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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9 votes
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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 ...
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
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 ...
Rahul Sarkar's user avatar
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 ...
Tobias Fritz's user avatar
  • 5,775
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, ...
0xbadf00d's user avatar
  • 161
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 $...
Alex M.'s user avatar
  • 5,207
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 ...
ryanriess's user avatar
  • 209
6 votes
1 answer
318 views

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}^...
Eduardo Longa's user avatar
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(...
user34104's user avatar
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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?
Mohammad Farajzadeh-Tehrani's user avatar
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 $...
MathGeo's user avatar
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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^...
Daniel Santiago's user avatar
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{...
Praphulla Koushik's user avatar
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 ...
anonymous67's user avatar
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 $F_t$...
Totoro's user avatar
  • 2,515
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 ...
Jens Reinhold's user avatar
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 \...
user8469759's user avatar
18 votes
4 answers
3k views

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. ...
Praphulla Koushik's user avatar
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 ...
Dmitri Scheglov's user avatar
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 $...
Ali's user avatar
  • 4,067
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 ...
geometricK's user avatar
  • 1,851
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 ...
wonderich's user avatar
  • 10.3k
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 ...
Andrews's user avatar
  • 79
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 \...
Eduardo Longa's user avatar
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 ...
Cusp's user avatar
  • 1,703
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 ...
Fernanda's user avatar
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 ...
Totoro's user avatar
  • 2,515
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 ...
Kafka91's user avatar
  • 641
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 $\mathcal{O}(K_S)$ ...
user267839's user avatar
  • 5,948
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 $D^n$?
ABIM's user avatar
  • 5,019
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\...
ABIM's user avatar
  • 5,019
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 ...
Eduardo Longa's user avatar
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 $W$?
Kafka91's user avatar
  • 641
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 ...
user156275's user avatar
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 $G$ ...
Kafka91's user avatar
  • 641
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. ...
Zineb mazouzi's user avatar
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^...
Anonymous amateur's user avatar
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 $\...
erz's user avatar
  • 5,385
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 ...
Jan Vysoky's user avatar
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
Ali Taghavi's user avatar
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 ...
ABIM's user avatar
  • 5,019
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 \...
user1952770's user avatar
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 ...
wonderich's user avatar
  • 10.3k
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/...
Nikhil Sahoo's user avatar
  • 1,175
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 = \...
Rahul Sarkar's user avatar
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 ...
Ian Teixeira's user avatar
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 ...
Jens Reinhold's user avatar
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^\...
Arrow's user avatar
  • 10.3k
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 $...
Eduardo Longa's user avatar
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 "...
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