2
votes
0answers
38 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
1
vote
3answers
77 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
1
vote
2answers
44 views

Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
1
vote
2answers
63 views

Topologies on n-manifolds

In the study of n-manifolds (real and imagined), is there any reason to spend much time learning about topologies other than the usual topology?
1
vote
1answer
47 views

Alexandrov embedded disc

We say that a compact surface $\Sigma$ is Alexandrov embedded via an immersion $f:\Sigma \rightarrow \mathbb{R}^3$ if there exist, $X$ a 3 manifold and an immersion $F: X \rightarrow \mathbb{R}^3$ ...
1
vote
1answer
47 views

Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
1
vote
0answers
32 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
0
votes
1answer
55 views

Question about Alternating forms

So I understand the definition of an alternating form on $\mathbb{R}^m$, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so?
0
votes
2answers
52 views

What is the difference between a manifold and a topological manifold?

Is it the case that a topological manifold is just a topological space but we have not mentioned any specific structure on it ?
1
vote
1answer
35 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
1
vote
0answers
25 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
1
vote
2answers
148 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
1
vote
2answers
83 views

Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
1
vote
1answer
39 views

Integrable Manifolds

I'm trying to understand why the line of slope y passing through (x,y) is an integral manifold. My intuition tells me that there exists a point in the slope field where the distribution cannot be ...
0
votes
1answer
63 views

Link between a topological space and a manifold

A topological space is defined as a non-empty set $X$ together with a given collection of subsets $T$ (topology) of $X$, such that, (i) any union of these subsets is one of the subsets. (ii) any ...
1
vote
0answers
73 views

n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
3
votes
1answer
46 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
0
votes
0answers
31 views

Antipodal map and parallel transport on $S^3$

I'm reading Thurston and Levy's "Three-dimensional geometry and topology" where they have an informal explanation of what life in $S^3$ would look like. For the following detail that I am concerned ...
0
votes
0answers
15 views

Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
0
votes
1answer
96 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
10
votes
1answer
117 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
1
vote
1answer
27 views

Suppose $A \subset S$ and both $A$ and $S$ are regular surfaces. Show that $A$ is open in $S$

Suppose $A \subset S$ and both $A$ and $S$ are regular surfaces. Show that $A$ is open in $S$ (w/ respect to subspace topology on $\mathbb{R}^3$. Note that the definition of a regular surface $S$ is ...
1
vote
1answer
40 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
6
votes
2answers
142 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
3
votes
1answer
80 views

Isometry vs. measure preserving?

Consider functions between two measured metric spaces. What is the relation between an isometry and a function which preserves the measure of subsets? This question arose in my head as I thought ...
1
vote
2answers
57 views

Compactness of a certain set coming from the Ricci Flow

I was wondering about the following situation: Suppose we have a solution to the Ricci Flow on a compact manifold $M$, $g(t)=g_t$, on a compact time interval $[0, \delta]$, and we consider ...
2
votes
1answer
95 views

Brouwer's fixed point theorem

Theorem: If $f:D^n\rightarrow D^n$ is continuous then there is $x \in D^n$ such that $f(x)=x$. To prove the theorem we assume that $f$ is cts but has no fixed point, that is $f(x)\neq x$ for all ...
0
votes
1answer
28 views

Relative Compactness $\Rightarrow$ Compactness

I try to figure out: $(\overline{A}^U\text{ compact in }U )\Rightarrow( \overline{A}^X\text{ compact in }X)$ ...while $U\in\mathcal{T}$ It's clear for the case: $\overline{A}^X\subseteq U$ But else, ...
1
vote
2answers
78 views

How to choose $f\in C^{2}(\mathbb{R})$ with compact support and takes value 1 on connected compact set?

Let $0< \delta < \pi$. My questions: (1) How to construct(choose/method) $f\in C^{2}(\mathbb R)$(= First two derivatives ($f' \ \text{and} \ f''$) of $f$ on $\mathbb R$ exists and are ...
0
votes
0answers
54 views

Chart on a manifold

I have the following question. If I consider a manifold, for example a torus T see as space of identification $[0,1]\times [0,1]$ why I can't cover it with only one chart? what fails if a chart ...
1
vote
1answer
29 views

Minimizer and invariance of normal projection energy of a knot

While reading the paper, A simple energy function for knots, I understand that the authors have proved the two conditions of the first page for the normal projection energy of a knot. But I failed to ...
0
votes
1answer
58 views

On the “regularity” of the boundary of an open set

Let $M = \mathbb{R}^2$ (or more generally, let $M$ be a topological manifold) and let $\Omega$ be an open set in $M$. I'm considering the following regularity conditions for the boundary of $\Omega$: ...
1
vote
1answer
66 views

Product manifolds

I have a question on the product of two manifolds. I have $M, N$ two real manifolds (with a smooth differentiable structure), with $\partial M=0$. I have showed that $M\times N$ has a natural induced ...
2
votes
2answers
57 views

homeomorphism between maninifolds

Exist a local homeomorphism between the manifolds with boundary $[0,1) \times [0,1) $ and $\mathbb{R}^{2}_{+}$? I don't think that a local homeomorphism like this can exist..
7
votes
2answers
147 views

When does homeomorphism imply diffeomorphism?

In $R^n$, suppose $U$ and $V$ and two homeomorphic open sets. Then, is $U$ diffeomorphic to $V$? If not, can we impose stronger conditions such that this true?
0
votes
1answer
39 views

Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
1
vote
1answer
50 views

Using transition maps as a comparison tool between charts on a manifold.

In the wikipedia article http://en.wikipedia.org/wiki/Chart_%28topology%29#Transition_maps we read A transition map provides a way of comparing two charts of an atlas. To make this comparison, we ...
0
votes
1answer
30 views

Support of form and embedded varieties

I need help with some inclusions. Let $i: S \rightarrow M$ be an embedding between two oriented varieties of dimension k and n respectively. Assume that the $i(S)$ is closed and that $\omega\in ...
1
vote
0answers
54 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
0
votes
1answer
56 views

Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
2
votes
1answer
69 views

Is the map from the circular half cone to the $xy$ plane a local isometry?

This is a text book exercise. And I think that this map is not a local isometry. But, I don't know how to show this question. Please help me explaining this question. Thanks a lot. I posted its ...
1
vote
1answer
58 views

The question related to a regular surface.

Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$. I know ...
1
vote
0answers
25 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
1
vote
0answers
90 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
0
votes
1answer
59 views

Verify this is not orientable.

Verify this is not orientable. Möbius transformation: $$U=\{(t,\theta) \mid \frac{-1}{2}\lt t\lt \frac{1}{2}, 0\lt \theta \lt 2\pi \}$$ $\sigma (t, \theta)=<((1-t\sin (\theta/2))\cos (\theta), ...
3
votes
1answer
40 views

$S^1\times S^1$ diffeomoprhic to torus of revolution.

I am searching for an diffeomorphism between $$S^1\times S^1$$ and the torus of revolution $$\{(x,y,z)\in\mathbb{R}^3|z^2+(\sqrt{x^2+y^2}-a)^2=r^2\}.$$ I know it's true, I am just looking for an ...
4
votes
1answer
91 views

Characterize a continuous function in terms of its graph

(I believe) a map $\varphi: G_1 \to G_2$ is a group homomorphism iff $\operatorname{Graph}{(\varphi)}$ is a subgroup of $G_1 \times G_2$ (similarly for the categories of vector spaces, algebras, ...
0
votes
1answer
31 views

prove the existance of $U$ and $V$ Neighbourhoods of $X_0$ and $F(X_0)$ so that $F_{|U}$ is a diffeomorphism in $V$

we have $E = M_n(R)$ and : $$F : E \to E , F(X) = X^2 + X - I$$ we need to prove the existance of $U\in \mathscr V(X_0)$ and $V\in > \mathscr V(0_E)$ so that $F_{|U}$ (restriction ...
1
vote
1answer
86 views

Parametrization of $S^3$ embedded in $\mathbb R^4$?

I would like to know of any parametrization of the standard 3-sphere: {$(x_1,x_2,x_3,x_4): x_1^2+x_2^2+x_3^2+x_4^2=1$} embedded in $\mathbb R^4$. I know of parametrizations for $S^1$, for $S^2$ , ...
5
votes
2answers
96 views

Applications of showing a set is both open and closed?

A general technique is as follows: To show that a property holds for a connected space, one can prove that the set of all points that satisfy this property is nonempty and forms a closed and open ...