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

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4
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1answer
50 views

Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
0
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0answers
33 views

Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...
0
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0answers
18 views

Is every compact manifold with boundary a manifold of bounded geometry?

If $M$ is a compact Riemann manifold with boundary, does it have bounded geometry, which means that the injectivity radius of the manifold is positive and every covariant derivative of the Riemannian ...
1
vote
2answers
53 views

Action of the fundamental group

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined? EDIT: Of course I want my action to be nontrivial, say ...
0
votes
2answers
27 views

Are $X=M \times [0,T]$ and $\partial X$ smooth compact manifolds when $M$ is smooth compact Riemannian manifold?

Let $X=M \times [0,T]$, where $M$ is a smooth and closed compact Riemannian manifold. I want to know if: $X$ is smooth compact manifold, and if $\partial X$ is smooth compact manifold? I am not ...
7
votes
1answer
137 views

Smooth map $S^1 \to S^2$ can not be surjective

Why cannot a smooth (or piecewise linear) map $S^1 \to S^2$ be surjective? There are space-filling curves, but the usual examples have very "twisty" definitions. UPD A bit of background for this ...
0
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0answers
30 views

Why do we require differential manifolds to be Hausdorff? [duplicate]

Among the requirements for a differential manifold $M$ is that it be connected and Hausdorff. What fails if a manifold is not Hausdorff?
0
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1answer
30 views

Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
5
votes
1answer
84 views

Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis ...
2
votes
0answers
15 views

Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
2
votes
2answers
45 views

What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...
0
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0answers
6 views

Maximally symmetric manifold with boundary and non-vanishing extrinsic curvature?

I was wondering if the following requirements are compatible: Given a $d$-dimensional manifold with boundary $M$ with $\partial M\neq \emptyset$ endowed with a metric $g$. The following conditions ...
2
votes
3answers
99 views

Homeomorphic manifolds have the same dimension

So I want to prove: If two manifolds $M$ and $N$ are homeomorphic then $dim(M) = m = n = dim(N)$. My idea was to use the property of the manifolds that they are locally homeomorphic to the ...
3
votes
1answer
47 views

Hermitian metric on $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space

Consider the line bundle $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space. Locally is descriebd by $\{U_a,g_{ab}\}$ where $U_a=\{z_a\neq0\}$ is the standard covering of the projective ...
2
votes
1answer
63 views

What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
2
votes
0answers
25 views

Tubular neighborhood by restricting the Riemannian exponential map

Let $M$ be a Riemannian manifold (possibly non-compact, possibly non-complete) and $N\subseteq M$ a smooth submanifold (possibly non-compact). Does there exist a continuous $\mu\colon M\rightarrow ...
1
vote
1answer
47 views

Several statements about $\mathbb{R}$ with chart defined by $f(x)=x^3$

I think I managed to show this statements but I am not sure about it. Since this is common problem in differentiable manifolds I was wondering if anybody has (or may write) a solution. Let $X$ be a ...
0
votes
1answer
42 views

A closed set is not a submanifold

Can someone explain me why the set $A:=\{(x,y)\in \mathbb R^2: x\geq 0\}$ is not a submanifold? I also got another (easy) question: In our lecture we are always talking about submanifolds. We ...
1
vote
0answers
42 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
3
votes
1answer
50 views

group cohomology of 3-manifolds

If $M$ is a closed aspherical 3-manifold with first fundamental group $G$, then cohomology groups of $G$ and $M$ are isomorphic because $M$ is a Eilenberg-MacLane space $K(G,1)$. In particular there ...
11
votes
0answers
144 views

Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
1
vote
1answer
59 views

Understanding tangent space basis

Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric. In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the ...
0
votes
1answer
31 views

Continuity in definition of Induced Functional Structure

I have a really simple question, however I am confused. Bredon's Topology and Geometry gives definition of Induced Functional Structure as follows: Suppose $F_x$ is a functional structure on space ...
3
votes
1answer
44 views

Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
4
votes
2answers
90 views

Is there a unique preferred connection on a general manifold?

I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined? Afais for Riemannian manifolds, there exists always exactly one Levi-Civita ...
3
votes
1answer
65 views

Explaining the definition of vector bundles

Recall the definition of a vector bundle: Let $M$ be a topological space. A $k$-dimensional vector bundle over $M$ is a topological space $E$ with a surjective continuous map $\pi\colon E \to ...
3
votes
1answer
34 views

Definition of submanifolds by regular values

Let $f: M \rightarrow N$ and $q \in N$ be a regular value, then $f^{-1}(q)$ is a submanifold of $M$. Now assume that $q \in N$ is not a regular value, but you pick $K:=f^{-1}(q) \cap \{p \in M; ...
0
votes
1answer
37 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
1
vote
0answers
32 views

Preimage of a small normal deformation under an embedding again a normal defomation?

Let $j\colon M\rightarrow W$ be a smooth embedding of smooth manifolds and assume $M$ and $N$ have Riemannian metrics, but $j$ is not necessarily an isometric embedding. Let $N\subseteq W$ be a ...
3
votes
1answer
35 views

Homology and Neighborhood

Let $X$ a connected manifold, $x \in X$ and $V$ a neighborhood of $x$. Assume $i:V \to X$ induce isomorphism between all homology groups. Does $X-p$ and $V-p$ still have the same homology groups ? ...
5
votes
2answers
53 views

Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$

PROBLEM: Construct a diffeomorphism $\phi:M\to M$ such that $\phi(p)=q$ and also $d\phi(X_p)=Y_q$, where $M$ is a connected smooth manifold and $p,q \in M$ , $X_p \in T_pM$ and $Y_q \in T_qM$ I know ...
2
votes
1answer
57 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
5
votes
0answers
56 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
3
votes
0answers
25 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
2
votes
0answers
32 views

Let $p=(5,0,-4)$ and $v \in T_{(5,0,-4)}M$. Compute $(F^{*}\omega)_p(v)$.

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5, x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. ...
2
votes
1answer
36 views

Compute in the chosen charts of $M$ and $S^1$ the expression of $DF_{(5,0,-4)}$

Let me show my work before presenting the problem itself. Let $M=\{(x,y,z) \in \mathbb{R}^3 : x+y=5 x+z=cos^2y\}$. We can easily see that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $1$. We ...
0
votes
1answer
28 views

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth?

When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition. I know, that we say, that $\partial\Omega$ has a ...
1
vote
1answer
54 views

Injectivity of the Differential of Smooth Map

I am trying to answer the following question: Let $M = \{(x,y)\in \mathbf{R}^2 : x^2 + y^2 < 1\}$. Define a smooth or $C^\infty$ function by $f\colon M \rightarrow \mathbf{R}^2$ as ...
4
votes
1answer
140 views

Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks ...
3
votes
1answer
51 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
0
votes
1answer
23 views

$M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$

PROBLEM: $M\subset\mathbb{R}^n$ is a open subset, $p\in M$ is arbitrary. Find $T_pM$ and $N_pM$. I know how to determine $T_pM$ and $N_pM$ for explicit examples, but I dont know how to handle this ...
3
votes
1answer
30 views

Existence of a vector field which dominates the first local vector fields given by the charts of a locally finite covering

Let $M$ be a smooth manifold, let $\{U_i,\psi_i\}_{i\in I}$ be locally finite family of charts and let $K_i\subseteq U_i$ be compact subsets. Does there exist a vector field $X$ on $M$, such that ...
1
vote
1answer
47 views

Open product neighbourhoods

Let $X\times Y$ be a topological space (if it helps anything, it can be assumed to be a smooth manifold and $U\subseteq X\times Y$ an open subset. Let $x\in X$ and $O\subseteq Y$ be open, such that ...
4
votes
1answer
74 views

Group action and smooth manifolds

I was wondering if it is for a compact (i.e. Hausdorff) smooth manifold $M$ sufficient to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
2
votes
0answers
77 views

Check Riemannian manifold's isometry to $\Bbb{R}^n$

Let $\mathcal{M}$ be the convex cone of symmetric positive definite $n\times n$ real matrices. $\mathcal{M}$ is an $\frac{n(n+1)}{2}$-dimenasional Riemannian manifold. Could you help me proving (or ...
2
votes
0answers
24 views

How to show that the inverse image of a face of a simple polytope is a connected manifold?

If $M^{2n}$ is a toric manifold over a simple polytope $P^n$ i.e; the orbit space of the action of the $(S^1)^n$ on $M^{2n}$ is an $n$ dimensional simple polytope $P^n$. Let $\pi : M^{2n} \rightarrow ...
2
votes
1answer
65 views

Is this one infinite dimensional manifold?

First of all, just to give context to the question: I've been reading some articles in Physics, and those articles imply without proof that one space is one infinite dimensional manifold. One of those ...
3
votes
0answers
66 views

$L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator

Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$. We know $w_k$ are ...
3
votes
0answers
73 views

Differentiating an endomorphism

Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(*,*) = \rho(*,J*)$$ is a Riemannian metric. Denote ...
0
votes
0answers
13 views

Two vector fields are cojugate but not take orbits

Let X and Y be C1 vector feilds on R^m. Suppose that 0 is an attracting hyperbolic singularity for X and Y. Show that there exists a homemorphism h of a neighborhood of origin which conjugate the ...