For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.
16
votes
0answers
137 views
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
13
votes
0answers
198 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
12
votes
0answers
123 views
Are locally homotopic functions homotopic?
Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
8
votes
0answers
165 views
Infinite dimensional constant rank theorem
Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
8
votes
0answers
111 views
How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?
I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
8
votes
0answers
159 views
Manifold of Density Matrices
Let $\mathrm{M}_{d\times d}\left(\mathbb{C}\right)$ denote the set of all $d\times d$-matrices with complex entries.
My goal is to show that the set
$\mathcal{M}:= \left\{ \rho\in \mathrm{M}_{d\times ...
7
votes
0answers
388 views
Fundamental group of a compact manifold
In an article I am currently reading, the author tells us that for compact (finite dimensional topological) manifolds X and finite groups $\Gamma$, the set $$\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$$ where ...
6
votes
0answers
70 views
Invariant submanifolds
Let $M$ be a smooth manifold, and let $N$ be a submanifold. Let $V$ be a smooth vector field on $M$ which generates a flow $\Phi_t$ on $M$. My intuition tells me (perhaps modulo some technical ...
6
votes
0answers
80 views
Is there any holomorphic version of the tubular neighborhood theorem?
This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'.
Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
5
votes
0answers
57 views
Homological definition of orientation at a boundary point?
For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
5
votes
0answers
79 views
Gluing manifolds
Is it true that gluing can be realized two ways? In the example of gluing two cylinders to obtain a torus one can imbed each cylinders into the torus like this:
A point $(\cos \phi , \sin \phi, ...
5
votes
0answers
88 views
Does every non-compact manifold admit an incomplete vector field?
I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess ...
5
votes
0answers
114 views
Invariance of Wall's self-intersection under the regular homotopy
For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
4
votes
0answers
55 views
Differentiable manifolds, Serge Lang
I have started reading "Introduction to differentiable manifolds" by Serge Lang. In this book, Lang takes a different approach, by immediately introducing manifolds on arbitrary Banach spaces. His ...
4
votes
0answers
40 views
Levi-Civita Connection for 2-dimensional Riemannian manifold
I'm trying to show the following. Suppose $(M, g)$ is a $2$-dimensional Riemannian manifold with connection $\nabla$. Suppose also that $\nabla$ is metric compatible, and that length extremizing ...
4
votes
0answers
43 views
non-constant curve c with $\frac{\mathrm{d}c}{\mathrm{d}t}= \lambda\frac{\mathrm{d}^2c}{\mathrm{d}t^2}$
I don't know how to solve the following problem and would appreciate some help.
Let $M$ be a submanifold of euclidean space and $c:[a,b] \to M$ a non-constant curve, such that the velocity field of ...
4
votes
0answers
169 views
Geodesics on the torus
[This is a follow-up to my question Is there a Möbius torus?]
Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus:
There are five ...
4
votes
0answers
115 views
de Rham Cohomology of Non-Flat Bundle
Let $E$ be a smooth vector bundle on a smooth manifold $M$. If $E$ is flat, there is a connection $\nabla$ which is a differential which we can use to define the de Rham cohomology of $E$.
If $E$ ...
4
votes
0answers
84 views
Heegaard Splitting of Brieskorn homology 3-spheres
For pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by
$\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$.
I want to know ...
4
votes
0answers
110 views
Schwartz Space on a Manifold
On $\mathbb{R}^n$, the Schwartz space is an incredibly nice space of functions, and in many ways is more natural than $C_c^\infty (\mathbb{R}^n)$. On a manifold $M$, it of course still makes sense to ...
3
votes
0answers
83 views
Simple exercise in cohomology
I know this is a simple exercise but I am stuck unfortunately.
Question:
Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
3
votes
0answers
35 views
A Simons' type inequality
I have a problem with the inequality (5) in the article 'Estimates for stable minimal surfaces in three dimensional manifolds' of R.Schoen. As the author suggests this inequality comes from 'well ...
3
votes
0answers
38 views
Any vector bundle on $\mathbb R$ is a trivial bundle
How to prove that any vector bundle on a Euclidean space is a trivial bundle? It is enough to prove it for the case of dimension $1$ and I hope it will be a nice exercise for me to generalize to the ...
3
votes
0answers
78 views
How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?
The diagonal $Q$ in $X\times X$ is the set of points of the form $(x,x)$. Show that $Q$ is diffeomorphic to $X$, so $Q$ is a manifold if $X$ is.
Can anyone please help me to solve this question I ...
3
votes
0answers
78 views
Construct a space with free involution and homological restriction
I'm looking for a space $X$ which satisfies the following conditions:
$X$ is a compact manifold.
$H_\ast (X;\mathbb Z)$, the integral homology groups $X$, are torsion free.
There is a free ...
3
votes
0answers
59 views
Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?
I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
3
votes
0answers
38 views
Show that the cosets of a closed isotropy group form a manifold
Suppose $G$ is a Lie group acting on the manifold $M$ and $p \in M$ is such that $G_p$, the isotropy group of $p \in M$, is closed in $G$. I'm trying to prove that $G/G_p$ has a manifold structure.
...
3
votes
0answers
78 views
Alternate pullback bundle construction
If $\pi : F \to N$ is a fiber bundle, and $\phi : M \to N$ (here, $M$ and $N$ are manifolds), then the standard way to define the pullback bundle of $F$ by $\phi$ is
$$\phi^* F := \{(m,f) \in M ...
3
votes
0answers
80 views
tangent space clarification
I was just wondering why we don't define the tangent space of a smooth manifold at a point $p$ to be $\{p\} \times \mathbb{R}^{n}$, rather than using derivations, germs, or equivalence classes of ...
3
votes
0answers
250 views
Ambient Isotopy
From Hatcher's (edit. Hirsch's) Differential Topology, p. 180.
The first of the isotopy extension theorems says;
Let $A\subset M$ be a compact submanifold and $F:V\times I \rightarrow S^{3}$ an ...
3
votes
0answers
120 views
Locally free sheaves on locally ringed spaces
One can define the notion of a locally free sheaf (of finite rank) on any locally ringed space.
If you restrict to the category of (noetherian?) schemes, this category is equivalent to the category ...
3
votes
0answers
71 views
Uncountably many 3 dimensional foliations on a 5 dimensional torus
This is a problem on a review for some upcoming quals:
Prove there are uncountably many 3-dimensional foliations on a 5-dimensional torus.
Unless I am looking at this wrong it seems like a pretty ...
3
votes
0answers
141 views
How to define a topological tunnel?
I would like to define a notion of a topological tunnel, but I don't know how
(or even if it is possible) to capture it topologically.
I am interested in closed 2-manifolds in $\mathbb{R^3}$.
Suppose ...
3
votes
0answers
110 views
Is there a theory that generalizes both varieties and manifolds?
As I understand it, many of the ideas that were introduced into algebraic geometry in the mid 20th century by french mathematicians were done by transporting over ideas from the theory of manifolds ...
2
votes
0answers
34 views
Stoke's theorem application to curl theorem. I did. Please can you check it?
Now, I need to apply stoke's theorem to curl theorem.
My teacher gave a hint.
Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$
$dim(M)=2$
M is the subset of $\Bbb ...
2
votes
0answers
51 views
Real projective space is Hausdorff
I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix??
This prove is correct or I need to add something ?? ...
2
votes
0answers
74 views
Show that the projection map is Orientation preserving iff n is even
My question is that
Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere
$U =${$x∈S^n |x^{n+1} >0$}.
It is a coordinate chart on ...
2
votes
0answers
71 views
Figure $\infty$ is immersion of circle
Where can I find prove of:
Figure $\infty$ is immersion of circle.
More thanks for a prove or a function between these manifolds.
2
votes
0answers
33 views
Submanifold with boundary of a manifold with boundary
Let $M$ be a smooth manifold.
(1) A subset $S$ of $M$ that with the subspace topology is a topological manifold (with or without boundary), together with a differential structure that makes the ...
2
votes
0answers
56 views
Topological Manifold with ball, removed and antipodal points identified orientable?
Suppose you have a compact, orientable $(2n+1)-$manifold $M$, as in $H_{2n+1}=\mathbb{Z}$. You take a neighborhood about a point homeomorphic to $\mathbb{R}^{2n+1}$, and remove a small ball $B$. So ...
2
votes
0answers
37 views
Torus biholomorphic to smooth cubic curve?
I am trying to understand that all compact genus 1 Riemann surfaces are biholomorphic to a smooth cubic curve. ( assuming that $\dim H^{1,0} = \dim H^{0,1} = \frac{1}{2} \dim H^1$ )
I think I ...
2
votes
0answers
57 views
Prove Poincare duality theorem with Morse theory.
First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
2
votes
0answers
55 views
Doubt in Spivak's examples of Manifolds
I've started to study Differential Geometry in Spivak's first volume of his Differential Geometry books. I like very much his approach since general topology isn't assumed, and since he gives many ...
2
votes
0answers
20 views
Exponential Families and Riemannian Symmetric Spaces
Suppose the $f_{X}(x|\theta)$ is a probability density function from an exponential family. Is it true that the Riemannian manifold which has the Fisher information as it's Riemannian metric is a ...
2
votes
0answers
70 views
Vanishing of local cohomology of constructible sheaves
Recall, that if $\mathcal{F}$ is a coherent sheaf on a variety and $Z$ is an l.c.i. subvarity of codimension $n$, then $H^i_Z(\mathcal F)$ vanishes for $i > n$.
Is there an analogous statement for ...
2
votes
0answers
35 views
Show that $\Gamma(TM) \cong_{\mathbb{R}} \mathsf{Der}_{\mathbb{R}} C^{\infty}(M)$
Let $M$ be a smooth real manifold. I want to show that we have an isomorphism of real vector space $\Gamma(TM)$ of all smooth sections of $TM$ (i.e. of vector fields on $M$) and of real vector space ...
2
votes
0answers
63 views
Pullbacks as manifolds versus ones as topological spaces
Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd).
Questions:
Suppose that we ...
2
votes
0answers
133 views
Topological proof that this set is a topological manifold
let $S \subseteq \mathbb R^3 \times \mathbb R^3$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$.
i am trying to show that this is a topological manifold without ...
2
votes
0answers
42 views
Combinatorial surfaces and manifolds
Before we can start some basic definitions to come into the topic:
Suppose $T$ is the standard closed triangles, the convex hull of the three basic vectors inside $\Bbb{R}^3$. Consider $T$ is the ...
2
votes
0answers
13 views
How to build the space BTOP
Can anybody explain how is the procedure for building the space BTOP, which classifies microbundles of topological manifold ? Is there any good (and easy to read) references on this subject ?
Thanks ...
