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Questions tagged [manifolds]

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

In mathematics, a manifold of dimension $n$ is a topological space that near each point resembles $n$-dimensional Euclidean space. More precisely, each point of an $n$-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension $n$. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.

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Hodge dual definition

I'm reading Christodoulou and Klainerman's book on the stability of Minkowski spacetime and came upon the definition of the Hodge dual on a surface $S$: for a 2-form $\xi$, for instance, $$ *\xi_{ab} = \epsilon_{ac}\xi^c_b. $$ But I've usually seen…
Chris
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Introduction to Manifolds, Loring W. Tu page 225

$$\ ϕ_{t*}\biggl(\frac∂ {∂x^j}\bigg|_ p \biggr) = \sum_i \frac{∂ϕ^i} {∂x^j} (t, p) \frac∂ {∂x^i}\bigg|_{ ϕ_t(p)}$$ Thus, if $\ Y = ∑b^j\frac∂{∂x^j}$ , then $$\ ϕ_{−t∗} Y_{ϕ_t(p)} = \sum_j b_j (ϕ(t,…
folo polo
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Coefficients Relative to a Smooth Frame

An exercise from Loring Tu's textbook asks the following question: Let $\pi:E\to M$ be a $C^\infty$ vector bundle and $s_1,\ldots,s_r$ a $C^\infty$ frame for $E$ over an open set $U$ in $M$. Then every $e\in\pi^{-1}(U)$ can be written uniquely as a…
Clayton
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Can an $n$-dimensional manifold live inside $\mathbb{R}^m$ for $m < n$?

This may be a silly question but I want to make sure my general intuition is correct. I don't think it is possible for an $n$-dimensional manifold to live inside $\mathbb{R}^m$ for $m < n$ because in the "best case scenario" we can move in all…
green frog
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Finite partition of a manifold into manifolds

Let $X$ be a topological $n$-manifold and $C \subset \mathcal{P}(X)$ a finite partition of $X$ into topological manifolds (with subspace topologies). Does there exist at least one $M \in C$ such that $\dim(M) = n$?
kaba
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Manifold: Why a sphere requires two $2$-dimensional coordinate systems to fully describe it, not just one?

I am reading "Information Geometry and its Application" by Shun-ichi Amari. The example of a sphere as a 2-dimensional manifold says that, and I quote: A sphere is the surface of a three-dimensional ball. The surface of the earth is regarded as a…
TNg
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Exercise about smooth maps in Lee

I'm looking at Exercise 2.6b (p.58), which is to prove that for smooth manifolds $M$ and $N$ and a continuous map $F:M\rightarrow N$, we have that $F$ is smooth iff $F^*(C^\infty(N))\subseteq C^\infty(M)$, where $F^*(f)=f\circ F$ is the pullback by…
Zev Chonoles
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Is there any manifold that is not a subspace of a finite dimensional euclidean space?

I mean, is there any topological space that is locally euclidean, Haudorff and second countable and can't be embedded into a finite dimensional Euclidean space. I think it's hard for me to find such spaces because manifolds are often described…
Lee Dae Seok
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How about the converse of the property that a product of manifolds is a manifold?

We know that the Cartesian product of two manifolds is a manifold, but is the converse true? Let us assume that we have $A$ and $B$ two second countable Hausdorff topological spaces, and $M = A \times B$. If we assume that $M$ is a $n$-manifold,…
Nicolas Boutry
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Surface of the cube, differentiable manifold

Provide the structure of a differentiable manifold on the surface of the cube: $\{x\in\mathbb{R}^{n+1}\colon \max\{|x_1|,\dotso,|x_{n+1}|\}=1\}$ Hello, I have a problem with this task. I do not know how to solve this task and I am not able to come…
Cornman
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Concept of Manifold

The concept of manifolds is freaking me out. For me it seems like a manifold is just a subspace embedded in a higher dimension. In order to clear out my confuision I have created a list and I would be glad if someone could tell me if my guesses are…
MrYouMath
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Prove that $T_pM^*=\mathfrak m_p/\mathfrak m_p^2$.

Let $M$ a manifold of dimension $n$ and $T_pM$ its tangent space. Let denote $T_p^*M=\mathcal L(T_pM,\mathbb R)$ it's dual. Let also denote $$\mathfrak m_p=\{f\in\mathcal C^\infty (M)\mid f(p)=0\}\quad \text{and}\quad \mathfrak…
MSE
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An orientable manifold of codimension 1 is the zero set of a differentiable function

I want to solve the following exercise from M. Spivak's Calculus on Manifolds: If $M \subseteq \mathbb{R}^n$ is an orientable $(n-1)$-dimensional manifold, show that there is an open set $A \subseteq \mathbb{R}^n$ and a differentiable $g:A \to…
user1337
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rank of function on connected manifold

Let $X$ be a connected $n$-dimensional manifold and $f:X\to X$ a differentiable function satisfying: $f\circ f =f$. Prove that for all $p\in X$ that $rk_pf\leq rk_{f(p)}f$ and subsequently that $rk \;f$ is constant along $f[X]$. Can anyone give me…
user178468
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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), if we can "walk around" at any point on an…
Jiang
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