# Tagged Questions

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

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### Why $I = [0,1]$ is a $1$-manifold and $I^2$ not?

I am stuck in this, I have no idea why! $[0,1]$ is a manifold with boundary, how to justify? Which are the charts? And how about $[0,1]^2?$ Why it is not a manifold? My definition of topological ...
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### Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
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### Calculate the area of $\partial G$ of $G = (z > x^2 + y^2$ and $x^2 + y^2 + z^2 < z)$ in $\mathbb{R}^3$.

Calculate the area of $\partial G$ of $G = (z > x^2 + y^2$ and $x^2 + y^2 + z^2 < z)$ in $\mathbb{R}^3$. I'm not really sure how to approach this. I've tried using spherical coordinates but I ...
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### Connectedness and dimension of a manifold

Let $S=\{(x,a_3 , a_2, a_1 , a_0) \in \mathbb R^5 : x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 =0\}$ I want to show that $S$ is a connected manifold, and find the dimension of $S$. It seems that each $x$ ...
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### Applications of Banach's fixed point theorem on Differential Geometry

Does anyone know any simple application of Banach's fixed point theorem on Differential Geometry. I am looking for something involving manifolds.f
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### Simple properties of wedge product [closed]

How to prove a) $\omega \wedge \eta =(-1)^{kl}\eta\wedge\omega, \omega$ is $k$-tensor and $\eta$ is $l$-tensor. b)$f^*(\omega \wedge \eta)=f^*(\omega)\wedge f^*(\eta)$ where $f:V\rightarrow W$ ...
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### Pullback of the metric on $\mathbb S^n$ on $\mathbb R^n$.

Let $\varphi:\mathbb R^n\longrightarrow \mathbb S^n$ the inverse of the stereographic projection, i.e. $$\varphi(y)=\left(\frac{2y}{\|y\|^2+1},\frac{\|y\|^2-1}{\|y\|^2+1}\right).$$ What I'm trying to ...
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### Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...
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### Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold.

Prove: The boundary of the set $\{(x,y,z) \in \mathbb{R}^3 | z > \sqrt{x^2+y^2}\}$ isn't a smooth manifold. The boundary is defined by $z = \sqrt{x^2+y^2}$. I'm trying to think how to approach ...
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### Definition of a cubic coordinate system

I'm looking at "Foundations of Differentiable Manifolds" by Frank Warner, and have a question about one of the basic definitions at the beginning of the book. He writes: A coordinate system $(U,\phi)$...
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### Compact cohomology group of connected n-dimensional connected oriented manifold

I know how to show $H_c^n(M)\simeq\mathbb{R}$, where M is a oriented connected n-dimensional manifold, by showing the integration map is isomorphism. However, I found in the book that this is a ...
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### Hausdorffness of the Attaching two manifolds along their boundaries

In professor John Lee's Introduction to Topological Manifolds, there is a proof(page 75- page 76 of 2nd edition) about the attaching two manifolds along their boundaries. But the hints to the proof of ...
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### Identification of the holomorphic tangent space with the real tangent space

I'm reading Griffiths and Harris and just want to check I'm interpreting a passage correctly. Let $M$ be a $n$ dimensional complex manifold. We define $T_p^\mathbb{R}M$ as the tangent space of the ...
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### Lagrangian Multipliers exercise

Let $M = \{(x, y, z) \in {\rm I\!R}^3 : F(x,y,z) = 0\}$ and let $F(x,y,z) = (3x^2z + y^2 + z^3-1, \, x + z-1)$ . Does the function $f(x, y, z) = x$ have any extrema in $M$? We are asked in advance ...
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### Integrate $\int_{\partial G}(x,y,z) \times N dS$ for $G \subset \mathbb{R}^3$ with a smooth, regular boundary.

Integrate $\int_{\partial G}(x,y,z) \times N dS$ for $G \subset \mathbb{R}^3$ with a smooth, regular boundary. $N$ the outward-pointing normal to $\partial G$, and the integral is evaluated per ...
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### What is the topological degree of the constant map?

What is the topological degree of the constant map? To me it does not make any sense, once $f$ being the constant map has no regular values. So, how to proceed?
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### constructing a manifold structure for a plane in $\mathbb{R}^3$ [closed]

Any help on this problem would be greatly appreciated. thanks! Let M be the plane in $\mathbb{R}^3$ with normal vector (a,b,c)$\neq$0. Construct a manifold structure each topological space (M,$\tau$) ...
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### What is the tangent space o SO(n) [closed]

It should be the kernel of the map $H\mapsto A^TH+H^TA$ at some $A$ such that $A^TA=I$ . But I cant find this Kernel can someone help me?
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### Real Analitic Manifolds, Tubular Neighborhood, Radius of Convergence

Given a Real Analytic Manifold isometrically embedded into an Euclidean Space. Gicven the maximum value of the radius of a Tubular Neighborhood "around" the manifold: what relation does it have with ...
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I would like to ask the following: How does the Injectivity Radius relate to the Radius of Convergence (of the analytic function to its power series) of any local (parametrization) map in the Real-...
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### Determining a derivation on the unit sphere of the $\mathbb{R}^3$

Let $S^2 \subseteq \mathbb{R}^3$ be the unit sphere in $\mathbb{R}^3$ (which is a smooth manifold of dimension $2$). Let $\phi = (x_1, x_2): S^2 \backslash \{N\} \to \mathbb{R}^2$ be the ...
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### The reason behind the definition of manifold

I was going thorough the definition of a manifold and needless to say it wasn't something that I could digest at one go. Then I saw the following Quora link and Qiaochu's illustrative answer. It was ...
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### Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$?

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$? ($k > -2$). Using the divergence theorem, I got that the flux is: $\frac{3\pi}{k}(1-(-1)^k)$ and ...
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### Simplification of Levi-Civita in an orthonormal frame

I have been struggling to understand how picking an orthonormal frame for the tangent space of a Riemann surface with local coordinates ${x_1,x_2}$ simplifies the matrix of one forms associated to its ...
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### Orientation on the boundary

If $M$ is an oriented without boundary manifold, and $\mu$ is it volume form, is true that the boundary of $M\times [0,1]$ is $M \cup M$, right? It is true also that the orientantion on the boundary ...
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### push forward of the levi civita connection

Let $M$, $M'$ be riemann manifolds with levi-civita connection $\nabla$,$\nabla'$. If $\phi$ is an isometry (global so diffeomorphism too) I want to show: $\nabla'_{X'} Y'=D\phi (\nabla_X Y)$ where ...
For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
Let $M$ be a compact subset of $\mathbb{R}^3$ with smooth boundary $S=\partial M$. Consider M with the standard orientation $\mu=\mu_{0}$ from $\mathbb{R}^3$ and $S$ with the boundary orientation \$\...