1
vote
3answers
47 views

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
1
vote
1answer
29 views

A bounded domain can be considered as a compact manifold?

A bounded domain $\Omega$ with smooth boundary $\Gamma$ can be considered as a compact connect Riemannian manifold?
1
vote
1answer
20 views

Cont. Function smooth iff composition with submanifold inclusion is smooth

I'm trying to proof the following: Let $X$ be a smooth manifold, $X_0$ an open subset of $X$, $i: X_0 \to X$ the canonical inclusion, $Y$ another smooth manifold and $f: Y\to X_0$ continuous, then ...
7
votes
0answers
75 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 ...
0
votes
0answers
39 views

finite-dimensional continuous vector bundle

Let $M$ a compact metric space and $\pi: F \rightarrow M$ a finite-dimensional continuos vector bundle over $M$, endowed with a continuous Riemannian metric. I was wondering if it will be true that: ...
3
votes
2answers
96 views

Is this set a manifold?

For which $ ( \alpha , \beta ) \in \Bbb R^2$ set: $\{ (x_1,x_2,x_3,x_4) \in \Bbb R^4 | x_1+x_4= \alpha, x_1 x_4 - x_2x_3 = \beta \}$ is a manifold? I made a Jacobian matrix: $ \begin{bmatrix} ...
0
votes
1answer
95 views

Question about statement of Rank Theorem in Rudin

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. ...
1
vote
1answer
46 views

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$.

Let $f:\mathbb{R^n} \to \mathbb{R^m}$, if $f$ is a linear transformation, prove that $Df(a)=f$. My try : By definition of derivative of a function $f:\mathbb{R^n} \to \mathbb{R^m}$ , If I know ...
1
vote
1answer
94 views

Set of matrices differentiable manifold? [closed]

Let $X$ be a set of matrices $2\times 3$, that for all $A$ from $X$ rank $A=1$. Is $X$ a manifold? If not find a maximum subset in $X$, which is a manifold and its dimension.
0
votes
2answers
70 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
96 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
140 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 ...
2
votes
1answer
28 views

sufficient condition for being an integral factor

Let $ f: \mathbb {R}^m \rightarrow \mathbb {R}-\{0\} $ function $C^{\infty}$ class and $w$ a one-form $C^{\infty}$ class in $\mathbb {R}^m $. If $\alpha=w-\dfrac{1}{f}dx_{m+1} $ satisfies $\alpha ...
1
vote
1answer
43 views

Existence of a nonzero vector to form

Let $ f: \mathbb {R}^m\times \mathbb {R}^m \rightarrow \mathbb {R}^m $ an alternate form of grade two. If $ m $ is odd, prove that there exists $ v\neq 0 $ such that $ f (u, v) = 0 $, for all $ u \in ...
1
vote
1answer
39 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
57 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
60 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
59 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
123 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
votes
1answer
130 views

The double cone is not a surface.

My question is that A double cone ( also named as "circular cone") is not a surface. I know its reason. But I cannot show this mathematically. Suppose $\sigma : U \to S\cap W$ Is a surface ...
0
votes
1answer
42 views

Show that a paraboloid is asurface .

That I know about paraboloid is all in the picture. I wrote its surface patch. (Hopefully, it is correct) From there, what do I need to do in order show that a paraboloid is a surface. ...
1
vote
1answer
46 views

Taylor expansions on manifolds..

can one consider Taylor expansions of functions defined between smooth manifolds? If so, is there a reference for learning more about it? Thanks
7
votes
1answer
234 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 ...
1
vote
1answer
49 views

Why is it that the vector space of all derivations has the basis $\partial/\partial{x_1} \ldots \partial/\partial{x_n}$?

So I have seen stated that two definitions of tangent spaces (w.r.t manifolds) are equivalent. But I am having some difficulty proving they are indeed equivalent. It looks like it boils down to me ...
1
vote
1answer
258 views

Proof of the General Stokes Theorem in Munkres

In "Analysis on manifolds" Munkres proves the general Stokes theorem $ \int_{\partial M}\omega = \int_Md\omega $ in the case where the support of $ \omega$ can be covered by a single coordinate patch ...
1
vote
1answer
81 views

torus filling curve

I'm trying solve this problem but I didn't many ideas how to do it. So, if someone can give me a hint or the step of a solution I would greatly appreciate it. This is the problem: "Let ...
1
vote
1answer
56 views

Integers or cantor set submanifold of the real numbers?

I'm trying to see whether $\mathbb{Z}$ or the cantor set $C$ are submanifolds or $\mathbb{R}$. Actually, I thought that $\mathbb{Z}$ was not a submanifold. As every subset of $\mathbb{Z}$ is ...
4
votes
2answers
78 views

Understanding the proof for: $d(f^*\omega)\overset{!}{=}f^*(d\omega)$

Consider this Proposition: Let $U\subset\mathbb{R}^n$ and $V\subset\mathbb{R}^n$ be open sets and $\phi:U\to V$ be differentiable. For all $k\in\mathbb{N}_0$ and $\omega\in \Lambda^k(V)$ it is true ...
3
votes
1answer
110 views

Tietze–Urysohn's lemma in $\mathbb{R}^n$

Let $F_1$ and $F_0$ be closed subsets in $\mathbb{R}^n$, $F_0\cap F_1=\varnothing$. How to build a $C^{\infty}$- function $f:\mathbb{R}^n\to \mathbb{R}$, such that $f|_{F_1}=1$, $f|_{F_0}=0$ and ...
4
votes
1answer
109 views

Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
3
votes
3answers
234 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
1
vote
0answers
111 views

Deformation retract

How to prove that $r_t$ is a deformation retract $M^a=\lbrace q\in M ; f(q)\leq a\rbrace$ We have the definition : $r_t$ is a difformation retract if: $r_t$ is a continius ,onto application ...
2
votes
1answer
64 views

Many partitions of unity on sufficiently “nice”; what does this mean?

In a class I am taking, we are told that are manifolds have "many" partitions of unity (we assume paracompact, Hausdorff, second-countable). However, the course content is not related to this subject. ...
3
votes
1answer
258 views

Question about theorem 3.2 from Morse theory by Milnor

The demonstration of the theorem 3.2 in the book Morse theory by Milnor THEOREM $\mathbf{3.2.}$ Let $f:M\to\bf R$ be a smooth function, and let $p$ be a non-degenerate critical point with index ...
2
votes
1answer
181 views

An other question about Theorem 3.1 from Morse theory by Milnor

In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
1
vote
2answers
92 views

Diffeomorhism of manifold

This is one of the exam questions of the previous semester. I have studied these. But I didn't do this. Please show me how to solve this question. Thank you for help
0
votes
1answer
69 views

Problem about tangent vector and the inclusion map of the unit circle.

It is so complecated for me. Please can you show how to solve. Thank you.
0
votes
1answer
73 views

The differential $i∗ : TpS_{2} → TpR_{3 }$ maps $ ∂/∂u|p,∂/∂v|p $ into $TpR_{3}. $ Find $(α_{i}, β_{i}, γ_{i})$

Hi! This was my homework. Prof. sent its answer. But I didnt understand how can this answer be reached? Please can someone explain this?
5
votes
2answers
184 views

Why is $\partial\partial M=\varnothing$?

Why is the border of the border of an oriented differentiable $n$-dimensional Manifold $M$ empty, that is $$\partial\partial M = \emptyset?$$
0
votes
1answer
67 views

Lie bracket of vector fields on $\Bbb R^{n}$

Please show how to solve? I am stack with lie bracket. Thank you.
0
votes
1answer
50 views

Is $S$ a regular submanifold of $\Bbb R^{3}$?

$$S=\{(x,y,z) \mid x^{2}+y^{2}=z^{2}\}$$ $g: \Bbb R^{3}\to \Bbb R$, $S=g^{-1}(0)$ Is $S$ a regular submanifold of $\Bbb R^{3}$? I'd be grateful for a clear and explicit explanation of why this is ...
0
votes
1answer
92 views

Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
2
votes
2answers
114 views

Smooth maps between Euclidean spaces

There is a question that has bothered me for quite a long time. When we define smoothness of a function(or a map from one space to another), we define it as "has continuous partial derivatives of all ...
5
votes
1answer
317 views

Line integral and integration of differential forms

The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $. Let $ \gamma:(a, b) ...
2
votes
2answers
786 views

Does the Implicit mapping theorem imply the inverse mapping theorem?

Does the Implicit mapping theorem imply the inverse mapping theorem?
1
vote
2answers
72 views

About regular surfaces

I never had seen this exercise, but I'm confused again, I don't know what I have to use. I have the surface $S=\{(x,y,z)\in \mathbb{R}^3|xy+xz+yz=1,x>0,y>0,z>0\}$, is $S$ regular?. Then, if ...
0
votes
0answers
47 views

Problem with a proof of one of characterization of manifold in $\mathbb R^n$

Let $M \subset \mathbb R^n$, $k \in \mathbb N$. Assume that $M$ is a $k$-dimensional manifold in $\mathbb R^n$, i.e. for each $x \in M$ there exists an open set $W \subset \mathbb R^k$ and a smooth ...
1
vote
0answers
45 views

Manifolds question

Let $M$ subset of $R^{n+p}$ be the zero set of a $C^\infty$ mapping $g:R^{n+p} \rightarrow R^{p}$. Assume that the Jacobi matrix of $g$ has rank $p$ everywhere on $M$. Show that $M$ is an ...
5
votes
2answers
833 views

Spivak's proof of Inverse Function Theorem

I am having trouble with Spivak's proof of the Inverse Function Theorem in his Calculus on Manifolds: 2-11 Theorem (Inverse Function Theorem). Suppose that $f: \mathbb{R}^n\to\mathbb{R}^n$ is ...
3
votes
1answer
206 views

Graph and manifold

I needed help to prove the following: Let $k, n, m$ be elements of the natural numbers and $g : R^m \to R^n$. Prove that the graph of $g$ is an $m$-manifold of class $C^k$ if and only if $g$ is of ...