For questions on manifolds of dimension $n$, a topological space that near each point resembles $n$-dimensional Euclidean space.

learn more… | top users | synonyms (1)

10
votes
0answers
518 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
10
votes
0answers
1k 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 ...
9
votes
0answers
116 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
7
votes
0answers
82 views

Is a compact, simply-connected 3-manifold necessarily $S^3$ with $B^3$'s removed?

Let $M$ be a compact, simply-connected 3-manifold (which is also smooth and connected). Is $M$ diffeomorphic to $S^3$ with a finite number of $B^3$'s removed? This seems like a handy fact, but I ...
7
votes
0answers
170 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
7
votes
0answers
144 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
6
votes
0answers
74 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
6
votes
0answers
108 views

Sobolev space on $M \times [0,\infty)$, $M$ compact closed manifold

Consider a manifold of the form $M \times [0,\infty)$, where $M$ is a closed compact manifold without boundary. So $M \times [0,\infty)$ is a semi-infinite cylinder. I want to know information about ...
5
votes
0answers
109 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
5
votes
0answers
44 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
5
votes
0answers
70 views

Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
5
votes
0answers
108 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
5
votes
0answers
90 views

Does there exist a vector field s.t. all orbits are dense on $\mathbb{R}^2$

Is there a complete vector field such that the all orbits are dense on a contractible manifold? For example, $\mathbb{R}^2$,the interior of a $n$-unit ball.
5
votes
0answers
162 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
5
votes
0answers
70 views

Domain invariance for smooth functions

The domain invariance theorem states that for an open set $U\subset \mathbb{R}^n$ and a continuous and injective mapping $f:U\to \mathbb{R}^n,$ the image $f(U)\subset \mathbb{R}^n$ is open. I've read ...
5
votes
0answers
159 views

An imcomprehensible proof on Arnold's Lectures on PDE, contact diffeomorphism

In Lectures on Partial Differential Equations, Arnold gives a theorem in the chapter Huygens' Principle in the Theory of Wave Propagation: Theorem 1 (the theory of support functions). The manifold ...
5
votes
0answers
90 views

Example of charts on $\mathbb{R}$ that are $\mathcal{C}^r$ compatible but not $\mathcal{C}^{r+1}$ compatible.

Is there a simple example of two charts where this is the case? I'm struggling to think up one.
5
votes
0answers
218 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
5
votes
0answers
101 views

A good reference for learning about super-differentiation & super-integration?

I've looked at a couple of books for basic information for super-differentiation & super-integration - Rogers Supermanifolds, and Khrennikovs Superanalysis. Unfortunately both books lack a clear ...
5
votes
0answers
342 views

Differentiable Manifold Hausdorff, second countable

Why do we generally require that a differentiable manifold be Hausdorff and second countable? Is this universally accepted in the definition? My Professor only required the Hausdorff condition for ...
5
votes
0answers
116 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
150 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
196 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
132 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 ...
5
votes
0answers
145 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 ...
5
votes
0answers
160 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
31 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
4
votes
0answers
27 views

What does it mean for a function f to be C^k “close” to a function g?

My impression is that $f$ is $C^k$ "close" to $g$ if $f$ is close to $g$, $f'$ is close to $g'$, ... $f^{(k)}$ is close to $g^{(k)}$. However, my professor is being a bit nebulous on what "close" ...
4
votes
0answers
33 views

Transversality and homotopic maps

I'm trying to solve some problems in differential topology, and I came across the following: suppose $f:M\times [0,1]\rightarrow N$ is a homotopy, where $M$ is a compact manifold, such that $f_0$ and ...
4
votes
0answers
65 views

Can a differential k-form be integrated on a manifold that is not k-dimensional?

For example, can you integrate a 2-form on some curve, a 1-dimensional manifold, or some 3-dimensional manifold? I know that Stokes's Theorem states that if you integrate $\omega \in \mathcal ...
4
votes
0answers
59 views

Is there a way to define the concept of manifolds so it looks more like “generalised affine spaces”?

What I have in mind is along the lines of this: Let $M$ a topological space, $V$ a normed vector space, and $$ \boxminus \colon M\times M \to V, $$ $$ \boxplus \colon M\times V \to M. $$ Then ...
4
votes
0answers
88 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
4
votes
0answers
61 views

Surgery and Euler Characteristic

I am trying to find out how a $(p,n-p)$ surgery affects the Euler Characteristic of an orientable, $n-$ dimensional, compact manifold. Call the initial manifold $M$ and the post-op manifold $M'$. This ...
4
votes
0answers
108 views

Flowing a vector along a vector field $X$ using the pushforward of the flow of $X$

On the three-sphere $S^3$, I'm given three vector fields $X$, $Y$ and $Z$, such that at each point $p\in S^3$, the tangent vectors $X_p$, $Y_p$ and $Z_p$ form an orthogonal basis of the tangent space ...
4
votes
0answers
98 views

Lie Bracket and vector fields

Could you please help me solve it? Let X and Y be smooth vector fields on $\mathbb{R}^n$. Suppose that $[X,Y]=0$ (Lie bracket). Show that around each point there exists local diffeomorphism f such ...
4
votes
0answers
44 views

Cone is a submanifold iff it is a vector subspace

Could you tell me how to prove the following? Let $\emptyset \neq M \subset \mathbb{R}^n $ be a cone ($\forall x \in M, t \in \mathbb{R} : tx \in M $ ), $0 \le d \le n.$ Prove that $M$ is a $d$ ...
4
votes
0answers
244 views

Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
4
votes
0answers
67 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
4
votes
0answers
148 views

Diffeomorphism invariant scalars of a Riemannian manifold

Let $(M,g_{ab})$ be a Riemannian manifold. I know of the following scalars that one can construct them out of the metric and its derivatives: Ricci scalar $R$ $R_{ab}R^{ab}$ $R_{abcd}R^{abcd}$ ...
4
votes
0answers
64 views

Equivalent definitions of Tangent space - 2

L.S., In my book Vector Analysis by Klaus Jänich, Three different 'versions' of the Tangent space of a point $p$ at a differentiable variety are being discussed. The 'geometrical': the set of ...
4
votes
0answers
135 views

Submanifold of a regular value of a manifold with boundary

Question: Suppose $M$ is a smooth manifold with boundary, $N$ is a smooth manifold, and $F:M\rightarrow N$ is a smooth map. Let $S=F^{-1}(c)$, where $c\in N$ is a regular value of both $F$ and ...
4
votes
0answers
91 views

For what kinds of manifolds $\dim T_pM=\dim M$ holds?

Does the truth that $\dim T_pM=\dim M$ hold only for differentiable manifolds or for all topological ones?
4
votes
0answers
211 views

Visualization of immersed submanifold

I am trying to visualize the difference between immersed submanifold and embedded submanifold. At first, I thought that, for example, if I can embed manifold $M$ in $\mathbb{R}^4$ and if my friend can ...
4
votes
0answers
188 views

Properties of the projection onto a nonconvex set

Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ ...
4
votes
0answers
130 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
107 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
53 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 ...
4
votes
0answers
55 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
200 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
379 views

an injective immersion between two compact manifold of same dimension

$f:M\rightarrow N$ be a injective immersion, where $M$ and $N$ are same dimensional manifold with out boundary, we need to show $f$ is a covering map. what I tried is, $df_x:T_x(M)\rightarrow ...