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

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43 views

Tangent bundle is orientable

I am having some trouble finishing a proof that the tangent bundle of any manifold is orientable. What I've done so far is calculate the transition function between two standard charts on the bundle. ...
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1answer
105 views

Orientation of hypersurface

Some books on mean curvature flow (e.g. Mantegazza, Ecker) state that an embedded hypersurface in $\mathbb{R}^{k+1}$ is orientable (Mantegazza page 3, Ecker page 110). In other words, they assume the ...
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1answer
26 views

Question on exact $n$ form with compact support

I've encountered with the following problem: Consider the map $$\int: \Omega_c^n(\Bbb R^n)\to\Bbb R$$ $$\alpha(x)dx^1\land...\land dx^n\mapsto \int_{\Bbb R^n}\alpha(x)dx^1\land...\land dx^n$$ ...
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2answers
174 views

A proper local diffeomorphism between manifolds is a covering map.

The following is an exercise taken from "Manifolds and Differenial Geometry" by Jeffrey M. Lee. Let $\widetilde M$ and M be (connected) $C^r$ manifolds. Let $f: \widetilde M \to M$ be a proper map ...
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0answers
19 views

Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
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1answer
88 views

Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
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1answer
43 views

Is the composition of a homeomorphism with itself orientation-preserving?

Just a short question about the degree of a homeomorphism. So, I understand that in the continuous setting we define the degree of a map $\ f: M \rightarrow M$ on a connected orientable manifold as ...
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1answer
63 views

Deleting a subset from a topological manifold

Consider an $n$-topological manifold $M$. We remove a subset $A$ from $M$. Are there cases where $M-A$ is no longer a topological manifold. In case we suppose that $M-A$ is still a manifold, what ...
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2answers
130 views

Classical tensor analysis and Tensors on Manifolds

I learned tensors the bad way (Cartesian first, then curvilinear coordinate systems assuming a Euclidean background) and realize that I am in very bad shape trying to (finally) learn tensors on ...
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2answers
55 views

Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
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1answer
69 views

Signature of $S^2 \times D^2$

Every closed connected oriented $4$-manifold has a signature, defined via a cohomological intersection form. In Turaev's book Quantum Invariants of Knots and 3-Manifolds the definition of a certain ...
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1answer
75 views

Confused about the possibility of different differentiable structures.

In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a ...
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0answers
38 views

Submersions define Foliations

Let $M$ be a $C^\infty$ manifold of dimension $m$. A $C^r$ foliations of dimension $n$ of $M$ is a $C^r$ atlas $\mathcal{F}$ of $M$ which is maximal (not needed) with the following properties: a) If ...
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1answer
59 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
63 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 ...
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1answer
64 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 ...
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2answers
56 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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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 ...
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1answer
74 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 ...
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1answer
19 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 ...
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1answer
56 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
70 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
51 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 ...
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1answer
51 views

An almost complex structure on real 2-dimensional manifold

Why an almost complex structure on real 2-dimensional manifold is integrable?
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40 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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65 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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61 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 ...
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2answers
66 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
51 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 ...
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1answer
27 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 ...
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1answer
72 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 ...
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1answer
22 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 ...
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1answer
52 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?
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1answer
70 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$$ ...
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1answer
61 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
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1answer
123 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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21 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 ...
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0answers
24 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
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1answer
57 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
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1answer
71 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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24 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 ...
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1answer
85 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 ...
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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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51 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 ...
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1answer
82 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 ...
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1answer
34 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
25 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& ...
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1answer
45 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
66 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?