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

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46 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
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29 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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1answer
33 views

Whitney sum of smooth vector bundles

I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth ...
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0answers
25 views

Embedding of manifolds of constant negative curvature

Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding ...
6
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1answer
92 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
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1answer
40 views

How to find tangent cone in singular point?

How to find tangent cone in singular point of surface? For example, considering surface in $\mathbb{R}^3$ given by equation $x^2z=y^2$, what is it's tangent cone in the origin? UPD:By tangent cone ...
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180 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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0answers
36 views

Are stable manifold for gradient flows embedded submanifold?

Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image ...
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2answers
95 views

Two definitions of integral on boundary $\int_{\partial\Omega}f$?

I have seen two definitions of an integral of a function $f:\partial\Omega \to \mathbb{R}$ from the boundary of an open set $\Omega \subset \mathbb{R}^n$ where the domain is Lipschitz. 1) ...
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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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0answers
82 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 $M$. ...
3
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1answer
60 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why is the following sequence exact? $$0\to \mathbb{R} \to C^\infty (M)\to A\to 0$$ Here $A$ is the set of global Hamiltonian vector fields.
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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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0answers
61 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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3answers
69 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'\\ ...
2
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1answer
43 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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1answer
49 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
37 views

Significance of bump function in the proof.

In the book Semi-Riemannian Geometry with applications to Relativity by Barrett O'Neill, in Chapter 2 (Tensors), he stated that: However, I don't get why he had to use a bump function in the proof. ...
3
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1answer
54 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 ...
2
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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
49 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 ...
10
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1answer
174 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
2
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2answers
109 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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1answer
48 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
32 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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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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3answers
46 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}, ...
3
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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 ...
2
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1answer
94 views

Proof of Euler Characteristic for Sphere

Theorem 1. All cell decompositions of a sphere $S$ have Euler characteristic 2. This is well-known, but I had this idea for an intuitive proof: for any cell decomposition $\Gamma$ with $V$ ...
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36 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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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 ...
0
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1answer
27 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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1answer
62 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
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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1answer
52 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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0answers
30 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 ...
0
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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
56 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 ...
2
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2answers
80 views

Homologous surfaces in three-manifolds

Let M be a 3-manifold. Let $S$ and $T$ be properly embedded surfaces in $M$ such that $[S] = [T] \in H_2(M, N(\partial S)) $. Is it true that we can isotope $\partial S$ so that it coincides with ...
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1answer
128 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
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1answer
243 views

How to define a topological tunnel?

I would like to define a notion of a topological tunnel, but I don't know how (or even if it is possible) to capture it topologically. I am interested in closed 2-manifolds in $\mathbb{R^3}$. Suppose ...
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1answer
38 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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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
56 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
41 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 ...
2
votes
1answer
67 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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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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0answers
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 $dF_x(v)=w$? ...
0
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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 ...