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].
1,233
questions
9
votes
1
answer
312
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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
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 ...
1
vote
1
answer
360
views
Number of connected components of the set of invertible matrices over the reals when some of the matrix entries are fixed
Let $\mathcal{M} = \text{GL}(n, \mathbb{R})$ denote the set of $n \times n$ invertible matrices over $\mathbb{R}$. We know that $\text{GL}(n, \mathbb{R})$ has exactly 2 connected components. I have ...
5
votes
1
answer
293
views
Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective
Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective?
I'm looking for conditions ...
2
votes
2
answers
417
views
Reference for the divergence theorem for embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary
I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $K$ of $\mathbb R^d$ with "$C^1$-boundary"$^1$.
I know that there are several generalizations of this theorem, ...
6
votes
2
answers
379
views
An abstract characterization of line integrals
Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
3
votes
0
answers
179
views
Moving on Riemannian manifolds
Let $a,b,c\in\mathbb{R}^n$ such that $c$ is inside the $n$-disk with $a$ and $b$ as south and north poles. Then as $c$ moves toward $a$ through the line segment joining $a$ and $c$, $c$ is also moving ...
6
votes
1
answer
318
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Is Gauss map of a free boundary convex disk a diffeomorphism?
I asked this question on MSE, but obtained no answer. Maybe this is the right place to post it.
Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^...
7
votes
1
answer
194
views
Manifolds with $w_1(TM)\cup w_1(TM)=0$ and $w_2(TM)=0$ but $w_1(TM)\neq 0$
For a generic dimension $d$, is there an nonorientable manifold $M$ (i.e. $w_1(TM)\neq 0$) with vanishing $w_1(TM)\cup w_1(TM)$ and $w_2(TM)$, i.e.,
$$w_1(TM)\cup w_1(TM)=0, ~~~~~ w_2(TM)=0, ~~~~~w_1(...
17
votes
2
answers
1k
views
If $M$ and $N$ are closed and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?
If $M$ and $N$ are closed smooth manifolds, and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true that $M$ and $N$ are diffeomorphic?
3
votes
1
answer
116
views
Existence of a special function
Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$.
Is there any smooth function $...
1
vote
0
answers
173
views
Examples of why conditions for Novikov compact leaf theorem are necessary
Let $M^3$ be a smooth, closed 3-manifold. Given a smooth codimension-one foliation $\mathcal{F}$ of $M$, the Novikov compact leaf theorem asserts that, when the universal cover $\widetilde{M}^3$ of $M^...
9
votes
1
answer
425
views
Results in “generalised smooth spaces” that did not hold in the case of smooth manifolds
Consider the category of smooth manifolds $\text{Man}$. I quote from n-lab page:
Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.
I understand that this category $\text{...
2
votes
1
answer
217
views
To what extent is a vector bundle on a smooth manifold determined by its restriction to the complement of a closed smooth submanifold?
The question is a follow-up to this one.
Let $N\subset M$ be a closed smooth submanifold of codimension $k\geq 1$ of a smooth manifold $M$. Let $E_1\rightarrow M$ and $E_2\rightarrow M$ be two vector ...
1
vote
1
answer
235
views
The existence of an isotopy in the manifold
Let $M$ be a connected closed smooth manifold and $B(1)$ the unit ball in $\mathbb R^n$. Suppose $f:M \times B(1) \to M \times \mathbb R^n$ is a smooth embedding.
Can we find an ambient isotopy $F_t$...
9
votes
0
answers
286
views
Rational cobordism classes of manifolds fibered over spheres
Let us fix positive integers $k, m$. Let $A^k_{4m} \subset \Omega^{\text{SO}}_{4m} \otimes \mathbb Q$ be the subgroup generated by oriented cobordism classes of manifolds fibered over $S^k$.
The ...
1
vote
0
answers
78
views
Can this problem be rephrased as optimization on a manifold?
I have question. I have a Riemmanian manifold $\mathcal{M}$, like an $n$-dimensional regular surface in $\mathbb{R}^n$. And I have a smooth scalar field defined on this manifold $f:\mathcal{M} \to \...
18
votes
4
answers
3k
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When (why) did we allow manifolds to be non-Hausdorff and/or non-second countable?
I was reading David Carchedi's answer for a question on Grothendieck topology for a non-small category. It "reads" like people "choose" if they allow manifolds to be Hausdorff and/or second countable. ...
4
votes
0
answers
137
views
Typical preimage of the commutator map
By Goto's theorem for any compact connected semisimple Lie group $G$ of dimension $n$, any element $x\in G$ is a commutator, namely $x=[y,z]$ for some $y, z\in G$. Another way to say it is that the ...
2
votes
1
answer
93
views
conformal changes to Lorentzian curvature
Let $(M,g)$ be a Lorentzian manifold and let $R$ be the curvature tensor. We say $R\leq 0$ if
$$ g(R(X,Y)Y,X) \leq 0\quad \forall \, X,Y \in TM.$$
My question is whether given a Lorentzian manifold $...
5
votes
1
answer
263
views
Levi-Civita connection from idempotents
Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
3
votes
0
answers
177
views
Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
2
votes
1
answer
212
views
$\mathbb Z$-graded manifold is isomorphic to the structure sheaf of supermanifold locally
In this paper, definition 4.4.1 about supermanifold and definition 4.6.1 about graded manifold:
Definition 4.4.1: An supermanifold $\mathcal{M}$ is a locally ringed space $(M,\mathcal O_M)$ which is ...
2
votes
0
answers
113
views
Is this $1$-form harmonic?
Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \...
2
votes
1
answer
277
views
Restriction of diffeomorphisms homotopic to identity to the boundary
Let $M$ be a smooth manifold with boundary $\partial M$. Let $Diff_0(M)$ be the group of all diffeomorphisms homotopic to identity. According to this article (Page 6, section "
Beyond mapping class ...
3
votes
0
answers
174
views
References on integration on non-compact manifolds
I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem.
In particular I have non-compact manifold $M$ and I have an integral (in ...
1
vote
0
answers
57
views
Spherical space-form as the boundary of an Euclidean ball
Let $M^n$ be a smooth compact manifold such that the boundary $\partial M$ is diffeomorphic to a spherical space-form $S^{n-1}/\Gamma$, where $\Gamma \subset O(n)$ is a finite subgroup acting freely ...
4
votes
1
answer
347
views
Difference between the diffeomorphism classification of a manifold $M$ and the set of equivalences of homotopy smoothings $hS(M)$
In Lopez de Medrano "Involutions on manifolds", a homotopy smoothing of a Poincaré space $X$ is a homotopy equivalence $f:M^n\rightarrow X$, where $M^n$ is a smooth $n$-dim. manifold (everything is ...
5
votes
1
answer
452
views
Residue of the canonical sheaf along subvariety
Let $S$ be a smooth projective surface over an
algebraically closed field $k$ and $C \subset S$ a singular curve. Let us denote by $K_S$ the class of canonical divisor of $S$ and $\mathcal{O}(K_S)$ ...
3
votes
0
answers
144
views
Diffeomorphisms fixing origin and boundary
Let $D^n$ be a disc in $\mathbb{R}^n$. Is there a known characterization of all the diffeomorphisms of $D^n$ fixing the origin and boundary of $D^n$?
3
votes
0
answers
56
views
Criteria for density of subgroup of diffeomorphism group
Let $C^{\infty,\star}(\mathbb{R}^d)$ denote the non-commutative topological group of smooth diffeomorphisms from $\mathbb{R}^d$ to itself with $\circ$ as multiplication and let $\emptyset\subset X\...
5
votes
1
answer
371
views
Every homotopy class contains at least a harmonic representative
Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is ...
2
votes
0
answers
69
views
Extend fibre bundle
Let $F\rightarrow E\rightarrow B$ be a smooth fibre bundle. Suppose $W$ is a smooth manifold such that $F=\partial W$.
When is it possible to extend the bundle to a bundle over $B$ with fibre $W$?
6
votes
0
answers
239
views
Signature of a non-compact manifold
Let $v_0,\dots,v_n\in\mathbb{Z}^2$ be integer vectors which satisfy the condition $\det\begin{pmatrix}v_{k-1}&v_k\end{pmatrix}=(-1)^k$, whose relevance will become apparent in a moment. We may ...
4
votes
2
answers
199
views
Extend (Lie) group action from the boundary to the entire manifold
Let $M$ and $W$ be smooth manifolds such that $\partial W=M$. Let $G$ be a group acting on $M$.
Can one generally extend the action of $G$ to $W$? If not, under which conditions on $W$ and/or $G$ ...
2
votes
3
answers
278
views
Space of representations of surface group into Lie groups
In the context of Goldman's paper The symplectic nature of fundamental groups of surfaces:
Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. ...
1
vote
1
answer
225
views
Continuity of $r\mapsto\int_{\Sigma\cap B_r(x)}f^2d\mu$
Let $\Sigma$ be an embedded smooth surface in $\mathbb{R}^3$, and let $f:\Sigma\to\mathbb{R}$ be a smooth function. Suppose $f$ is square-integrable on $\Sigma$, with
\begin{align}
0<\int_{\Sigma}f^...
9
votes
2
answers
709
views
Is limit of null-homotopic maps null-homotopic?
The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\...
3
votes
1
answer
641
views
Subsupermanifolds defined using ideal, transversal example
I am currently learning about algebraic viewpoint on closed embedded subsupermanifolds. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma just ...
6
votes
1
answer
187
views
A smooth map on a Banach manifold whose pointwise rank is finite but its rank is not globally bounded
Is there a connected Banach manifold $M$ and a smooth map $f:M \to M$ such that the rank of $Df_x$ is finite for every $x\in M$ but this rank is not uniformly bounded
5
votes
1
answer
375
views
Non-density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed with the ...
3
votes
1
answer
951
views
Geodesic convexity and the Geometric Hessian
This is an elementary question in differential geometry. We know that for a smooth real-valued function $f$ defined on an open geodesically convex set of a Riemannian manifold $ \mathcal{X} \subset \...
4
votes
0
answers
229
views
Any cobordism invariant made of "characteristic classes", on unorientable manifolds, must be a mod 2 class?
For any cobordism invariant (or simply bordism invariant) quantity $\omega$ that satisfy the conditions:
$\omega$ can be fully decomposed from the cup product of characteristic classes (such as ...
3
votes
1
answer
229
views
Finite-dimensional argument for Morse-Smale pairs?
Using the Sard-Smale theorem, it is relatively easy to show that Morse-Smale pairs on a manifold $M$ (i.e. pairs $(f,g)$ where $g$ is a metric on $M$, $f$ is a Morse function on $M$, and the stable/...
0
votes
0
answers
80
views
Existence of a Euler-like formula for the continuous image of $S^1$ in an orientable surface
Let $\mathcal{M}$ be a compact 2-manifold, and let $\gamma: S^1 \rightarrow \mathcal{M}$ be a continuous map (you can assume piecewise smooth if it is convenient), with the property that the set $A = \...
3
votes
1
answer
149
views
Model geometry uniqueness
Let $ M $ be a compact connected manifold with
$$
M \cong \Gamma \backslash G /H
$$
where $G $ is a Lie group, $ H $ a compact subgroup, $\Gamma $ a discrete subgroup, and $ G/H $ is connected and ...
20
votes
1
answer
539
views
Can every manifold be dominated by a parallelizable one?
A closed, oriented $d$-manifold $M$ is said to dominate another such manifold $N$ if there exists a map $M \to N$ of non-zero degree. (This notion should not be confused with the unrelated concept of ...
5
votes
0
answers
431
views
A struggle with jets and Grothendieck vs Ehresmann connections
Let $X\to Y$ be a $C^\infty$ submersion. Consider the following two sheaves.
The sheaf on $Y$ comprised of jets of sections of $X\to Y$.
The sheaf on $X$ given by the quotient of $\Delta_{X/Y}^{-1}C^\...
2
votes
0
answers
71
views
Embeddedness and homology of a limit of minimal surfaces
Consider the following theorem, proved in
this paper:
Theorem (Theorem 6.1). Suppose we have a sequence $(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$ of immersed free boundary minimal $...
7
votes
0
answers
664
views
Nash-Tognoli Theorem
The Nash-Tognoli theorem says that every closed, smooth manifold is diffeomorphic to a real algebraic variety.
Suppose I wanted to study the Ricci curvature of some class of manifolds.
Is there a "...