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

learn more… | top users | synonyms (1)

1
vote
1answer
57 views

Statistical Inference and Manifolds

I have just begun approaching the connection between statistical inference and differencial geometry. If I got it correctly, one of the most fundamental concept regards the connection between a $ P(...
7
votes
1answer
126 views

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable (Tensors, or k-linear forms)

Show that $f\in A_{n-1}(V)$ or $f\in A_n(V)$ is decomposable. $f\in A_k(V)$ is decomposable if there exists a $a_1,...,a_k\in V^\wedge$ such that $f=a_1\wedge...\wedge a_k$ In this case "let $n=dim(...
1
vote
0answers
55 views

Are there non-manifold objects in real world?

I'm a beginner in Computer Graphics and today, I encountered the concept of "manifold". And according to the brief interpretation in Wolfram MathWorld: (http://mathworld.wolfram.com/Manifold.html), ...
5
votes
1answer
109 views

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ - I've got the gist, not sure how to write

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ With $A_k(V)$ being the vector space of alternating k-tensors. for $f\in A_k(V)$ for some $v\in V$ we define $...
3
votes
0answers
136 views

Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
0
votes
1answer
61 views

Manifold projection to 2m+1 dimensional subspace is a manifold.

Let $M \subseteq \mathbb{R}^n$ be a m-dimensional manifold. Suppose $n>2m+1$. Show that there is a projection from $M$ to a (2m+1)-dimensional subspace of $\mathbb{R}^n$ so that the image is a ...
2
votes
0answers
59 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 ...
1
vote
1answer
170 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 ...
1
vote
0answers
29 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 ...
1
vote
1answer
105 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 ...
2
votes
0answers
68 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 $...
6
votes
1answer
159 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 $...
1
vote
2answers
50 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 ...
1
vote
1answer
125 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), \...
3
votes
1answer
71 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 ...
1
vote
3answers
117 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'\\ ...
1
vote
1answer
85 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 ...
4
votes
1answer
221 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
votes
2answers
80 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 ...
0
votes
2answers
95 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 ...
2
votes
1answer
232 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 ...
3
votes
0answers
38 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 ...
0
votes
3answers
58 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}, \frac{1}{...
1
vote
0answers
43 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 FM-...
0
votes
0answers
83 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 ...
1
vote
1answer
76 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 ...
0
votes
1answer
153 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} \...
2
votes
0answers
28 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
votes
1answer
98 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 $M$ ...
0
votes
2answers
34 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 $\mathbb{...
1
vote
1answer
97 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 ...
1
vote
0answers
101 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
votes
1answer
227 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 ...
1
vote
1answer
78 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 = ...
2
votes
1answer
70 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
88 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 ...
2
votes
1answer
89 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 ...
1
vote
0answers
50 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
votes
1answer
53 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 ...
0
votes
2answers
83 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 ...
0
votes
1answer
77 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 $p_1,\...
4
votes
2answers
240 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 ...
6
votes
5answers
2k views

Why the 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 ...
1
vote
1answer
44 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 ...
0
votes
3answers
308 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 ...
2
votes
1answer
59 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 W:g_1(y)=0,......
1
vote
0answers
31 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
0
votes
2answers
38 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 = f(x)...
0
votes
1answer
233 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 ...
2
votes
1answer
76 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 ...