5
votes
1answer
63 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
0
votes
0answers
18 views

What is the connection between $\sqrt g$ and $|\det \psi'|$?

My text defined integration on a manifold as follows Let $M\subset \mathbb R^n$ be an $m$-dimensional manifold, $\varphi:U\to V$ a local map $(U\subset\mathbb R^m, V\subset M)$ and $f:M\to\mathbb ...
3
votes
1answer
65 views

Recommendation on studying differential geometry

Below are what i studied so far: Rudin - Principles of Anlysis (only except one chapter, namely differential forms) Munkres - Topology (only point-set topology) Rudin - RCA (Only first 4 ...
0
votes
0answers
27 views

In the def. of a (k+l)-manifold with boundary, is it permissible for the domain of the coord. patch to be open in R^k x H^l?

Studying Munkres's "Analysis on Manifolds" on my own. One exercise (#5, p. 103), asks the reader to show that if $M$ is a k-manifold without boundary in $R^m$, and if $N$ is an l-manifold in $R^n$, ...
0
votes
2answers
62 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
6
votes
1answer
91 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
1
vote
1answer
52 views

Show that a set is a manifold.

Let $n \ge 3 $. How can I show that $M:= \{(x_1,...,x_n) \in \Bbb R^n \setminus \{(0,...,0)\} | x_1^2+...+x_n^2 = x_1 \cdot...\cdot x_n \}$ is a manifold of class $C^1$? Can anyone please tell me ...
1
vote
0answers
31 views

Do we need to pay attention to the codomain of a differentiable function?

I came across the following definitions: We call $M\subset \mathbb R^N$ $m$-dimensional $C^k$-submanifold of $\mathbb R^N$ if for all $a\in M$ there is an open neighborhood $U$ of $0$ in $\mathbb ...
0
votes
1answer
29 views

Lie brackets definition

Let $v,w$ be vector fields on a smooth manifold $M$ (i.e. $v : M \rightarrow TM = \lbrace (p, v_p) : p \in M, v_p \in T_p M \rbrace$). The Lie brackets of $v,w$ are defined as $$ [v,w](f)|_p = ...
2
votes
1answer
48 views

Tangent vector to a curve on a manifold

If one has a curve $\sigma : (-1,1) \rightarrow M$, where $M$ is a smooth manifold, the tangent vector in $\sigma(0)$ is usually defined as $$ \sigma'(0) (f) = \dfrac{d f \circ \sigma}{dt} \Big|_0,$$ ...
4
votes
0answers
32 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 $d$ ...
2
votes
1answer
61 views

How to decide if a given set is a manifold

Could you tell me how to decide if a certain set is a manifold? I know there already is a similar question here, but there we have fairly "visualizable" sets: a hemisphere and a square. What in ...
0
votes
0answers
46 views

Is a manifold over $\mathbb{R}$ normal?

We have manifold $G$ over the reals with its finite atlas ($g_i:U_i \to g_i(U_i) \subseteq \mathbb{R}, G=\bigcup U_i$). The atlas induces a topology in the normal way ($A \subseteq G$ is open iff ...
0
votes
1answer
44 views

Diffeomorphism of closure of open sets

Let $F:\overline{X} \to \overline{Y}$ be a map between the closure of two open Lipschitz domains $X$ and $Y$ in $\mathbb{R}^n$ (with boundaries). $F$ is such that it maps $X$ to $Y$ and it maps ...
0
votes
1answer
45 views

Triangle a manifold

Let $x,y,z \in \mathbb{R}^3$ and $\Delta:=\text{conv} \{x,y,z\}$ be a triangle. My question is: Is this triangle a $C^2$ submanifold in $\mathbb{R}^3$? The reason is, that I would need this fact in ...
0
votes
2answers
60 views

Orientability of surfaces in $\mathbb{R}^3$

I have a question: I'm currently reading a few things about orientability and understood this concept as the answer to the question: Given a surface and a unit normal vector field on it: Is there a ...
4
votes
1answer
88 views

The unsolved extension problem on manifolds.

I have been struggeling for quite a while with this problem: Let $M \subset \mathbb{R}^n$ be a compact $C^k-$ submanifold and $\phi_i: B_i(0) \rightarrow M$ be the associated set of charts ...
1
vote
1answer
132 views

How do 1-d compact submanifolds look like?

I was wondering whether it is true that every 1-d compact submanifold $\subset \mathbb{R}^n$ that is $C^1$ is a closed curve that is also $C^1$, cause I cannot think of more examples. Therefore, I ...
1
vote
1answer
36 views

How to show that the metric in the tangent space is independent from the chart you take?

I want to prove that for vectors $v_1,v_2 \in T_aM$ the euclidean length and distance is independent from the chart we are using, where $M$ is a submanifold in some $\mathbb{R}^n$ My problem is that ...
3
votes
1answer
50 views

Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
0
votes
2answers
53 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
2
votes
1answer
55 views

Constructing submanifolds. Did I understand this right?

I just want to know whether I understand the construction of a submanifold in some $\mathbb{R}^n$ properly. Please correct everything that you think could be wrong. As far as I know so far, it is ...
1
vote
1answer
114 views

Inner product on tangent space and metric tensor

In our class we talked about integrating on submanifolds and as a short side remark our teacher told us that by knowing the metric tensor, it is possible to define an inner product on a tangent space ...
1
vote
1answer
29 views

Significance of rank of frechet derivative in definition of manifold?

In studying manifolds, the stipulation that the derivative be full rank is confusing to me on an intuitive level. Can anyone please explain how I should think about this intuitively? What does it mean ...
7
votes
1answer
210 views

Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
3
votes
0answers
34 views

Lie differentiation operation for manifolds

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. Let ...
6
votes
1answer
136 views

Tangent space for product of submanifolds

Suppose that $X_1$ is an $n_1$-dimensional submanifold of $\mathbb{R}^{N_1}$, and $X_2$ is an $n_2$-dimensional submanifold of $\mathbb{R}^{N_2}$, and let $X=X_1\times X_2$. Let $p_1\in X_1$ and ...
4
votes
1answer
95 views

Product of submanifolds is a submanifold

Suppose that $X_1$ is an $n_1$-dimensional submanifold of $\mathbb{R}^{N_1}$, and $X_2$ is an $n_2$-dimensional submanifold of $\mathbb{R}^{N_2}$. Prove that $X_1\times X_2\subseteq ...
3
votes
1answer
41 views

Derivative of function between sets of matrices

Let $M_{k,n}$ be the set of all $k\times n$ matrices, $S_k$ be the set of all symmetric $k\times k$ matrices, and $I_k$ the identity $k\times k$ matrix. Let $\phi:M_{k,n}\rightarrow S_k$ be the map ...
0
votes
3answers
89 views

What dimensions are possible for contours of smooth non-constant $\mathbb R^n\to\mathbb R$ functions?

While for $n=2$ it is pretty clear that the contours of a non-constant $f:\mathbb R^n\to\mathbb R$ are either extrema (and therefore points) or (the union of) 1-dimensional isolines, for $n=3$ I am ...
1
vote
0answers
36 views

problem with submersion

Given $\varphi:\mathbb{R}^{m+n}\longrightarrow \mathbb{R}^m$ is $C^{ k}$ class. If there $a\in \mathbb{R}^{m+n}$ with $\varphi^{\prime}(a)$ is surjective. Then there a mergullo $f:V\longrightarrow ...
6
votes
1answer
187 views

A problem from Spivak's Calculus on Manifolds

Notation As Spivak suggests, given $A\subset\mathbb R^n$, boundary $A$ denotes the topological boundary of $A$, i.e. $\overline A\cap\overline{A^c}$. Problem 5-3(a): Let $A\subset\mathbb R^n$ be ...
2
votes
0answers
88 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 $$ ...
2
votes
1answer
72 views

Diagonal Inclusion Map of a manifold $X$

This question actually comes from the question I asked before: Derivative map of the diagonal inclusion map on manifolds And I repeat it as follows: Let $f: X\longrightarrow X\times X$ be the ...
4
votes
1answer
109 views

Horn and spindle tori

I was trying to prove that the horn torus and the spindle torus are not manifolds by definition(locally diffeomorphic to some Euclidean space.). I have no idea how to do this, but I attempted it in ...
6
votes
1answer
177 views

Derivative map of the diagonal inclusion map on manifolds

I was trying to work through a problem(#10 of $\S$1.2) in Guillemin and Pollack's book $\textit{differential topology. }$ The problem is given as follows. Let $f: X\longrightarrow X\times X$ be ...
2
votes
2answers
76 views

Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
-1
votes
2answers
129 views

The euclidean space $\Bbb R^n$ is orientable as a manifold.

I know that The euclidean space $\Bbb R^n$ is orientable as a manifold. I think that it is orientable because it has a nowhere vanishing $n$-form. But I am not sure. Please can you explain ...
2
votes
0answers
58 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 ...
3
votes
0answers
278 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
1answer
70 views

Real Projective Space

How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help me.
1
vote
1answer
40 views

Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
votes
1answer
91 views

What is overlop

I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
2
votes
2answers
90 views

An open cover that is not locally finite

I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
2
votes
0answers
128 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
1answer
86 views

Manifolds with boundary and definition

Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
2
votes
1answer
82 views

Boundary orientation for a cylinder

Please help me.I am think that I can use stokes theorem but ı could not apply.This question is very benefical for me to learn the subject please help me :(
4
votes
1answer
151 views

The open Möbius Band is not orientable

Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
2
votes
1answer
135 views

Why is the cylinder surface on $\Bbb R^3$ orientable?

Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?
2
votes
1answer
85 views

$F(x,y) = (x^2 +y^2,xy)$. compute $F^{∗}(u \, du+v \, dv)$

Let $F : \Bbb R^2 → \Bbb R^2$ be given by If $u$,$v$ are the standard coordinates on the target $\Bbb R^2$, compute $F^{∗}(u \, du+v \, dv)$. $$F(x,y) = (x^2 +y^2,xy).$$ I am confused so much. I ...