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

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9
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397 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 ...
9
votes
0answers
900 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 ...
7
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0answers
122 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
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0answers
256 views

Munkres' Question on Manifolds

In Munkres' 'Analysis on Manifolds' on pg. 208 there's a question which reads: QUESTION: Let $f:\mathbb R^{n+k}\to \mathbb R^n$ be of class $\mathscr C^r$. Let $M$ be the set of all the points ...
7
votes
0answers
116 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 ...
7
votes
0answers
99 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 ...
5
votes
0answers
105 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
61 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
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91 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
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0answers
85 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
143 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
59 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
136 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
86 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
173 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
87 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
108 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
137 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
177 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
125 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
134 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
151 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
59 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
92 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
78 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
38 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
177 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
50 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
113 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
327 views

Product of manifolds & orientability

I'm studying orientability of manifolds currently and I'm having trouble to prove the following: $M\times N$ is orientable iff $M$ and $N$ are orientable. I am able to prove that the product is ...
4
votes
0answers
57 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
126 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
90 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
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0answers
189 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
139 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
126 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
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0answers
97 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
51 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
180 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
136 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
18 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
3
votes
0answers
30 views

quaternion vector bundle and quaternion grassmannian

Let $\mathbb{H}$ be quaternion numbers. Let $G_n(\mathbb{H}^\infty)$ be the grassmannian of $n$-subspaces of $\mathbb{H}^\infty$. Then ...
3
votes
0answers
33 views

The unit tangent bundle for submanifold $M^{m}\subset \mathbb{R}^{n}$ is a (2m-1)-dim submanifold

This is homework so no answers please Here is the problem: Show that $UM:=\{(x,v)\in T\mathbb{R}^{n}:x\in M^{m}, v\in T_{x}M^{m},|v|=1\}$ is a (2m-1)-dim submanifold of $T\mathbb{R}^{n}$. My ...
3
votes
0answers
30 views

How many charts?

My question is rather vaque but I hope that You will feel what I would like to know. A manifold (say: real, $n$-dimensional) is something which locally looks like an open set in $\mathbb{R}^n$. A rank ...
3
votes
0answers
66 views

Manifolds of Non-Standard Dimension

Can there exists (non-trivial) manifolds of non-standard dimensions? Certainly, there do exist manifolds of dimension $n$ for any $n \in \mathbb{N}$ (as well as manifolds of countably many ...
3
votes
0answers
44 views

Brief summary of simplicial, CW and manifold notions

I tried to summarize the relations between the following notions of: a manifold (smooth, topological and PL), simpilicial complex, CW complex. However I found some inconsistencies, which may be not a ...
3
votes
0answers
44 views

References on the relations between Top, Diff and PL

I have heard many times informal statements like "differentiable and pl manifolds are essentially the same for such and such dimensions", but I would like to know what they mean exactly and how such ...
3
votes
0answers
32 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 ...
3
votes
0answers
50 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...