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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1answer
45 views

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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28 views

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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1answer
37 views

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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28 views

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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54 views

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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1answer
35 views

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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64 views

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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1answer
25 views

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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2answers
53 views

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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0answers
17 views

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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25 views

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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24 views

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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1answer
33 views

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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1answer
46 views

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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2answers
60 views

Why $h$ has zero topological degree?

I am trying to prove that $f,g : M^n \to S^n$, both $C^1$ (indeed just $C^0$ is enough) with the same topological degree are homotopic. I saw on a book that the trick is as follows: Take $W = M\...
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Tangent space in the book “Differential Forms and Applications”

In the book "Differential Forms and Applications", the author defines the tangent space of $\mathbb{R}^{3}$ at $p$ ($p \in \mathbb{R}^{3}$) as $\mathbb{R}^{3}_{p}=\{q-p; q \in \mathbb{R}^{3}\}$. My ...
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1answer
33 views

Induced metric on a one-sheet hyperboloid

I am trying to find the induced metric on a one-sheet hyperboloid. Suppose we use cylindrical coordinates $(r, \theta, z)$ for the ambient space in which the hyperboloid is embedded. The hyperboloid ...
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29 views

Partially gluing manifolds along boundaries

I have a question about attaching two manifolds along their boundaries, inspired by my research I do as a graduate student in computer science. Before I state my question, let me preface it with ...
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32 views

Orientability of differantiable manifold of orthogonal matrices

I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal ...
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Sketching the global phase portrait for a version of the Lotka-Volterra system

I'm trying to sketch the phase portrait for a version of Lotka-Volterra given by $$\begin{cases} \dot{x} = x(3-x-2y)\\ \dot{y} = y(2-x-y) \end{cases}.$$ I can sketch this just fine except for the ...
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1answer
63 views

Trying to understand Heegaard diagrams

I have been looking through Rolfsen's "Knots and Links" and I have come across some questions that I am confused about regarding Heegaard diagrams. Let $H_1$ and $H_2$ be genus $g$ handlebodies and ...
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2answers
42 views

Finding the stable and unstable manifold of this system

Consider the system $$\begin{cases}\dot{x} = x \\ \dot{y} = -y + x^2\end{cases}$$ This has fixed point $\overline{X} = (0,0)$, which is a saddle point. The aim is to find the equation of the stable ...
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1answer
18 views

Statement verification - Stable and unstable manifold theorem

Let $\dot{X} = f(X)$ have hyperbolic fixed point $\overline{X}$ and linearisation $\dot{X} = Df(\overline{X})X$. Then there exists a stable manifold $W^s_{\overline{X}}$ of dimension $d_s$ and an ...
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1answer
37 views

Tangent space to the intersection of two manifolds

Let $M, N \subset \mathbb{R}^n$ be two manifolds such that, for every $p \in M \cap N$, $(T_pM)^\bot \cap (T_pN)^\bot = \{0\}$. How do I determine the tangent space of $M \cap N$? I found some places ...
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1answer
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the heat equation for mappings between closed Riemannian manifolds

Let $M$ be a closed (smooth) Riemannian manifold. Then we have the following existence and uniqueness theorem for the heat equation on $M$, which is considered more or less a standard result: Let $0&...
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1answer
64 views

Frobenius theorem

I came across the following conclusion in a textbook, but can't really understand it. I would be grateful if anyone could elaborate: Assume that we have two linearly independent vector fields $V_{1},...
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Laplacian of a submanifold in an Euclidean space

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$ ($n<m$). Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. ...
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1answer
46 views

Show $\mathbb{C}P^n$ is a $2n-$manifold [in singular homology theory]

There is a Theorem in the book that says: The space $\mathbb{C}P^n$ is CW complex of dimension $2n$. I wonder some questions: Is there any Theorem or result that if a space has CW complex structure,...
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2answers
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Is the Lie derivative $L_{X}(\omega \wedge \mu)$ an exact form?

Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an exact ...
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1answer
35 views

Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
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86 views

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 ...
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1answer
80 views

Is Whitehead's manifold with a point removed homotopy equivalent to a sphere?

A contractible open subset of $\mathbb{R}^n$ need not be homeomorphic to $\mathbb{R}^n$. The Whitehead manifold is an open subset of $\mathbb{R}^3$ which is contractible but not homeomorphic to $\...
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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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1answer
28 views

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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Injectivity Radius vs. Radius of Convergence in Analytic Manifolds

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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23 views

Immersions-possible error in Dieudonné III?

Below I refer to [D] Dieudonné Treatise on analysis III [B] Bourbaki VARIETES DIFFÉRENTIELLES ET ANALYTIQUES [M] Michor Topics in differential geometry In [D,16.7.7], we can read: Let $f \...
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1answer
30 views

Transversality: what is wrong with this counter example to persistence for small perturbations?

Let $M$ and $N$ be differentiable manifolds in $\mathbb{R}^{n}$, and let $p \in \mathbb{R}^{n}$. We say that $M$ and $N$ are transversal at $p$ if $$T_{p} M + T_{p}N = \mathbb{R}^{n}.$$ By dimension ...
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Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
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Converting a vector $v \in \mathbb{R}^2$ given in Polar coordinates to Cartesian coordinates

I know that switching inbetween Polar coordinates and Cartesian coordinates in $\mathbb{R}^2$ can, on suitable open subsets of $\mathbb{R}^2$, be done via $(x, y) = (r cos \theta, r sin \theta)$. Let $...
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1answer
31 views

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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389 views

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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0answers
25 views

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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1answer
35 views

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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1answer
33 views

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 ...
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51 views

Fundamental class for tangent bundle

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 ...
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1answer
37 views

Prove there exists a smooth unit normal at the boundary of the following manifold

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 $\...