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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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Manifold definition clarification

When dealing with topological manifolds, the definition specifies that for every point in the manifold, there should exist at least one neighborhood around that point that is homeomorphic to an open subset of Euclidean space. However, does this…
Ajay Varghese
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Darboux Theorem for closed 2-forms of constant rank

I'm looking for a proof to this theorem stated in Abraham-Marsden book, foundation of mechanics. Let $\omega$ be a closed 2-form such that $\hat{\omega}:TM\to T^{*}M$, given by $\hat{\omega}(u)(v)=\omega(u,v)$, has constant rank. Then, there exist…
no name
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Manifold based on Grassmann numbers?

A manifold is locally homoeomorphic to Euclidean space. Is there a notion of a manifold that has Grassmann coordinates instead of Euclidean coordinates?
dennis
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Tu's An Introduction to Manifolds Applying the Lemma Repeatedly Confusion Page Seven

On page 7 of Loring Tu's book called "An Introduction to Manifolds", he makes a jump and a probable mistake. It pertains to his proof of Taylor's remainder Theorem. https://www.math.toronto.edu/~jeffrey/matd67/tu.pdf Tu states "Applying the lemma…
Jason Broadway
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Concept map for manifolds

What exactly is manifold? What concepts do I need to learn in order to take on manifolds and concepts related to it?
danny gotze
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Question about the pull-back of vector fields

I'm working on differential manifolds, and at a moment in my lesson there is a pull-back of a vector field on an open sets of ${\mathbb R}^n$ to a differential manifold $M$ by a function from charts of $M$. Do that make any sence ? Because this…
Johny06
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Do any subset of a differential manifold can be a differential manifold?

I know that all subsets $S$ of a manifold $M$ are not a submanifold of $M$, but can it always be given a structure of differencial manifold ? I feel like yes, taking the induced topology and the restriction to $S$ of charts from $M$. Am I right ?
Johny06
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How to define an open subset containing a boundary point of a manifold

According to the definition of a manifold with boundary, there are two types of coordinate charts of a $n$-dimensional manifold with boundary: interior charts and boundary charts. A boundary chart is a homeomorphism $\phi: U \rightarrow V$ where $U$…
Ka Fat Chow
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$\int_M\omega $ integral on a manifold with an atlas with two charts.

Let $M$ a manifold with Atlas $\{(U,\varphi ),(V,\psi)\}$ where $U\cap V\neq \emptyset$. If $A\subset U$, then for $f:M\to \mathbb R$, $$\int_A f:=\int_{\varphi (A)}f(\varphi ^{-1}(x))\,\mathrm d x.$$ But what happen if we integrate over $M$ ?…
joshua
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Prove 2 manifolds are homeomorphic iff they are diffeomorphic

Suppose $M_1$ and $M_2 $ each $C^{\infty}$ manifolds and admits an atlas with once chart. Prove $M_1$ and $M_2$ are homeomorphic iff they are $C^\infty$ diffeomorphic. One direction is obvious. But given that those manifolds are homeomorphic I'm…
DirichletIsaPartyPooper
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Metrizable manifolds without paracompactness?

I'm studying manifolds (by myself) and I can't understand why in order for a manifold to be metrizable, it needs to be paracompact (that's what I have read). The definition of a manifold is: Manifold is a topological space which is Hausdorff, second…
mxaxc
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Higher dimensional Manifolds

I know these manifolds can be expressed as constraint equations. For example if we consider a sphere embedded in a 3D Euclidean space, the sphere can actually be expressed in a set of intrinsic coordinates restricted to the manifold.For example we…
Jasmine
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Issues showing that X is a manifold.

Show that $ X = \{ (x,y) \in \mathbb{R}^2 : (x^2 +xy + y^2 -1)(x^2 + y^2 -8) = 0 \}$ is a non connected manifold of dimension $1$ in $\mathbb{R}^2$ First I can say that after I have shown that it is a manifold of dimension $1$, it is clear that it…
vitalmath
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Is the outward pointing normal always consistent with a orientation preserving coordinate system?

In Folland's Advanced Calculus, when discussing Stokes' Theorem, the surfaces are oriented by a choice of normal (either inwards or outwards). Visually, it's easy to see which direction is inwards and outwards, especially for simple shapes such as…
Snowball
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Invertibility of the Derivative of a Invertible Function

In Spivak's Calculus, when discussing manifolds with boundary, and trying to make a point that a point cannot be both on the boundary and not on the boundary on page 113 Spivak notes: Since $\text{det} (h_2 \circ h_1^{-1})' \neq 0$, this…
Snowball
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