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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Does composition on the right by a volume-preserving diffeomorphism preserve homotopy class?

Let $M, N$ be smooth manifolds with $M$ orientable and compact. Let $\sigma$ be some volume form on $M$ and consider the set $\mathcal{M}$ of smooth maps from $M$ to $N$ in a fixed homotopy class. Now ...
Dorado Toro's user avatar
5 votes
0 answers
112 views

S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group

I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...
onefishtwofish's user avatar
4 votes
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101 views

First usage of the terms pseudo-isotopy and concordance in manifold theory

I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in ...
Ryan Budney's user avatar
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Proving an equality of differential forms by assuming some perhaps topological condition

Let say I want to show two differential forms $\omega_1$ and $\omega_2$ on a smooth manifold $M$ are equal. Of course it suffices to show $\omega_1=\omega_2$ locally, i.e. the equality holds over ...
Ho Man-Ho's user avatar
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1 answer
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Diagonalization of symmetric matrices of functions

I asked this question some time ago in MSE but I didn't recieved any feedback. https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions This problem arised ...
user1234567890's user avatar
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Existence of a smooth extension

In the three dimensional Euclidean space $\mathbb R^3$ let us define the hypersurface $$ S= \{(x,y,z) \in \mathbb R^3\,:\, z^2= x^2+y^2\}.$$ Suppose that $f \in C^{\infty}(S)$. Does there exist $u\in ...
Ali's user avatar
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2 votes
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Are these the only first eigenfunctions on a hemisphere?

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
Eduardo Longa's user avatar
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33 views

$1$-parameter family of metrics preserving the normal direction

Let $(M^n,g)$ be a compact Riemannian manifold with boundary, $n \geq 2$, and let $N$ be the unit outward normal to $\partial M$. Denote by $S^2(M)$ the symmetric covariant $2$-tensors, by $S^2_0(M)$ ...
Eduardo Longa's user avatar
8 votes
0 answers
150 views

Changing coordinate to smoothen a function

Let $U\subset \mathbb{R}^2$ be an open neighborhood of the origin $0$, and let $f:U\to \mathbb{R}$ be a continuous function which is smooth on $U\setminus\left\{0\right\}$. Let's say that $f$ is ...
user49822's user avatar
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Is the square of a primitive cohomology class always primitive?

Let $M$ be a closed manifold (in my case $\dim M=3$). Take $\alpha\in H^1(M;\mathcal{Or})$, where $\mathcal{Or}$ is the orientation local system for $M$ with coefficients $\mathbb Z$. Suppose $\alpha$ ...
Andrey Ryabichev's user avatar
8 votes
2 answers
468 views

Compactly-supported harmonic tensors

Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$...
B.Hueber's user avatar
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1 answer
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How to construct this embedding of semi-infinite cylinder into itself?

In order to remove a double point $q=g(p_1)=g(p_2)$ of a immersion $g:M^n\to\mathbb{R}^{2n}$ of a non-compact connected manifold with dimension $n\geq2$, Whitney suggets that it can be taken an ...
Ludwik's user avatar
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Implicit function theorem in Riemannian manifold and Wasserstein space

My question is about to what extent can we extend the implicit function theorem to Riemannian manifolds. In the Euclidean space, consider a bivariate function $F \colon \Theta \times \mathcal{X} \...
Steve's user avatar
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Is the intersection of such a triple of minimal surfaces in the 3-ball a single point?

Let $S_1,S_2,S_3$ be three simple closed curves on the 2-sphere $\mathbb{S}^2$. (With no smoothness or rectifiability assumption) For each $i$, let $M_i$ denote the minimal surface (i.e. disc) bounded ...
Agelos's user avatar
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6 votes
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Difference between parallel transport and ambient projection

Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
Jason Li's user avatar
1 vote
0 answers
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Proving the canonical form of a vector field near a regular point on the boundary [closed]

I'm trying to solve Theorem 9.35 of Lee's Introduction to smooth manifolds: Let $M$ be a smooth manifold with boundary and let $V$ be a smooth vector field on $M$ that is tangent to $\partial M$. If $...
Livetrack's user avatar
4 votes
1 answer
546 views

Basic question on the de Rham theorem

There is a modern nice proof of the de Rham theorem based on sheaf theory. The de Rham theorem says that for a smooth manifold $M$ there is a canonical isomorphism $$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
asv's user avatar
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3 votes
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123 views

Isotopy-classes of oriented Schoenflies spheres in $S^4$ is a group under oriented connect-sum

Given two oriented, smoothly-embedded copies of $S^3$ in $S^4$ (called Schoenflies spheres), one can take an oriented connect-sum of the pairs $(S^4, M_1) \# (S^4, M_2)$. This puts a monoidal ...
Ryan Budney's user avatar
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4 votes
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Hodge decomposition on non-compact manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition $$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
B.Hueber's user avatar
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1 answer
106 views

When is compactness of fiber components an open condition?

Consider a smooth map $f:M\rightarrow N$ between smooth manifolds. Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that ...
Nikhil Sahoo's user avatar
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2 votes
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100 views

Existence of eigen basis for elliptic operator on compact manifold

Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting ...
asv's user avatar
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10 votes
1 answer
555 views

Hodge decomposition in elliptic complexes

EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a ...
asv's user avatar
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2 votes
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Question about stable manifold theorem and Frobenius integrability theorem

I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms) For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...
WaoaoaoTTTT's user avatar
8 votes
0 answers
172 views

A reference on a result by Steve Schanuel

In the Author Commentary section of the TAC reprint of the paper of 1968 Diagonal arguments and cartesian closed categories., Bill Lawvere wrote: ‘Nilpotent infinitesimals fall far short of even one-...
Evgeny Kuznetsov's user avatar
5 votes
3 answers
313 views

Poisson equation on manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation $$\Delta u=f$$ does have a solution on $C^{\infty}(\mathcal{M})$ ...
B.Hueber's user avatar
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1 vote
0 answers
56 views

Poisson equations for tensors on compact Riemannian manifold

Let $({M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well known that the Poisson equation $$\Delta f=S$$ where $\Delta:C^{\infty}({M})\to C^{\infty}({M})$ denotes ...
B.Hueber's user avatar
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1 answer
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When can vector fields span the tangent space at each point?

If the tangent bundle of a smooth manifold is a smoothly trivial smooth fiber bundle, is it a trivial smooth vector bundle? Since this question got no answer in MathExchange, I am migrating it here. ...
kathy4k's user avatar
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8 votes
2 answers
277 views

Two different spin structures of the real projective space $\Bbb RP^3$

It is known that every orientable 3-manifold has a spin structure, because its tangent bundle is trivial. Also it is known that if a manifold $X$ has a spin structure, then the number of distinct spin ...
user302934's user avatar
2 votes
1 answer
115 views

Sufficient condition for the union of two submanifolds to be a submanifold

I have two smoothly embedded orientable surfaces $S_1,S_2\subset S^3 \times [0,1]$ with boundary such that $(i)$ $S_1\cap S_2$ is a smoothly embedded surface without boundary and $(ii)$ $\overline{...
Euler Characteristic's user avatar
1 vote
0 answers
77 views

Integral flow that can commute to Laplacian operator

Firstly, considering the vector field in $ \mathbb{R}^3 $, $ X=x_2e_1-x_1e_2 $, we can see that $$ \phi(t,x)=\phi(t,x_1,x_2,x_3)=(t,x_1\cos t+x_2\sin t,-x_1\sin t+x_2\cos t,x_3) $$ is the ...
Luis Yanka Annalisc's user avatar
3 votes
0 answers
110 views

Applications of Strong Whitney Embedding

I am looking for applications of the strong Whitney's embedding theorem that have an advantage over weak theorems. That is, applications where it's important that the dimension of the Euclidian space ...
Ludwik's user avatar
  • 225
7 votes
0 answers
117 views

Are these two concepts of a differential form on the loop space equivalent?

Notation: Let $X$ denote a smooth manifold (without boundary) and define $LX = C^{\infty}(S^1, X)$ to be the loop space on $X$. In the context of loop space homology and the supersymmetric path ...
ChenIteratedIntegral's user avatar
2 votes
0 answers
69 views

Leaves of bounded genus

Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such ...
Eduardo Longa's user avatar
4 votes
1 answer
147 views

What integral formula is being used here?

I am trying to read the paper "Simple closed geodesics on convex surfaces" by E.Calabi and J. Cao and a certain passage is unclear for me. Before, let me contextualize and set up some ...
Eduardo Longa's user avatar
1 vote
1 answer
160 views

Decomposition of tensor field on hypersurface

Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...
B.Hueber's user avatar
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2 votes
1 answer
224 views

What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?

Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE, $$ d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0 $$ where $\left(W_t\right)_{t \geqslant 0}$ ...
Student's user avatar
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6 votes
1 answer
346 views

Threefolds with the same Betti numbers and the same Chern numbers

By a threefold, I mean a compact complex manifold of dimension three. My question is a simple one: Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the ...
user69559's user avatar
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4 votes
1 answer
152 views

How to formalize this isotopy?

I'm studying the H-Cobordism theorem following the Lectures of John Milnor, and in the proof of the Whitney trick for cancel pairs of self-intersection points I have the next problem with an isotopy ...
Ludwik's user avatar
  • 225
2 votes
0 answers
101 views

Unstably dualizable maps

Call a map between compact, connected framed $n$-manifolds $f:M \rightarrow N$ unstably dualizable if there exists an $f':N \rightarrow M$ such that the following diagram commutes up to homotopy: $$\...
Connor Malin's user avatar
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4 votes
2 answers
169 views

Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following. Use pseudo-differential operators on $M$...
kuuga's user avatar
  • 71
3 votes
0 answers
81 views

The tangent bundle of $G \times_H M$

Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to ...
Lukas's user avatar
  • 198
11 votes
1 answer
245 views

Does the Lie algebra of vector fields $\mathfrak{X}(M)$ determine the diffeomorphism class of a manifold $M$?

Let $M_1,M_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M_1)\cong \mathfrak{X}(M_2)$ as Lie algebras. ...
Amr's user avatar
  • 1,031
0 votes
1 answer
65 views

Invariance of mutual information under injective functions

Let $X\colon \Omega\to\mathcal X$ and $Y\colon \Omega\to \mathcal Y$ be two random variables. In M.S. Pinkser's Information and information stability of random variables and processes mutual ...
Paweł Czyż's user avatar
1 vote
0 answers
112 views

Fourier transform of functions mapping manifolds, is there a definition?

$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form $$ f : \mathbb{R} \to \SO(3)^n $$ Since $\SO(3)$ is a compact group so is $\SO(3)^n$. Now if ...
user8469759's user avatar
3 votes
0 answers
97 views

"Practical" references on mapping spaces as infinite-dimensional manifolds

I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
B.Hueber's user avatar
  • 701
10 votes
1 answer
348 views

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 ...
Michael Albanese's user avatar
12 votes
1 answer
330 views

Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
Tobias Fritz's user avatar
  • 5,128
5 votes
1 answer
238 views

Reference for Calderon-Zygmund $L^p$ inequalities on the sphere

The following question is motivated from Chapter 2 (Generalized Hodge Systems in 2D), particularly Section 2.3 ($L^p$ theory for Hodge systems in 2D) of Christodoulou and Klainerman's book, The global ...
Chris's user avatar
  • 285
2 votes
0 answers
44 views

Normalizing self-intersections of immersions $f:M^n\to\mathbb{R}^{2n}$

In the proof of the strong Whitney embedding theorem, given a self-transverse immersion $f:M^n\to\mathbb{R}^{2n}$ of a compact manifold $M^n$ (thus, with only finitely many double points), a key for ...
Ludwik's user avatar
  • 225
10 votes
1 answer
467 views

Is every retraction homotopic to a smooth retraction?

I am not an expert in Differential Topology, so let me apologize if this question admits a straightforward answer. I checked some standard references, but I could not find one. Let $M$ be a smooth $n$-...
Francesco Polizzi's user avatar

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