$\sup_{t\in[0,\:\tau]}\left\|v(t,x)-v(t,y)\right\|\le c\left\|x-y\right\|\tag1\;\...
1
vote
1
answer
125
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Collection of local defining maps for smooth Euclidean submanifolds
Suppose the set $S \subset \mathbb{R}^{n}$ is a smooth submanifold of dimension $k$, that is [Lee, Proposition 5.16] for every $x \in S$ there exist an open set $W \subset \mathbb{R}^{n}$ and a smooth ...
1
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0
answers
57
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Rigidity case of a geometric theorem for $3$-manifolds with boundary
Let $(M^3,g)$ be a compact Riemannian $3$-manifold with boundary. In a paper by L. Ambrozio, he considers the set $\mathcal{F}_M$ of all immersed disks in $M$ whose boundaries are curves in $\partial ...
10
votes
2
answers
667
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4-dimensional cohomology $\mathbb{CP}^2$'s
Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?
8
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1
answer
226
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Separating spheres in $3$-manifolds of positive scalar curvature and mean convex boundary
Recently, A. Carlotto and C. Li proved a complete topological classification of those compact, connected and orientable $3$-manifolds with boundary which support Riemannian metrics of positive scalar ...
3
votes
1
answer
348
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Genericity of equivariant embeddings
I'd like to ask an equivariant version of this question.
Let $M$ be a closed manifold equipped with the action of a compact Lie group $G$. By the Mostow-Palais embedding theorem, $M$ can be embedded ...
22
votes
2
answers
1k
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If the universal cover of a manifold is spin, must it admit a finite cover which is spin?
If $M$ is non-orientable, then it has a finite cover which is orientable (in particular, the orientable double cover).
If $M$ is non-spin, then it does not necessarily have a finite cover which is ...
0
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0
answers
78
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Generalization of: The dimension of a projective $\mathbb{F}$-variety equals the smallest codimension of a disjoint linear subspace
Let $\mathbb{F}$ be an algebraically closed field. Consider the following definition of the dimension of a (quasi)projective $\mathbb{F}$-variety, given in Harris Algebraic Geometry: A First Course:
...
3
votes
1
answer
163
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Connected manifold without connected regular level set admits exactly two connected components
Let $M$ be a connected smooth manifold and $f \in C^\infty(M)$ such that $0$ is a regular value of $f$. Moreover, suppose that $f^{-1}(0)$ is connected. Is it true that $M \setminus f^{-1}(0)$ has ...
3
votes
1
answer
90
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Behaviour of mass for currents with disjoint supports
I am sorry if this is a basic question, but I don't think in MSE I will receive any answers.
Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral ...
1
vote
0
answers
127
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Nontrivial integer homology class implies orientability
I posted this question on MSE and I would like to see if my reasoning is correct.
Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and ...
10
votes
2
answers
1k
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If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic
I am currently reading Kervaire-Milnor's paper "Groups of Homotopy Spheres I", Annals of Mathematics, and I am trying to prove (or disprove) the following result. The more elementary the ...
9
votes
1
answer
307
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Every immersion can be deformed to have only transverse self-intersections
I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it.
Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true ...
15
votes
1
answer
429
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Status of a conjecture of C.T.C. Wall?
In Wall's paper Unknotting tori in codimension one and spheres in codimension two, he states the following conjecture:
Any $h$-cobordism of $S^3 \times S^1$ to itself is diffeomorphic to $S^3 \times ...