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

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Vectors in tangent space to a manifold independent of coordinates

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator $X$ acting on some function ...
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1answer
81 views

Is every hypersurface in $\mathbb{R}^n$ the boundary of an open domain?

We know if $\Omega \subset \mathbb{R}^{n}$ is a bounded $C^k$ domain, then its boundary $\partial\Omega$ is a $C^k$ compact hypersurface of dimension $n-1$. Is it true that every $m-$dimensional ...
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1answer
79 views

Why klein Bottle is 4-D?

I am wondering that Klein Bottle is 4-D. Can any body tell me how it is possible? I can give coordinates for each point of the Klein Bottle with 3 values. Then how it can be 4-D? What is immersion? ...
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1answer
56 views

Variation on Stokes Theorem for Manifolds (2)

Let $\omega \in \Omega^0(\mathbb{R}^{2}\setminus\{0\})$ be a $0$-form such that $d\omega=0$. Is the following statement true: For any compact, oriented, $0$-dimensional submanifold $M$ of ...
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1answer
58 views

Bundle metric and connection on trivial vector bundle

I read this: Let $(M,g)$ be a compact Riemannian manifold and let $W$ be a vector bundle (rank $n$) over $M$ with $h_W$ a bundle metric of $W$ and $D$ a bundle connection of $W$. I choose $W$ ...
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30 views

Differentiability of a function on a manifold is independent of the coordinate chart

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a ...
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17 views

What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap ...
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32 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
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57 views

If M is a manifold of dimension $ n \neq0$ then M has no isolated points.

I am in doubt whether the following statement is true or false: "If M is a manifold of dimension $ n \neq0$ then M has no isolated points." The idea that made me find the true statement was as ...
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2answers
66 views

Variation on Stokes Theorem for Manifolds

Let $n >1$ and $\omega \in \Omega^{n-1}(\mathbb{R}^{n+1}\setminus\{0\})$ such that $d\omega = 0$. Is the following statement true: For any compact, oriented, $(n-1)$-dimensional submanifold $M$ ...
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38 views

Existence of a fixed-point free map in a manifold.

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map. I know ...
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1answer
44 views

Is the Riemannian distance function Lipschitz on a hypersurface?

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$ of dimenion $n$. Is it true that there exists a constant $C$ such that $$d(x,y) \leq C|x-y|$$ for all $x, y \in M$? Here $d$ is the Riemannian ...
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1answer
30 views

What is $T^0_0(M,W)$ where $W$ is trivial vector bundle over a compact manifold $M$?

Let $W=(M \times \mathbb{R}, pr, M)$ be the trivial vector bundle over a compact manifold $M$, and define $$V=T^0_0(M,W) := T^0_0M \otimes W,$$ and $V$ is called "the vector bundle of $W$-valued ...
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27 views

Time Derivative of the integral over a singular k-cube

I am stuck on this question, and was wondering if someone could provide a hint of where to start? I can't see the first step.
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1answer
75 views

Does the Tangent Space Vary Continuously with The Points On a Manifold?

I recently read about Grassmannian manifolds. The following question naturally comes to mind. Let $GR_k(\mathbf R^n)$ is the grassmannian manifold of $k$ dimensional linear subspaces of $\mathbf ...
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2answers
101 views

How to visualize the gradient as a one-form?

I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I still visualize gradients as vector fields instead of the ...
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62 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
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1answer
43 views

Why is the image of a smooth embedding f: N \rightarrow M an embedded submanifold?

I'm reading An Introduction to Manifolds (Tu) and got confused on p.123 Theorem 11.13. Let me briefly explain what was done before that. The author defines an embedding between two manifolds $f: ...
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1answer
32 views

Negative Gauss Curvature

Let S be a manifold of dimension 2, compact and orientable. Suppose its border is made of k geodesic circumferences, with $k \geq 3$. Show that there exists a point in S with negative Gauss ...
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1answer
38 views

Find an explicit atlas for this submanifold of $\mathbb{R}^4$

I'm having a hard time coming up with atlases for manifolds. I can prove using the implicit function theorem that $M = \{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=x_3^2+x_4^2=1\}$ is a ...
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2answers
46 views

Give an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.

I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping. Thanks.
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59 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
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1answer
22 views

What is the significance of incompatible coordinate charts for a manifold?

For reference, here is my definition of a "manifold". A $\,C^\infty$ manifold is a topological manifold together with all the admissible charts of some $C^\infty$ atlas. When considering the ...
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1answer
45 views

Locally exact vs globally exact

Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots ...
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2answers
55 views

Given a nowhere zero vector field $Z$, does there exist a one-form $\gamma$ such that $\gamma(Z) = 1$?

Take $M$ a smooth manifold, and $Z$ a vector field on $M$ such that $Z(p)\neq0$ for all $p\in M$. Is there a one form $\gamma \in \Omega^1(M)$ such that $\gamma(Z)=1$? I started to work locally, but ...
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0answers
48 views

Riemannian Connection

How can we see for the Riemannian connection, connection 1-form with its first index lowered $\omega_{ab}=\delta_{ac}{\omega^c}_b$ is antisymmetric in a, b, i.e. $\omega_{ab}=-\omega_{ba}$. Thanks.
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1answer
67 views

Sum of wedge products with one forms equals 0

Let $M$ be a smooth manifold and $n \leq dim(M)$. Let $\omega_1,...,\omega_n$ be 1-forms on $M$, such that for every $q \in M$ their evaluations $(\omega_1)_q,...,(\omega_n)_q$ at $q$ are linearly ...
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34 views

Sard's Theorem with different Measures

From what I can tell Sard's theorem is formulated in terms of the Lebesgue measure. Is there a form of Sard's theorem for more general measures (in particular, those which are not absolutely ...
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27 views

$k$-jets , submanifolds

I am trying to solve this problem: Let $J^k (n,p)$ be the set of maps $\sigma:\mathbb{R}^n\rightarrow\mathbb{R}^p$, where $\sigma_i$ (the coordinate functions) is a polynomial of degree $\leq k$ with ...
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77 views

Manifolds, isolated singularities, open map

I can't figure out how to solve the following question: 1) Let $M$ and $N$ be $C^k$-manifolds, such that $\dim M = \dim N = n>1$, and let $f:M\rightarrow N$ be a $C^k$-function. Show that if the ...
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56 views

The induced map on the de Rahm cohomology of a surjective submersion.

Let $M,N$ be two smooth manifolds and $f: M \rightarrow N$ a surjective submersion (so $f$ and $f_*$ both surjective everwhere). It is straightforward to show that then the pullback of $k$-forms: ...
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1answer
42 views

Fiber bundles and manifolds

Each vector bundle is an example of a fibre bundle with some extra structure. This extra structure provides the algebraic object consisting of all sections (continuos or smooth) of given bundle. When ...
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1answer
52 views

Orthornormal basis and Dual basis

If $e_a$ is an orthonormal basis for vectors and $\theta^a$ the dual basis for coordinate vectors. How to prove that metric is expressed as $ds^2=\delta_{ab} \theta^a \theta^b$ and ...
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20 views

Characteristic parts of a submanifold to a differential operator

Given the curve $\gamma(t):=(\cos(t),\sin(t))$ for $t\in(0,\pi)$, the differential operator $L(u):=uu_{xy}+0.5u_yu_{yy}+u^2$ and some prescribed Cauchy data $u|_\Gamma=z|_\Gamma$ and $\partial_n ...
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1answer
16 views

Is $(-\infty,0)\times S$ for a compact closed manifold $S$ a “manifold with boundary and cylindrical ends”?

I read the following definition from this paper. Definition: Let $N$ be a Riemannian manifold with boundary $\partial N$. We say $N$ is a manifold with boundary and cylindrical ends if there ...
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1answer
31 views

Confused over k-chains and their boundaries.

I am writing a short report on de Rham cohomology, and I'm approaching it from a geometric perspective, much like (and with reference to) this article (written by a MSX member) ...
4
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1answer
170 views

Parametrised vs Regular Surfaces

Two types of surfaces in $\mathbb{R}^3$ are usually studied in introductory books on differential geometry: Parametrised or immersed surface: Is an immersion $F:U\rightarrow\mathbb{R}^3$ from an ...
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27 views

Principal orbit type

I have trouble understanding the proof of Proposition 1.2.5 on p.17 in Audin's Torus Actions on Symplectic Manifolds: Let $G\curvearrowright M$ be a smooth action of a compact Lie group $G$ on a ...
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52 views

Local expression of a differential form

During the course, we have defined differential forms as maps $$ \omega : D(M) \times \cdots \times D(M) \rightarrow \mathcal{C}^\infty(M), $$ where $D(M)$ are the global derivates of the manifold ...
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1answer
125 views

Spivak Calculus on Manifolds, Theorem 5-2

In the proof Theorem 5-2 of Spivak Calculus on Mannifolds how is \begin{align*} V_2\cap M=\{f(a):(a,0)\in V_1\}? \end{align*} (That $\{f(a):(a,0)\in V_1\}=\{g(a,0):(a,0)\in V_1\}$ is clear.) Edit: ...
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4answers
198 views

Tangent vectors: arrows vs. derivatives

I have a very hard time accepting the differential-geometric definition of a vector as a derivative operator, $$v = v^{\mu} \partial_{\mu}.$$ I want to make sure that the following line of reasoning ...
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1answer
56 views

Covering space of an orientable manifold

I want to show that every covering space of an orientable manifold is an orientable manifold. My definition of orientability is throw homology. It's a new notion for me, I need help... Thank you.
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2answers
69 views

Equivalent definitions of a surface

do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post. Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or ...
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Opens in a manifold

I am studying for my exam of manifolds and I dno't understand why the following is true. Let $(U,\phi)$ be a local map at $p$ in the manifold $M$, and let $\phi(p)=0$. Then there exists an open set ...
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2answers
70 views

Isomorphism of $(\mathbb{R^3},\times)$

I am studying lie algebras and I have difficulties doing this exercise: Show that $\mathbb{R^3}$ with vector product is isomorphic to the space of antisymmetric 3x3 matrices with operation ...
2
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2answers
58 views

Closure of an open set in manifold

I have a question. During a proof of a proposition, the following is stated: Let $K$ be a compact set in a manifold $M$ of dimension $n$. Then there exists an open set $U$ such that $K\subset U$ and ...
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2answers
64 views

Manifold with $\pi_1(M)=F_n$

We may construct a 3-manifold $M_n$ with $\pi_1(M_n)\cong F_n$ (i.e. the free group on $n$ generators) as follows: consider the complement of $n$ pairs of open 3-balls in $\mathbb{R}^3$. For each ...
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1answer
99 views

Proving a nowhere vanishing vector field on 2D manifold implies $TU\cong M\times S^1$

So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is ...
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1answer
51 views

Is $M \times (0,\infty)$ a manifold of bounded geometry?

If $M$ is a compact Riemannian manifold, is $M \times (0,\infty)$ a manifold of bounded geometry? I think it is, since $M$ is compact and $(0,\infty)$ is simply flat.
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Solving system of first-order PDEs with Frobenius theorem

I've been stuck trying to solve this system: $$\ \frac{\partial u}{\partial x} = \frac{-2xy^2}{u} + 3y $$ $$\ \frac{\partial u}{\partial y} = \frac{-2x^2y}{u} + 3x $$ Which must satisfy $ \ u(0,0) = ...