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].
871
questions
1
vote
0answers
49 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 \...
10
votes
4answers
527 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. ...
3
votes
0answers
59 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 ...
1
vote
0answers
31 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 $...
1
vote
0answers
57 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
0answers
140 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
1answer
127 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 ...
2
votes
0answers
97 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
1answer
72 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
0answers
53 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
0answers
50 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
1answer
110 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 ...
3
votes
1answer
197 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
0answers
75 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
0answers
52 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
1answer
186 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
0answers
44 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
0answers
159 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
2answers
133 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
3answers
184 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
1answer
143 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
2answers
427 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
1answer
591 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
1answer
149 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
7
votes
1answer
269 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
1answer
132 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
0answers
87 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 ...
2
votes
1answer
97 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
0answers
72 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
0answers
72 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 ...
19
votes
1answer
394 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 ...
4
votes
0answers
277 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
0answers
60 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 $...
6
votes
0answers
139 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 "...
4
votes
0answers
159 views
Smoothability of open 4-manifolds
F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5.
He first proves the generalized annulus conjecture:
Suppose $h:D^j\times \...
5
votes
0answers
62 views
Minimizing area in relative homology class
A well known result in geometric measure theory asserts that if $(M^{n+1}, g)$ is a closed Riemannian manifold and $\alpha \in H_n(M)$ is a nonzero homology class, then there exists a closed embedded ...
2
votes
1answer
85 views
Cut out an open ball from a 2-manifold and glue the boundary
I have a possibly elementary question. Let $\mathcal{M}$ be a manifold with $\text{dim} \; \mathcal{M} = 2$. Let $U \subseteq \mathcal{M}$ be homeomorphic to $\overline{\mathcal{B}(0,1)}$, and let $\...
11
votes
4answers
905 views
Book on manifolds from a sheaf-theoretic/locally ringed space PoV
I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...
4
votes
0answers
84 views
Exhaustion of a manifold by open $n$-cells
Suppose that a differentiable manifold $M^n$ has an exhaustion $M^n=\bigcup U_i$ with $U_i \subset U_{i+1}$ such that each element $U_i$ is diffeomorphic to $\mathbb R^n$.
Can we prove that $M^n$ is ...
1
vote
0answers
38 views
On the center-stable manifold theorem for sets
Suppose I have a dynamical system $f:S \to S$ where $S \subset \mathbb{R}^n$ and $S$ is compact and $f$ is twice differentiable. Assume there exists a function $V$ such that $V(f(x)) < V(x)$ unless ...
2
votes
0answers
137 views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
0
votes
0answers
61 views
If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
1
vote
0answers
98 views
Confused about A. Kosinski's description about surgery in his book “differential manifolds”
Please excuse me, if MO is not the proper place for this question. I aksed the same question on M.SE
https://math.stackexchange.com/questions/3511134/confused-about-a-kosinskis-description-of-surgery-...
0
votes
0answers
116 views
Why is the divergence theorem used in the Eells-Sampson paper slightly different from that in a textbook?
I am reading Harmonic Mappings of Riemannian Manifolds by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for ...
2
votes
1answer
264 views
Relate the solid angle and surface measure of a surface
Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$ with a global chart$^1$ $(U,\phi)$. If $u\in U$ and $x=\phi^{-1}(u)$, let $\nu_M(x)$ denote the unique unit normal vector of $M$ ...
5
votes
1answer
182 views
Generalized Schoenflies - formalizing step in proof?
[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.]
I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, ...
2
votes
0answers
63 views
For a manifold of positive curvature, can we lower bound the distance between unit normals?
Suppose $M \subset \mathbb R^d$ is a $C^2$ $(d-1)$-manifold. In particular I am interested in when $M$ is the set $\{x \in \mathbb R^d: f(x) \le 1\}$ for some $C^2$ function $f:\mathbb R^d \to \...
2
votes
0answers
69 views
Definition of Lie derivatives of sections of natural vector bundles - product preservation needed?
Section 6.15 of Natural Operations by Kolár, Michor, and Slovak defines the Lie derivative of a section of a natural vector bundle along a vector field. Set-theoretically, the definition is clear. ...
2
votes
0answers
176 views
Can a non-compact manifold become compact by cutting it?
I'm trying to understand a step in a proof, where one starts with a non-compact manifold $V$ containing a trapped (2-sided, closed) surface $\Sigma$ that's non-separating. In order to complete the ...
1
vote
0answers
71 views
Estimate the gradient of a function on a complete manifold
There is a function $f$ on the smooth, complete manifold with $f(x_{t})=f(x)+td(x,y)$, $t \in [0,1]$, where $x$ and $y$ are fixed points on the manifold, $d(x,y)$ is the geodesic distance, $x_{t} = {\...
