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

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2
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1answer
49 views

Triangulations, PL-triangulations and related conecpts

I'm confused about various definitions of triangulations and piece-wise linearity. I read, for example, on wikipedia "..the question of whether all topological manifolds have triangulations is an ...
5
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1answer
56 views

The graph of $x\mapsto |x|$ cannot be the image of an immersion.

How can one prove that the set $\{(x,|x|)\in \mathbb{R}^2 \mid x\in \mathbb{R}\}$ cannot be the image of an immersion of a smooth manifold? This was my homework exercise in a course about ...
5
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1answer
56 views

General introduction to orbifolds?

Where should I go to learn about orbifolds? I am interested in a general introduction that gives precise definitions and clear explanations. I have a fair background in topological and smooth ...
1
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2answers
43 views

How many 2-manifolds can be covered by a single chart?

Every 1-manifold is either homeomorphic to a sphere or to the real line. Therefore, one can trivially say that all 1-manifolds (without boundary) covered by a single chart are equivalent to ...
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0answers
28 views

Does a metric has a neighboorhood in which signature is the same?

suppose $M$ is a manifold and $g$ is a metric with a specific signature on it. there are at least 2 topologies for metrics that I am aware of. Fine $C^k$ topologies and the topology of metrics as ...
2
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1answer
66 views

Atlas/chart for a Hyperboloid

given is the following hyperboloid: $$H = \{(x,y,z) \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\},$$ where a,b,c are free parameters. I have to find an $C^\infty$-atlas for H. In ...
1
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1answer
18 views

natural projection on a slice

I'm currently studying Warner's book "Foundations of Differentiable Manifolds and Lie Groups". Within the proof of the Frobenius Theorem he is constructing a slice $S$ of a coordinate system ...
2
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1answer
50 views

A question about the condition of Frobenius theorem

I puzzled about the condition of Frobenius theorem: Condition FR1: Let $X$ be a manifold, $E$ is a subbundle of $TX$,vector fields $ ξ,η $ lie in $E$(i.e. $ ξ(x),η(x)\in E_x $),then bracket $[ξ,η]$ ...
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0answers
56 views

What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
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1answer
48 views

A question about tangent subbundle

Let $x$ be a manifold, $E$ is a subbundle of $TX$ , my question is : Can you give example such that vector fields $\xi ,\eta$ lie in $E$,but bracket $[\xi ,\eta]$ does not lie in $E$ in some point ...
2
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1answer
47 views

An almost complex structure on real 2-dimensional manifold

Why an almost complex structure on real 2-dimensional manifold is integrable?
3
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0answers
34 views

Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
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0answers
60 views

Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't ...
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0answers
51 views

Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
1
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2answers
55 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
0
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1answer
46 views

Calculate geodesic path on matrix manifold

I have a matrix which is change with time. Let me denote it as A(t). I know t=0 it is A(0) and I know t=1 it is A(1). A is symmetric positive semi-definite matrix. What I want to do is find the ...
0
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1answer
22 views

$C^{\infty}$ vector field along an immersion has local $C^{\infty}$ extensions

Let $\psi:M\longrightarrow N$ be a $C^{\infty}$. A smooth vector field $X$ along $\psi$ (i.e. $X\in C^{\infty}(M,TN)$ $\textrm{and } \pi\circ X=\psi$) has local $C^{\infty}$ extenstions in $N$ if ...
2
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1answer
65 views

kropholler's conjecture and $3$-manifolds group [closed]

Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial $H$-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that ...
0
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1answer
20 views

Boundedness of Riemannian curvature gradient

I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows: Let $B(x_{0},\gamma)$ be a ...
0
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1answer
49 views

Does the set of $n$ by $n$ matrices of rank $q$ form a manifold?

I'm not sure whether the space of all rank-$q$ square matrices of dimension $n$ is a submanifold. I have totally no clue. Can somebody help?
1
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1answer
65 views

Question Symmetric 2-Tensors on Vector Fields

I have a question on a direct computation. How would one compute the following $$ (dx \odot dy ) \left(\frac{1+uv+xy}{1+xy} \frac{\partial}{\partial v}, \frac{\partial}{\partial y} \right) $$ and$$ ...
2
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1answer
51 views

Graph theoretic view on manifold triangulations

To make the question (hopefully) clearer, I reformulated it: Some triangulation $T$ of a smooth manifold $M$ is a piecewise linear manifold, because smooth manifolds are topological manifolds. Such a ...
2
votes
1answer
106 views

Is a Variety a manifold?

Is it true that every smooth variety (over $\mathbb{R}$ or $\mathbb{C}$ ) is a (real or complex) manifold? I have tried to show this using the implicit function theorem but I am not getting anywhere. ...
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0answers
20 views

Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
0
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0answers
22 views

Extension of a bounded operator on manifold

I have a problem, which is quite urgent. I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. I wish to extend this operator to an ...
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0answers
23 views

Cap-Independence

I'm just trying to figure out something regarding Cap-Independence. The problem reads $\partial S:= r(t)=(\cos t,\sin t,\sin 2t)$, $0\le t \le 2\pi;$ $\phi=z\,dzdx-6y^2dxdy$ ($\partial S $ ...
2
votes
1answer
52 views

Orientability of algebraic manifolds

Is algebraic manifold always orientable? For example, unorientable Mobius strip $M$ can be represented as $$x(u,v)= \left(1+\frac{v}{2} \cos \frac{u}{2}\right)\cos u$$ $$y(u,v)= \left(1+\frac{v}{2} ...
3
votes
1answer
39 views

Graph of a continuous function is a smooth manifold? [duplicate]

Let $f:(a,b)\to \mathbb{R}$ be a continuous function and define $\Gamma(f) = \{(x,f(x)):x\in (a,b)\}$. The two maps $\Psi: \Gamma(f)\to (a,b)$ given by $(x,f(x))\mapsto x$ and $\Phi: (a,b)\to ...
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0answers
20 views

Question on framed bordism classes definition

I was reading recently about cobordism, and in specific about the Thom-Pontraygin theorem which states $\pi_{k}(S^n)$ is isomorphic to the cobordism classes of framed $n$-manifolds in $R^k$. In ...
6
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1answer
83 views

Orientability of Stiefel manifold $V_2(\mathbb R^4)$

What is an easy proof of orientability of Stiefel manifold $V_2(\mathbb{R}^4)$ (pairs of orthonormal vectors from $\mathbb{R}^4$ - subset of $\mathbb{R}^8$)? All proofs I found deal with Lie groups ...
1
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1answer
33 views

Pushforward of the tangent space at a point of a submanifold

Let $\iota : S \hookrightarrow M$ be an inclusion that serves as an injective immersion between real manifolds of dimension $k$ and $n$, respectively. Fix $p \in S$. Then we have a linear embedding ...
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0answers
37 views

Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
7
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1answer
81 views

What kind of tools do we have to detect when a manifold is a product of other manifolds?

What sort of tools are out there that can detect when a manifold is a product of other manifolds? For example, comparing the homology of the circle to the torus, the homology of the torus gets more ...
0
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1answer
33 views

Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $ x+y-z+u=1 $ $ 2x+u=2 $ $ z -u=0 $ I don't ...
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0answers
24 views

Induced measurable subbundle

Let $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ measurable and suppose that the application $$ \displaystyle{\begin{array}{rccl} h:&Z&\longrightarrow& ...
1
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1answer
41 views

Equivalence between vector field and generator of a group of translations

I've been reading Olver's Applications of Lie groups to differential equations and did not understand the excerp where the autor explains that "[...] every nonvanishing vector field is locally ...
2
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0answers
55 views

Ample tangent bundle

I am looking for definition of ample tangent bundle or positive tangent bundle and why on a complex manifold , positive bisectional curvature means ample tangent bundle?
1
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1answer
43 views

Typo in Spivak Calculus on Manifolds?

In the proof of theorem 5.2 of Spivak's calculus on manifolds, he invokes the inverse function theorem for a function g that he defines there without ever assuming that it was continuously ...
2
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1answer
28 views

Does a chart $\varphi$ of $S^2$ exist with $S^2=\varphi(T)$?

Let $n,k\in\mathbb N$ with $1\le k\le n$. A $k$-dimensional ($C^1$-) submanifold of $\mathbb R^n$ is a non-empty set $M\subseteq\mathbb R^n$ with the property that for all $a\in M$ there exists an ...
0
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1answer
31 views

The orthogonal group has a unique manifold structure

We proved the following theorem : Assume that $\psi:M^c\longrightarrow N^d$ is $C^{\infty}$. Let $n\in N$, $P=\psi^{-1}(n)$ be non-empty, and let $d\psi:M_m\longrightarrow N_{\psi(m)}$ is surjective ...
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0answers
35 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
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0answers
62 views

Why are affine subspaces also sometimes called linear manifolds?

According to Wikipedia, an affine subspace is a subset of a vector space closed under affine linear combinations. That is, linear combinations whose scalar coefficients sum to 1. It's not clear to ...
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1answer
35 views

Why is every tangent vector part of a vector field?

I am reading a book that defines tangent vectors and vector fields on a manifold $M$ as derivations: A vector field is defined as a linear function and derivation $C^{\infty}(M)$ to $C^{\infty}(M)$. ...
0
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1answer
37 views

A notational confusion on gradient

Given a parametrized function $f_{w}: \Bbb R ^{m} \to \Bbb R ^{k}, w \in \Bbb R^d$, I see in a book the following notation $\bigtriangledown ^ {w} f_{w}(.)$ denote its gradient w.r.t. $w$. What is the ...
1
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1answer
46 views

Inequality for second fundamental form

Suppose that all eigenvalues of the second fundamental form $A=\{h_{ij}\}$ of manifold $M$ are strictly positive, then there is some $\epsilon>0$ such that the inequality $$h_{ij}>\epsilon ...
0
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1answer
40 views

Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
2
votes
1answer
36 views

Wedge product of basis elements of cohomology

Let $M$ be a compact, connected, oriented 4-manifold without boundary. If $H^2(M)\cong \mathbb{R}^2$ and I have a basis $\{[\omega_1],[\omega_2]\}$ for $H^2(M)$, is it the case that $[\omega_1\wedge ...
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0answers
38 views

Can a 1-manifold $M$ without boundary in $\mathbb{R}^2$ have a “corner”?

If $α:U→V$ is a coordinate patch on a manifold without boundary, $M$, about the point $p$, one of the conditions on $α$ is that $Dα(x)$ have rank 1 for each $x \in U$. According to Munkres in Analysis ...
3
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2answers
87 views

Show $M_1\cap M_2$ submanifold iff $N_x(M_1)\cap N_x(M_2) = \{0\}$ and dimensions

Let $M_1,M_2 \subseteq \mathbb{R}^3$ two-dimendional submanifolds of $\mathbb{R}^3$ such that for every point $x\in M_1\cap M_2$ $$N_x(M_1)\cap N_x(M_2) = \{0\}.$$ Show $M_1\cap M_2$ is an ...
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0answers
41 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...