发布时间：2019-05-19 10:06

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I read lots of journal papers that had used Dual laplacian, but didn't find any theory. So plz help me witht dual laplcian and give some link for study materials Thanks

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38 views

I know that the integral homology group of the manifold $M$ is given by $$ H_j(M,\mathbb{Z}) $$ I also have tried that $H_j(T^3,\mathbb{Z})$ is given by $$ H_0(T^3,\mathbb{Z})=\mathbb{Z}, $$ $$ H_1(...

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I would like to understand the solution to the following Ode, can we solve that? This there any idea that we can analysis something on that? $\frac{d^2}{dx^2}u(x)+\sinh(u)=0$. Thanks.

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The text I am using "Nonlinear PDEs - A Dynamical Systems Approach" (Hannes Uecker) defines a stable manifold as follows Definition : Let $u^*$ be a fixed point of the ODE $\dot{u} = f(u)$ with ...

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I'm reading the article "Local and simultaneous structural stability of certain diffeomorphisms - Marco Antônio Teixeira", and on the first page says "Denote $G^r$ the space of germs of involution at $...

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10 views

I'm trying to do this problem: Let $f: S^2 \to \mathbb{R}^3$ given by $f(x,y,z)=z$. For the regular values $-1<t<1$, find the orientations of $f^{-1}(t).$ The hint is to find a positively ...

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21 views

I am going through Guillemin and Pollack and have reached some difficulty with orientation. The way it does preimage orientations confuses me, and likewise the problems on the orientation of ...

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45 views

Suppose I have a complex vector space with basis $\{v_1, ..., v_p, w_1, .., w_q\}$ and the standard Hermitian form of type $(p, q)$. I want to prove that the space $D$ of all the dimension $q$ sub-...

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54 views

I have a doubt about the fact that a derivative of $f:M\to \mathbb R$ of a $\mathcal C^1$ manifold is well defined... Indeed, let $a\in M$ and $(U,\varphi )$ a chart from a $\mathbb C^1$ atlas s.t. $a\...

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33 views

The following question is stated on an exercise sheet of Riemannian Geometry. We look at the pseudo Riemannian metric, defined on $M = \mathbb{R}^2 \ 0 $ by \begin{align*} < \partial_x, \...

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7 views

What are the (technical) differences between Sobolev spaces on domains $\Omega \subset \mathbb R^n$ or (compact) manifolds such as two-dimensional spheres?

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53 views

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-...

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33 views

Every compact connected 2-manifold (I define this as a surface) is homeomorphic to a 2-sphere, a connected sum of tori or a connected sum of projective planes. Since the fundamental groups of the ...

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84 views

My book is An Introduction to Manifolds by Loring W. Tu. Let $S = \{x^3-6xy+y^2=-108\}$, and let "submanifold" and "$k$-submanifold" mean, respectively, "regular" and "regular $k$-submanifold". As in ...

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My book is An Introduction to Manifolds by Loring W. Tu. As can be found in the following bullet points Can a topological manifold be non-connected and each component with different dimension? Is $[...

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Let $SL(n,\Bbb{C})$ be the group of matrices of complex entries and determinant $1$. I want to prove that $SL(n,\Bbb{C})$ is a regular submanifold of $GL(n,\Bbb{C})$. An idea is to use the ...

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37 views

My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. As part of Proposition 11.13(i), I'm trying to compute the degree of the "interchanging" $T: J \times K \to K \times J, T(x,y)...

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My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. This is the definition of local index: Corollary 11.10 says if $f$ isn't surjective, then $\deg(f) = 0$, I guess by empty ...

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42 views

I know how to show this if $X$ and $Y$ are euclidean spaces using IFT but wanted to confirm proofs about the abstract case. Q) a) $X$, $Y$ are smooth manifolds and $f:X\rightarrow Y$ is smooth. Show ...

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If $H$ is a closed subgroup of Lie group $G$, then show that $\mathfrak{h}=0$ if and only if $H$ is discrete, where $\mathfrak{h}$ is the Lie algebra of $H$. We know that $\mathfrak{h}=\{X\in \...

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Is the subgroup $S=\{m+n\alpha|\;m,n\in \mathbb{Q}\}$, where $\alpha$ is a fixed irrational number, locally compact in $\mathbb{R}$ ? Approach: I can see that $S$ is dense in $\mathbb{R}$. But I am ...

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42 views

I want to use the technique from hatcher section 3.2 to compute the cup product structure of a punctured torus (with $\mathbb{Z}$ coefficient), but I found that I still don't know how to do this when ...

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Recently I have been reading a lot about $\mathbb{Z}_2$-actions on topological spaces. Mainly I was focused on surfaces such as the sphere, torus and Klein bottle and here the existence of a ...

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The definition of Differential Manifold or Smooth Manifold include $\text{Second countability}$ and $\text{Hausdorffness condition}$. My question is why we include Second countability and ...

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217 views

Suppose that we have a Riemannian metric $ds^2=Edu^2+2Fdudv+Gdv^2$ on a local coordinate neighborhood $(U;(u,v))$ prove that for the following vector fields: $$e_{1}=\frac{1}{\sqrt{E}}\frac{\partial}{...

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I have a question about the manifold, especially when the manifold is as well a vector space of finite dimensional $k$. Actually, let $(v_1, \dots, v_k)$ be a basis of F as a vector space. I would ...

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145 views

The unit sphere $n$ dimensional is the set $$\mathbb{S}^n=\bigg\{(x_1,x_2,\dots, x_{n+1})\in\mathbb{R}^{n+1}\;|\;\big(x_1^2+x_2^2+\cdots+x_{n+1}^2\big)^{1/2}=1\bigg\}.$$ For all $i=1,\dots, n+1$ ...

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33 views

Let $M$ be a smooth $n$-manifold and let $U\subseteq M$ be any open subset. Define an atlas on $U$ $$\mathcal{A}_{U}=\big\{\text{smooth charts}\;(V,\varphi)\;\text{for}\; M\;\text{such that}\;V\...

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This is a problem from Lee 17.12: Suppose $M$ and $N$ are compact, oriented, smooth n-manifolds, and $F:M\rightarrow N$ is a smooth map. Prove that if $\int_M F^*\eta \neq 0$ for some $\eta \in \...

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I have a hard time seeing if the derivative of a vector field along a curve or parallel transport is the main purpose of introducing the connection on a vector bundle. Anyone have some idea about ...

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I am self-learning integration on manifolds, and I'm trying to find an answer to the following question. For the manifold $M=\{(x,y) \in \mathbb{R} : (x,y) \neq (0,0) \}$, let $f: M \rightarrow \...

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I am reading Munkres’ Analysis on Manifolds, and I am having trouble understanding the comment after the following statement. Let $A$ be an open set in $\mathbb{R}^k$; let $\eta$ be a $k$-form ...

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38 views

Let $M$ be a compact smooth $3$-manifold, and $h: M\to \mathbb{R}$ a function such that $\{0\}$ is a regular value of $h$, and define $\Sigma = h^{-1}(0).$ Moreover, we will denote $\mathfrak{X}^r(M)$ ...

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The weak formulation of the Poisson equation of Dirichlet type in Euclidean space reads For given source function $f \in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such that \begin{equation} \int_{\...

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Munkres book on Manifolds constructs a wedge product by defining the following sum on $f$ (an alternating $k$-tensor on $V$) and $g$ (an alternating $l$-tensor on $V$): $$(f \wedge g)(v_1,...,v_{k+l}) ...

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I'm trying to prove that the universal cover of $S^1 \times S^2$ is $\mathbb{R}^3 \setminus \{0\}$. I know that the universal cover of $S^1$ is $\mathbb{R}$ and the universal cover of $S^2$ is $S^2 $. ...

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I’m having difficulty solving this problem. Could you tell me how to prove this? I showed the intersection with two variables, but still don’t see how to prove that it’s a manifold. ↓the problem and ...

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In wikipedia there is a proof for 3-manifolds that I don't understand. It says that if $M$ is an irreducible manifold and we express $M=N_1\sharp N_2$, then $M$ is obtained by removing a ball each ...

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Consider a smooth map $\Delta :M \to N$. Let $q\in N$ be a regular point. I want to understand how I go about examining the topology of $\Delta^{-1}\{q\}\subseteq M$. In the example of the sphere, $\...

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I know as a matter of fact, that $\mathbb{R}$ compactifies to a circle $S^1$. So there should, in my visualization, exist a single infinity. If I want to go from $S^1$ back to $\mathbb{R}$ I have to ...

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I just started studying Information Geometry and its applications by Amari. Right in the first chapter, the author talks about parallel transport in Dually flat manifolds. Just some quick notation: ...

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For each nonnegative integer $n$, the Euclidean space $\mathbb{R}^n$ is a smooth $n$-manifold with the smooth structure determined by the atlas $\mathcal{A}=(\mathbb{R}^n,\mathbb{1}_{\mathbb{R}^n})$. ...

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Denote $x = (x_1,...,x_n)$. I'm trying to prove the following: $$\int_{S^{n-1}}x_1^2dS =\int_{S^{n-1}}x_k^2dS \; , \; 2\leq k\leq n $$ Intuitively this equality is due to the symmetry of the ...

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Why does the Jacobian have constant sign for connected sets? I've seen in two separate proofs now (having to do with manifold orientation) that the Jacobian has constant sign for a connected set, but ...

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Let $X$ be a (smooth) vector field on a manifold $M$ and let $\gamma$ be its integral curve passing through $m$ at $t=0$ and finally let $T:U\times (-c,c)\to M$ be the local group of transformations ...

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29 views

Denote $\mathbb{R}^0=\{0\}$. Proposition. A topological space $M$ is a $0$-manifold if and only if it is a countable discrete space. Proof. $(\Rightarrow)$ Suppose that $M$ be a topological ...

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I was trying to understand the definition of a manifold. This question arised: is every manifold $M$ the inverse image of some $\Delta : \mathbb{R}^n \to \mathbb{R}$. The implicit function theorem, i ...

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The Classification Theorem for surfaces says that a compact connected surface $M$ is homeomorphic to $$S^2\# (\#_{g}T^2)\# (\#_{b} D^2)\# (\#_{c} \mathbb{R}P^2),$$ so $g$ is the genus of the surface, $...

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I'm currently working through Do Carmo's book Riemannian Geometry and came across the following question: Let $M$ be a Riemannian manifold with the following property: given any two points $p, q \in ...

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Let $H$ be a genus $g$ handlebody embedded in $S^4$ and let $X = S^4 - N(\partial H)$ where $N(\partial H)$ is an open tubular neighborhood of the boundary of $H$. What is $X$? In the case where $g=...

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