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

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45 views

Clarification of vector valued forms (sections and tensor products)

I am seeking a bit of clarification on vector valued forms. Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately, Let ...
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1answer
53 views

The curvature of the connection one form - misunderstanding

Let $(P,M,G,\pi,\cdot)$ be a principal bundle. Let $\omega$ be the connection one form for a connection $H\subset TP$. Let $X,Y$ be smooth vector fields on $P$. Then the curvature $\Omega$ of the ...
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2answers
61 views

Definition of differential on manifolds

I'm studying some differential Geometry at the moment and I'm getting a bit stuck with the definition of the differential. It's defined as follows \begin{array}{cl} \phi_{\star,m} : T_{m}M ...
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29 views

Intuition about twisted homology

I will need to do quite some reading on twisted invariants this year, such as twistings of Reidemeister Torsion or Alexander Polynomials. I already had some insights into said (twisted) invariants, ...
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1answer
54 views

Prove that an atlas is $C^{\infty}$

The sphere $S^2$ can be covered by the following $6$ subsets (hemispheres) $$ O_i = \{(x^1, x^2, x^3) \in \mathbb{R}^3 | x^i > 0, i = 1, 2, 3\}$$ Each of these subsets can be mapped by the unit ...
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1answer
37 views

Higher homology groups relative a lower dimensional subspace

One often works with reduced homology, which (in the case of say, smplicial homology) is defined as the homology relative a point. Now at every grade except zero, the reduced homology objects are ...
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0answers
18 views

How two disjoint solid 2-tori linked?

I have 3-manifold which is a union of two disjoint solid 2-tori, How I can decide if they are meet or no? Also if they are meet how I can know the way they are linked?? Thanks in advance
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57 views

$F$-related vector fields

I'm having a bit of difficulty understanding $F$-related vector fields. I think I understand conceptually what's going on (taking a vector field on a manifold $M$ and getting a smooth vector field on ...
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2answers
82 views

Lie Groups/Exponential map identity

I have come across this identity a few times and I have absolutely no idea why it holds. $g^{-1}\exp(tX)g=\exp(t(\text{ad}_{g^{-1}}X))$ Would any one be able to explain exactly why this holds or ...
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1answer
31 views

Step in a proof about alternating operators

The theorem is that if $f$ is a $k$-linear function on a vector space $V$, then the $k$-linear function $Af$ is alternating. $\def\sgn{\operatorname{sgn}}Af=\sum (\sgn \sigma)\sigma f$ Proof: ...
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1answer
32 views

Definition of connection on vector bundle

A connection on a vector bundle $E$ is a map $ D:\Gamma(E)\rightarrow \Gamma(T^*(M)\otimes E)$ satisfying 1) For any $s_1,s_2\in \Gamma(E)$, $D(s_1+s_2)=Ds_1+Ds_2$ 2) For $s\in \Gamma(E)$ and ...
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0answers
18 views

Inverse Mapping Theorem and open mapping

Apparently, one consequence of the Inverse Mapping Theorem is that if $f: U \rightarrow \mathbb R^n$ is a continuously differentiable mapping defined in an open subset $U \subset \mathbb R^n$ such ...
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0answers
31 views

Fibers of polynomial as submanifold

Let $f\in \mathbb{R}[X_1,...,X_n]$ be a homogeneous polynomial of degree $d$. Let $F_{a}=\{(x_1,...,x_n)\in\mathbb{R}^n : f(x_1,...,x_n)=a\}$. For which $a$ $F_a$ is a submanifold in $\mathbb{R}^n$ ? ...
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1answer
55 views

Tangent cone and tangent space

Let $M$ be a set in $\mathbb{R}^n$ such that $M$ is locally a graph of some differentiable function (not necessarily $C^1$ ). Let $p\in \bar M$. We can define a tangent cone $C(M,p)$ as a set of all ...
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1answer
36 views

What are properties of dynamical systems in non-integer dimension spaces?

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc. A 3D dynamical system ...
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33 views

A question on Grassmannian manifolds and universal line bundles

Problem. Fix an $ n \in \mathbb{N} $. Is it true that there exists a $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the pullback of ...
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1answer
74 views

Proving existence of local diffeomorphism

Consider the setup from here: Do these vector fields span an integrable distribution? For any pair of points $p, q \in U$, show that there is a local diffeomorphism $F: U(p) \to U(q)$, such that ...
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1answer
73 views

Elementary tensors [duplicate]

I need to determine whether the following function is tensor on $\Bbb R^4$ and express it in terms of elementary tensors. Can someone please help me with it? I do not know what elementary tensor means ...
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0answers
18 views

Hausdorff Dimension of a k-dimensional submanifold, hence $L^n(M) = 0$

If $M \in P(R^n)$ is a k-dimensional submanifold, show that the Hausdorff-Dimension of M is k. Also, is there a quicker/easier way to show, that $L^n(M) = 0$, where L is the Lebesgue-Measurement.
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1answer
30 views

Is the unit square a submanifold/manifold?

In my course we have just been introduced to and will only be dealing with regularly embedded submanifolds. Let $M = [0,1]\times [0,1] \subset \mathbb{R}^2$. I don't think it's a submanifold. If it ...
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1answer
23 views

Construct a diffeomorphism $[a,b] \rightarrow [c,d]$ with slope 1 at $a$ and $b$

I'm trying a problem from An Introduction to Chaotic Dynamical Systems regarding bump functions. At this point, we have successfully constructed, for any $\alpha < a < b < \beta$ a bump ...
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2answers
63 views

A reference for a result by A. Casson

I was reading this article about the disproof of Triangulation conjecture: it says that A. Casson disproved this conjecture in dimension 4 in the '80s In 1982, Michael Freedman, then at the ...
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1answer
53 views

$M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Rightarrow C \neq 0$

I'm having some troubles on showing that $M=\{(x,y,z) \in \mathbb{R}^3 : xyz=C\}$ is a manifold $\Leftrightarrow C \neq 0$ I have already proved $\Leftarrow$ but I can't see how to prove ...
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0answers
66 views

Homotopically equivalent manifolds and product with $\mathbb{R}$

I know that some manifolds which are homotopically equivalent become homeomorphic after taking the product with $\mathbb{R}$, e.g. $\mathbb{T}^{2}$ minus a point and $\mathbb{S}^{2}$ minus three ...
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2answers
59 views

Can manifolds be defined over fields other than $\mathbb{R}$ and $\mathbb{C}$?

The question is one in the title. By other fields, I mean fields like $\mathbb{Q}$. But I have never seen manifolds defined over fields other than $\mathbb{R}$ and $\mathbb{C}$. Even if one defines, ...
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1answer
46 views

Showing that $f: \mathbb{R}^2 \to \mathbb{R}^n$ can't be a homeomorphism for $n>2$

Without using the Invariance of Domain result, I want to show that such an f cannot exist. Here is what I did: Assume that there is an $f: \mathbb{R}^2 \to \mathbb{R}^n \ , n>2$ that is ...
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1answer
28 views

Derivative of a function - how to compute for those examples

I'm taking a Diferential Manifolds course and I don't understand how to compute $DF_a$ in order to apply the following theorem: Let $F:U \rightarrow \mathbb{R}^m$ be a $C^\infty$ function on an ...
2
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1answer
21 views

matrix Lie group embedding as a manifold

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into ...
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1answer
29 views

Show that the covering space of a smooth manifold is a smooth manifold.

I indeed found this question Is a covering space of a manifold always a manifold. However I do not know the concepts here used. As far as I know I just need to present a suitable atlas for the ...
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38 views

A question on diffeomorphisms and their relation to active coordinate transformations

I've been reading Sean Carroll's notes on General Relativity, http://arxiv.org/pdf/gr-qc/9712019.pdf . I've got to chapter 5 (page 133) and am reading the section on diffeomorphisms in which Sean ...
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36 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
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1answer
35 views

Half-space is not a manifold

We define the half-space $H^n$ as the set containing all tupels $(a_1,\ldots,a_n)$ such that all $a_i\geq 0$. I know that this isn't a manifold - intuitively this is clear - but how can I formally ...
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1answer
38 views

Working with smooth functions defined on a manifold

I am having trouble working with smooth functions defined on a manifold. Is the following line of reasoning valid or not? Let $f$ be a smooth function defined on a manifold $M$ with a local maximum ...
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0answers
18 views

Geometric of 3-manifold

I have the 3-manifold $X$ $X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $ What is the geometric structure of $X$ from the 8 Thurston Geometries any help ...
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1answer
54 views

Möbius band inside projective plane

How can I see inside the projective plane the Möbius band? I need to know how the Möbius Band appears inside the projective plane. I know it is easy using identifications and algebraic topology. ...
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0answers
40 views

Lie algebras of GL(n,R) and differentials

This question comes from a proof in John Lee's Introduction to Smooth Manifolds, page 194. I am questioning a line in the proof of the following proposition: The composition of the maps ...
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2answers
88 views

What is manifold in Geometry?

What is manifold in geometry? WE always use this word like non-manifold geometry but I was wondering what is manifold in the first place. I got some definition online but couldn't understand. A ...
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27 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
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27 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
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1answer
30 views

Prove that the set of all vector fields $V(S^1)$ is a free $C^{\infty}(S^1)$-module

I need to prove that the set of all vector fields, $X:S^1\to TS^1$ name it: $V(S^1)$, is a free $C^{\infty}(S^1)$-module. So i need a basis $\frac {d}{dx_1},...,\frac {d}{dx_n}$ for $V(S^1)$.It's easy ...
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0answers
56 views

Thurston's Geometric structure for 3-manifold

I have an orientable 3-manifold $X$ , such that $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ I would like to know the geometric structure on X. My ...
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0answers
31 views

How can we calculate $Xf$?

Let $X$ be the vector field $x \dfrac{∂}{∂x} + y \dfrac{∂}{∂y}$ and $f (x, y, z)$ the function $x^2 + y^2 + z^2$ on $R^3$. Compute $Xf$. Could you give me some hints how we can calculate it?
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1answer
22 views

Outward pointing normal vector to a $k$ manifold in $\mathbb{R}^n $

What do we mean when we say that $\mathbf{N}$ is normal to the the manifold:$$\mathbf{X}:\mathbb{R}^k\to\mathbb{R}^n$$? How do we determine it? How to verify that it's an outward pointing normal?
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3answers
48 views

Length of Circle on a Sphere

I want to calculate the length of the circle $\theta=\pi/4$, where $\theta$ is the latitude, on the unit sphere. I know that the length of a curve $\gamma (t), t \in [0,T]$ on a manifold is given by ...
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0answers
16 views

The orientation of a parametrized $k$-manifold

Can someone explain me what do we mean by "orientation of a $k$-form manifold? Is this definition consistent with the earlier definitions of orientations of curves and surfaces?
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35 views

Centroid of manifold

The centroid of a compact manifold is defined by the following equation: $c(Y_a)$ is the centroid of the parametrized manifold $Y_a$ is the point in $\Bbb R^n$ whose $i^{th}$ coordinate is given by ...
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0answers
32 views

Relative singular homology $H(M,\partial M)$ for a manifold $M$?

Let $M$ be an orientable manifold. What can be said about the relative homology $H(M,\partial M)$? Perhaps one can calculate the homology using excision?
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1answer
14 views

Geometric interpretation of the evaluation of Poincaré dual with a fundamental class

Given oriented, closed submanifolds $X^k$ and $Y^{n-k}$ in an oriented, closed $n$-manifold, is there a nice geometric interpretation of the evaluation $\langle \operatorname{PD}([X]),[Y]\rangle$? I ...
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1answer
57 views

connected manifolds are path connected

prove every connected manifold is path connected manifold . my thought: connected space : Let $ X$ be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint nonempty ...
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35 views

How to find the tangent space of $O(n)$ by considering $O(n)$ as the pre-image of the map $A \mapsto AA^T$ at identity?

Why is the tangent space of $O(n)$ at $H$ equal to $T_H O(n) = \{ M \in \mathbb{M} ( n, \mathbb{R} ): (DF(H))(M) = 0 \}$, where $$F: \mathbb{M} ( n, \mathbb{R} ) \cong \mathbb{R}^{n^2} \to ...