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
68 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
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 ...
0
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1answer
51 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
67 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
55 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
117 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
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
55 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
55 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
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 ...
1
vote
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
46 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
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 ...
0
votes
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& ...
1
vote
1answer
44 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
votes
0answers
61 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
vote
1answer
47 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
29 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
33 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
37 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 ...
0
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0answers
68 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 ...
0
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1answer
38 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
vote
1answer
47 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
votes
1answer
50 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
38 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
votes
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 ...
0
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0answers
47 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 ...
3
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0answers
32 views

A singular $n-$cube and a circumference defined the border than 2-cube

This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100): If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer ...
5
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2answers
430 views

Spin manifold and the second Stiefel-Whitney class

We know that: Spin structures will exist if and only if the second Stiefel-Whitney class $w_2(M)\in H^2(M,\mathbb Z/2)$ of $M$ vanishes. Can someone use simple words and logic to show why the ...
0
votes
1answer
34 views

Is there a local flow that is diffeomorphic at any time?

This question is regarding p.223-224 of Loring Tu's Introduction to Manifolds (Second edition). Without proof, the author previously assumed the following. Theorem. Let $M$ be a manifold and $X$ be ...
0
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0answers
52 views

Calculate the geodesic of Z = XY between two points

I have only learned about calculus and linear algebra, so I don't know about differential algebra. I got to know about the concept of "geodesic" recently. What I need to know is this: Suppose I ...
2
votes
1answer
77 views

Connection vs Curvature

Why is twice a connection usually referred to the curvature: $\overline{\nabla}\circ\nabla=F^\nabla$ Is there an axiomatic definition of curvature, e.g. it is module-linear operator etc?
0
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1answer
28 views

one problem on multivariable claculus

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes ...
0
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1answer
28 views

Dual isogenies of complex tori in Birkenhake-Lange

Let $f: X\to Y$ be an isogeny of complex tori, of degree $n$. On page 13 of Complex Abelian Varieties, Birkenhake-Lange show that there is a dual isogeny $g: Y\to X$. Basically, they show ...
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0answers
32 views

Degree One Map induces Surjections on Homology

Is the following statement true: If $f:M\to N$ is a degree one map of compact closed manifolds, then $f$ induces surjections $f^*:H_q(M)\to H_q(N)$. I found this claimed on ...
1
vote
1answer
31 views

Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
0
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0answers
57 views

Stokes theorem on manifold without boundary

I'm struggling to understand why the integral should vanish: $\partial M=\varnothing:\quad\int_Md\omega$ For example: $0=\int_{B_1(0)}d(ydx)=\int_{B_1(0)}1dy\wedge dx=\mu(B_1(0))\neq 0\text{ ?}$
0
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2answers
86 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
2
votes
1answer
63 views

Spivak Calculus on Manifolds, problem 1-2

I am confused about the hint Spivak adds to problem 1-2 in his Calculus on Manifolds: When does equality hold in Theorem 1-1(3)? Hint: Re-examine the proof; the answer is not “when $x$ and $y$ ...
1
vote
1answer
33 views

Branched covering of a manifold [duplicate]

What would be the definition of a branched covering of a manifold? I am not familiar with branched coverings at all.
0
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1answer
58 views

Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
1
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0answers
37 views

Manifolds and magnetic potential

Assume we have a particle in $\mathbb{R}^3$, which we will subject to different fields independently. It will have some potential energy $U\in \mathbb{R}$ defined as some constant minus its kinetic ...
2
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1answer
54 views

Is a non-compact Riemannian manifold a “measure space”?

One can define $L^p$ spaces for measure spaces with a given measure. Is a non-compact (i.e., it has a boundary) bounded Riemannian manifold a measure space? I am thinking of the manifold $(0,T) \times ...
1
vote
2answers
173 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...