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].
1,180
questions
2
votes
1
answer
114
views
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 ...
- 61
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 ...
- 253
4
votes
0
answers
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 ...
- 41.9k
0
votes
0
answers
48
views
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 ...
- 939
5
votes
1
answer
221
views
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 ...
- 183
0
votes
0
answers
88
views
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 ...
- 3,863
2
votes
1
answer
94
views
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 $2$...
- 2,001
2
votes
0
answers
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 $S^2_0(M)$ ...
- 2,001
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 ...
- 1,893
2
votes
0
answers
206
views
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 $\alpha$ ...
- 713
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 $\mathbb{R}$...
- 701
0
votes
1
answer
83
views
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 ...
- 225
3
votes
0
answers
121
views
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} \...
- 1,107
4
votes
1
answer
152
views
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 ...
- 1,834
6
votes
1
answer
200
views
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 ...
- 63
1
vote
0
answers
50
views
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 $...
- 11
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
$$H^i_{dR}(M,\mathbb{R})\simeq H^i_{...
- 20.3k
3
votes
0
answers
123
views
Isotopy-classes of oriented Schoenflies spheres in $S^4$ is a group under oriented connect-sum
Given two oriented, smoothly-embedded copies of $S^3$ in $S^4$ (called Schoenflies spheres), one can take an oriented connect-sum of the pairs $(S^4, M_1) \# (S^4, M_2)$. This puts a monoidal ...
- 41.9k
4
votes
0
answers
234
views
Hodge decomposition on non-compact manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
- 701
3
votes
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 ...
- 1,107
2
votes
1
answer
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 ...
- 20.3k
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 ...
- 20.3k
2
votes
0
answers
62
views
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 ...
- 31
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-...
- 764
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 $C^{\infty}(\mathcal{M})$ ...
- 701
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 ...
- 701
3
votes
1
answer
127
views
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.
...
- 253
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 ...
- 475
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{...
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 ...
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 ...
- 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 ...
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 ...
- 2,001
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 ...
- 2,001
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 $...
- 701
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 $\left(W_t\right)_{t \geqslant 0}$ ...
- 591
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 ...
- 1,483
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 ...
- 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:
$$\...
- 3,891
4
votes
2
answers
169
views
Equivalence between two Sobolev norms on manifolds
On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.
Use pseudo-differential operators on $M$...
- 71
3
votes
0
answers
81
views
The tangent bundle of $G \times_H M$
Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to ...
- 198
11
votes
1
answer
245
views
Does the Lie algebra of vector fields $\mathfrak{X}(M)$ determine the diffeomorphism class of a manifold $M$?
Let $M_1,M_2$ be two simply connected, connected, compact smooth manifolds without boundary and of the same dimension. Assume that $\mathfrak{X}(M_1)\cong \mathfrak{X}(M_2)$ as Lie algebras.
...
- 1,031
0
votes
1
answer
65
views
Invariance of mutual information under injective functions
Let $X\colon \Omega\to\mathcal X$ and $Y\colon \Omega\to \mathcal Y$ be two random variables. In M.S. Pinkser's Information and information stability of random variables and processes mutual ...
- 231
1
vote
0
answers
112
views
Fourier transform of functions mapping manifolds, is there a definition?
$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form
$$
f : \mathbb{R} \to \SO(3)^n
$$
Since $\SO(3)$ is a compact group so is $\SO(3)^n$.
Now if ...
- 49
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 ...
- 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 ...
- 18k
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 ...
- 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 ...
- 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 ...
- 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 $n$-...
- 63.9k