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].

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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 ...
Anton Petrunin's user avatar
1 vote
0 answers
80 views

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 ...
Mozibur Ullah's user avatar
2 votes
1 answer
187 views

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 ...
L.F. Cavenaghi's user avatar
2 votes
0 answers
188 views

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 ...
Dubious's user avatar
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4 votes
1 answer
483 views

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 ...
asv's user avatar
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12 votes
0 answers
252 views

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 ...
Anibal Medina's user avatar
4 votes
0 answers
639 views

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 ...
Behnam Esmayli's user avatar
3 votes
0 answers
123 views

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 ...
Carlos Esparza's user avatar
4 votes
1 answer
434 views

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 ...
Ali Taghavi's user avatar
1 vote
1 answer
123 views

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 $S$....
Nicolas Boumal's user avatar
4 votes
0 answers
139 views

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 ...
BinAcker's user avatar
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0 answers
236 views

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 $...
Eduardo Longa's user avatar
1 vote
1 answer
171 views

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 ...
A beginner mathmatician's user avatar
1 vote
0 answers
281 views

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 $...
Rachid Atmai's user avatar
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0 votes
1 answer
131 views

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 ...
alexpglez98's user avatar
4 votes
0 answers
201 views

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 \...
dohmatob's user avatar
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2 votes
0 answers
73 views

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 ...
Eduardo Longa's user avatar
6 votes
1 answer
173 views

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 ...
BGJ's user avatar
  • 439
2 votes
1 answer
210 views

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 ...
Clement Yung's user avatar
1 vote
0 answers
84 views

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))^...
BinAcker's user avatar
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1 vote
0 answers
65 views

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} ...
alexpglez98's user avatar
5 votes
2 answers
846 views

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 \...
alexpglez98's user avatar
4 votes
0 answers
229 views

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 ...
Eduardo Longa's user avatar
6 votes
1 answer
272 views

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 ...
Arrow's user avatar
  • 10.3k
13 votes
2 answers
710 views

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 \...
Paul Cusson's user avatar
  • 1,735
5 votes
1 answer
493 views

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 ...
Kafka91's user avatar
  • 641
3 votes
2 answers
2k views

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 ...
user267839's user avatar
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3 votes
1 answer
217 views

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 ...
Malkoun's user avatar
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3 votes
1 answer
516 views

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 $E$ ...
Arrow's user avatar
  • 10.3k
2 votes
0 answers
325 views

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 ...
Nabla's user avatar
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9 votes
1 answer
549 views

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-...
Adam Levine's user avatar
17 votes
2 answers
983 views

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 ...
Danny Ruberman's user avatar
3 votes
2 answers
274 views

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 ...
Steve's user avatar
  • 494
5 votes
1 answer
585 views

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 ...
Z Wu's user avatar
  • 340
0 votes
0 answers
57 views

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 $D \cap \partial M = \partial D$ ...
Eduardo Longa's user avatar
1 vote
0 answers
71 views

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 ...
0xbadf00d's user avatar
  • 161
2 votes
0 answers
211 views

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\;\...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
125 views

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 ...
node's user avatar
  • 329
1 vote
0 answers
57 views

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 ...
Eduardo Longa's user avatar
10 votes
2 answers
667 views

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?
Nick L's user avatar
  • 6,923
8 votes
1 answer
226 views

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 ...
Eduardo Longa's user avatar
3 votes
1 answer
348 views

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 ...
geometricK's user avatar
  • 1,851
22 votes
2 answers
1k views

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 ...
Michael Albanese's user avatar
0 votes
0 answers
78 views

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: ...
Ben's user avatar
  • 1,010
3 votes
1 answer
163 views

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 ...
TheGeekGreek's user avatar
3 votes
1 answer
90 views

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 ...
Eduardo Longa's user avatar
1 vote
0 answers
127 views

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 ...
Eduardo Longa's user avatar
10 votes
2 answers
1k views

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 ...
user676464327's user avatar
9 votes
1 answer
307 views

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 ...
Eduardo Longa's user avatar
15 votes
1 answer
429 views

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 ...
user124543's user avatar

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