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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Smooth function on product manifolds

suppose $M, N$ are smooth manifolds and $f:M\times N\rightarrow \mathbb{ R }$ is a function such that $\forall y\in N$ the function $x\mapsto f (x, y)$ is smooth and $\forall x\in M$ the function $y\mapsto f (x, y)$ is smooth as well. Does this…
Braten
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Exponential maps of $\mathbb{R}^n$ and $T^n$

This is a question from Lee's book: The definition of the exponential map in terms of the one parameter subgroups seemed quite abstract, so I think that fully understanding this exercise will be helpful. As far as I understand, if we find the one…
Soap
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Examples of higher dimensional TQFTs

1-dimensional TQFT's assign to every 1-manifold (disjoint union of circles) a vector space and to every surface a linear map between the vector spaces that correspond to the boundary manifolds. So this is a functor from the cobordism category of…
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Find the linear tangent map $T_{I_n} \ f$

I want to find $T_{I_n} \ f$ where the map is $\ f : X \mapsto X^2$ with $X\in SL_n(\mathbb{R})$. By definition of the linear tangent map : $T_{I_n}\ f : T_{I_n}SL_n (\mathbb{R})\to T_{f(I_n)}SL_n(\mathbb{R})$. But here notice that…
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Necessity of closedness in going from precompactness in chart to precompactness in the manifold

I have a doubt about the last part of the proof of Lemma 1.10 in Introduction to smooth manifolds by John Lee, so I will put just the last part of the proof. Lemma 1.10 Every topological manifold has a countable basis of precompact coordinate…
George
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Resolving singularities in elliptically fibered fourfolds

Consider a Calabi-Yau fourfold given as an elliptic fibration $\pi : X_4 \to B_3$ over a complex threefold. The discriminant locus $\{\Delta=0\}$ describes the locus in $B_3$ over which the elliptic curve degenerates. We can (usually, though perhaps…
diracula
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Why is a discrete subset of a compact space finite?

I have a question about the proof of Lemma 5.13 in John Lee's text, the proof is shown below. My question is about the phrase underlined in red, why is a discrete subset of the compact set finite? Now, the author never gives a definition for…
Khoa ta
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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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Why the product of two manifolds is paracompact?

Some authors define a manifold as a paracompact Hausdorff space that is locally Euclidean. Also it is said that a product of two manifolds is a manifold. However, we know that product of a two paracompact spaces is not necessarily paracompact. So…
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Prove that the unit circle, $S^1:=\{(x,y)\in \mathbb{R}^2: x^2+y^2=1\}$, Is a one-dimensional manifold

Prove that the unit circle, $$S^1:=\{(x,y)\in \mathbb{R}^2: x^2+y^2=1\}$$, Is a one-dimensional manifold in $\mathbb{R}^2$ by using the following coordinate chars, known as stereographic projections, and completing the following steps. a)Show that…
combo student
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A little problem on Lie Brackets relating it with the commutator of matrices

I am trying to solve this problem: There is a hint saying that I can use the fact that for $$X=a_i\frac{\partial }{\partial x_i}\text{ with } a_i\in C^\infty(U)$$ and $$Y=b_i\frac{\partial }{\partial x_i}\text{ with } b_i\in C^\infty(U)$$ we…
Soap
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Why is a Dirac operator involutive only if the curvature and torsion are in the image of the kernel of the symbol under exterior multiplication?

I am trying to understand the classic paper by Atiyah, Hitchin, and Singer https://www.jstor.org/stable/79638 and I'm getting stuck on part of the proof of proposition 3.1. The proposition is determining the conditions for involutivity of the…
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What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are equivalent which I assume means that if $K$ is a knot in the…
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On Kervaire and Milnor's paper, "Groups of homotopy spheres: 1" (surgery on 3-manifolds)

Kervaire and Milnor's paper "Groups of homotopy spheres: 1" in Annals, is the condition "$k>1$" necessary in Lemma 6.3? If yes, why? (I couldn't find any part requiring the condition in their progress to obtain the lemma.) Lemma 6.3 is the…
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if the atlas of a topological manifold contain only one chart

if I have an atlas $\mathcal{A}$ for a topological $m$-manifold $\mathcal{M}$ which consists of only one chart; a homeomorphism $\phi:U \subseteq \mathbb{R}^m \to \mathcal{M}$. is it then true that $\mathcal{M}$ is necessarily smooth? I think so,…
QuantumEyedea
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