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

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168 views

Why Shape space is manifold?

In Shape analysis, often shape is considered as continuous parametrized closed curve and the shape space as Hilbert Riemannian manifold. Can any one help me to understand, why the shape space is ...
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2answers
29 views

Charts for level set manifolds & Multiplication map $F(A,B)=AB$ from $O(n)\times O(n)\to O(n)$ is smooth

This is homework so no answers please We have Multiplication map $F:O(n)\times O(n)\to O(n)$ defined as $F(A,B)=AB$ $F:O(n)\times O(n)\to O(n)$, where $O(n)=\{A\in M(n\times n):AA^{t}=id\}$. The ...
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1answer
40 views

Atlases on the topological manifold $\mathbb R$

I have been trying to produce an example of two incompatible atlases on $\mathbb R$. But no success. Could someone help me please? All my example seem compatible. For example, $A_1 = \{((-\infty,1), ...
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1answer
74 views

Exponential map and distance on a Riemannian Manifold

I'm trying to solve an exercise which I thought at first seemed simple but I'm having some trouble pegging down the error term. The question is to prove that on a Riemannian manifold we have ...
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0answers
58 views

Klein bottle visualization, parameterization, and isotropic version

Suppose a bright glowing orange mobius strip appeared in space for just an instant and then disappeared, except for its glowing orange edge, which remains suspended motionless in space for a moment, ...
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1answer
42 views

Correspondence between one-parameter subgroups of G and TeG

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
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3answers
66 views

does a commutative diagram implies pull-back?

Let $\xi=(E,p,B),\xi'=(E',p',B')$ be fibre bundles. Let $f: B\to B'$, $\bar f: E\to E'$ be maps such that the diagram commutes $\require{AMScd}$ \begin{CD} E @>\displaystyle\bar f>> E'\\ ...
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1answer
47 views

John Lee's Intro to Smooth Manifolds Inverse Function Theorem

In John Lee's "Intro to Smooth Manifolds" Chapter 7, p 160, we have a proof of the inverse function theorem. Here, in the middle of the page we have $F_2 = DF(0)^{-1} \circ F$. Is $DF(0)^{-1}$ a ...
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0answers
16 views

Distance from polynomial to Linear Manifold

How to calculate the distance form the polynomial $a(t) = 1+t^3$ to the Linear Manifold $H$ in the vector space $M_3$ $ H= \{ f(t) \in M_3 | f(1)=-1 \} $
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1answer
49 views

Second (and higher) derivatives of maps between manifolds

I'm trying to understand derivatives of maps between manifolds, and specifically something I read in Dodson and Poston's Tensor Geometry. I'll try to provide as much background as I can for those ...
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2answers
58 views

question about cover maps

Here's a problem I've had a hard time with If $f: M\rightarrow N$ is a cover map and $M$ is a m-manifold, will $N$ also be a m-manifold? A manifold is a space locally Euclidean space that is ...
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2answers
44 views

Prove this plane algeraic curve is not a differentiable manifold

Prove the algebraic curve $\{(x,y)~|~x^2(x+1)-y^2=0\}$ in $\mathbb{R}^2$ is not a differentiable manifold. Remark: It is evident that the given cubic curve has a singularity at $(0,0)$ which disable ...
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1answer
41 views

The formula for a distance between two point on Riemannian manifold

Given a Riemannian (say, connected) manifold $(M,g)$ one can produce the metric via a formula $d(x,y)=\inf_{\gamma}l(\gamma)$ where $\gamma$ is piecewise smooth curve joining $x$ and $y$. My question ...
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0answers
30 views

When is a pseudomanifold a manifold?

Under what conditions does a pseudomanifold become a manifold? I.e is there a nice conclusion we can make if our pseudomanifold has a certain homotopy type, or is possibly piecewiselinear to some ...
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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 ...
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3answers
43 views

Is the closure of an open connected subset of $\mathbb{R}^{n}$ a topological manifold?

If we remove the connectedness restriction, there are easy counter examples, as in: $\left(\frac{1}{2}, \frac{1}{1}\right) \cup \left(\frac{1}{4}, \frac{1}{3 }\right) \cup \left(\frac{1}{6}, ...
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0answers
28 views

Glueing cubes to manifolds with corner

I am interested in proposition 3.7 in Salvatore's `Configuration spaces with summable labels'. The result states that the bar construction on the Fulton-Macpherson operad is isomorphic to the ...
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0answers
35 views

Diffeomorphism between $\mathbb{R}^{2}/\sim$ (Torus ) and $\mathbb{S}_{1}\times \mathbb{S}_{1}$

This is homework so no answers please. Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. I ...
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1answer
66 views

Uniqueness of the nearest point in $N$

Let $(N,h)$ be a compact Riemannian submanifold of the Euciledean space $\mathbb{E}^q$. i.e.$N$ is a compact submanifold of $\mathbb{E}^q$ and $h$ is the Riemannian metric of $N$ derived from the ...
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1answer
26 views

How to prove a set is a submanifolds or not

I am studying about differentiable manifolds. My professor give me an example show that graph of a continuous function is a submanifold, but image of its is not in general. $$f: \mathbb{R} ...
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0answers
25 views

Why is the tangent space at the boundary of an $n$-manifold also $n$ dimensional?

This question is about the tangent space at the boundary of a manifold. My book says that the tangent space at the boundary of a manifold is defined in the same way that it is defined in the ...
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0answers
17 views

Smoothly compatible for $\mathbb{R}^{2}/\sim$ ,where $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$

This is homework so no answers please. Consider the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$ ,where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$. Then ...
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1answer
43 views

Equivalence between orientation of the tangent bundle and orientation of manifolds

If $M^{n}$ is a manifold then the following statement are equivalent. The tangent bundle $(TM,\pi,M)$ is an orientable $n$-dimensional vector bundle. $M$ has an $\lbrace (U,h)\rbrace$ atlas on $M$ ...
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29 views

Showing the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$, where $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$, is open

This is homework so no answers please. Showing the quotient map $\pi:\mathbb{R}^{2}\to \mathbb{R}^{2}/\sim$, where $(x_{1}y_{1})\sim (x_{2},y_{2})$ iff $x_{1}-x_{2},y_{1}-y_{2}\in \mathbb{Z}$ is ...
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2answers
16 views

Hopf map is continuous

Consider map $h:\mathbb{S}^{3}\to \mathbb{S}^{2}$ defined as $h(a,b)=(a\bar{b}+b\bar{a},ib\bar{a}-ia\bar{b},|a|^{2}-|b|^{2})$ Does it just follow by seeing h as map from $\mathbb{C}^{4}$ to ...
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1answer
55 views

Boothby's exercise $I.3.1$ on showing that a subset of $\mathbb{R}^{2}$ is not a manifold

First time here, so sorry for the rookie mistakes. I'm a $4^{th}$ year physics student taking Riemannian Geometry so my background on the subject is very small. I'm trying to solve Boothby's exercise ...
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0answers
81 views

Existence of smooth partition of unity

Suppose that $0<\alpha_1<\cdots<\alpha_n<1$. Let $M$ be a smooth paracompact connected manifold of dimension $d$. Let $(U_k)_{k\in\mathbb{N}}$ be a locally finite open covering of $M$. ...
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1answer
35 views

Proof that a map from an orientable surface to a non-orientable surface has even degree.

For a smooth map $f:M\to N$ from an orientable closed surface $M$ to a non-orientable closed surface $N$, we define its parity (also called modulo 2 degree, and denoted $\deg_2(f)$) as the parity of ...
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1answer
59 views

A question on Lie derivative

For a Lie derivative $\mathscr{L}_{X} Y$ of $Y$ with respect to $X$, we mean that for two smooth vector fields $X$ and $Y$ on a smooth manifold $M$ such that the following holds $$ \mathscr{L}_{X} Y = ...
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0answers
23 views

Computing time-dependent vector field

Let $H : \mathbb{R} \rightarrow \operatorname{Herm}(C^2) $ be a smooth function and $$ t \mapsto g(t) = e^{iH(t)} $$ be the associated curve of diffeomorphisms of $\mathbb{C}^2$. Compute the ...
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1answer
54 views

differential manifolds or algebraic topology [closed]

In our university we must catch a course at least in one of these courses: differential manifolds or algebraic topology . which one is harder to start at first , differential manifolds or algebraic ...
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1answer
40 views

Maps between Manifolds and Maximal Rank

I'm trying to prove a theorem from Olver's Applications of Lie Groups to Differential Equations. It's supposed to be an "easy" consequence of the Implicit Function Theorem but I honestly can't see ...
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1answer
46 views

Tangent space to $\mathbb{R}P^{n}$

I could not find any other question here related to this. If I have missed out, then this could be voted as a duplicate(Sorry if it is!). I was just trying to figure out the tangent space to the ...
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1answer
66 views

Equivalent topologies on Real projective space $RP^{n}$

This is homework,so no answers please. Prove that the topology on $RP^{n}$ given by the standard smooth structure (lines through the origin in $\mathbb{R}^{n+1}/\{0\}$ and $\tau_{1}$) is equal to ...
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32 views

Given points $x,y$, and tangent vectors $v,w$, is there a diffeomorphism $x\mapsto y$ and $v\mapsto w$?

Suppose $M$ is a smooth manifold, and you're given two points $x,y$, and associated tangent vectors $v$ and $w$. I'm curious, does there exist a diffeomorphism $F$ such that $F(x)=y$ and $dF_x(v)=w$? ...
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1answer
31 views

Real/Complex Manifolds - Transition Maps

I'm trying to understand how real/complex structure is imposed on a manifold, especially the likes of smooth manifolds. I can read the definitions and work with them, but I want to understand ...
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2answers
44 views

connected sum of two surfaces

I was reading Massey's textbook on Algebraic topology and the author claims that if $S_2$ is a 2-sphere then $S_1 \# S_2$ is homeomorphic to $S_1$. I don't know why that is true and since I'm very ...
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1answer
43 views

Braid Groups on Manifolds

I am studying braid groups on manifolds and am getting confused. In a geometric definition, one needs to first choose a simple curve $\theta$ on a given manifold $M$ and well-ordered points ...
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2answers
106 views

Codimension 1 homology represented by Embedded Submanifold

I'm looking for a reference for the following statement: Given an oriented manifold $M$ and a class $\xi \in H_{n-1}(M)$, there is some embedded oriented submanifold $F^+ \hookrightarrow M$ such that ...
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4answers
807 views

Why surface of the sphere is not a Euclidean space?

In wiki, I see a description in manifold: "Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in ...
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1answer
29 views

Knots as boundaries

Boundary of a 2-manifold is a closed curve (or a set of closed curves), so I was thinking of reversing this process. In 2D space, a simple closed curve in a plane can be thought of as a boundary of ...
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3answers
60 views

What exactly is a 0-form?

From what I understand, a k-form in the real numbers is essentially a mapping $\mathbb{R^k} \rightarrow \mathbb{R}$, but I can't seem to find a corresponding definition for a "0-form". Wikipedia seems ...
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1answer
38 views

Simultaneous coordinate representation of a submanifold and its sub-submanifold

Suppose $Z\subset X\subset Y$ are manifolds and $z \in Z$. Prove that there exist an independent function $g_1,...g_l$ on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W =\{y\in ...
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0answers
21 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
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1answer
23 views

Geometric translation of a theorem about stability of equilibrium point

In the book Nonlinear Systems by Hassan Khalil, there is a theorem about the stability of equilibrium point ‎‎ which asserts that : Theorem :‎ Let‎ ‎$X = 0 $ ‎be an equilibrium point for‎ ‎$‎\dot{x} ...
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1answer
53 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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1answer
50 views

Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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1answer
38 views

trace map is continuous

Prove that $tr: M_n(k)\to k$ is continuous. I did continuity of determinant map using induction, but how to prove trace map is continuous. please give a thorough answer. My analysis is not too good. ...
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0answers
25 views

Orientation of the intersection of manifolds

From Guillemin and Pollack Differential Topology: Compute the orientation of $\mathbf{X}\cap\mathbf{Z}$ in the following examples by exhibiting positively oriented bases at every point: a) ...
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0answers
36 views

Compact manifold with smooth structure

Is the following surface smooth and compact, when all its partial derivatives are continuous? How to tell about self-intersections without visualization? $x=\cos(a) + \cos(a + b) + \cos(a + b + ...