Questions tagged [manifolds]

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

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Submanifold of the space of holomorphic functions

Let $H$ be the space of holomorphic functions from the unitary disk $\Delta \subseteq \mathbb{C}$ to $\mathbb{C}^2$ mapping $0 \in \mathbb{C}$ to $0 \in \mathbb{C}^2$. Let $x \in \mathbb{C}^2$ be a ...
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Is a volume defined in manifold with spacetime-signature?

A volume in $R^n$ is easily calculated by multiplying the lengths of the $n$ dimensions. I'm wondering how the different sign of time acts on the volume in a manifold with spacetime signature (1,3). ...
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Constructing cut-off function on geodesic ball

Let $M$ be a complete non compact Riemannian manifold, $p \in M$ be a fixed point, $r > 0$ be an arbitrary (as large as one may want it to be) constant and $B_r(p) = \exp(B_r(0))$ the geodesic ball ...
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1 answer
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Can we know that a map is a submersion with using only its level sets

Let $M$ be a manifold of dimension $2n$, $f:M\rightarrow\mathbb{R}^n$ a smooth map. We know that if $f$ is a submersion, then for any $c\in f(M)$, $f^{-1}(c)$ is a $n$-submanifold of $M$. But what ...
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Is there manifold, whose cohomology or homology groups are Lie groups?

Is there manifold, whose cohomology or homology groups are Lie groups? I have scanned to introductory article https://en.wikipedia.org/wiki/Cohomology and the examples mentioned there does not include ...
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$M$ is a real analytic variety and $p \in M$ such that $M \setminus \{p\}$ is a manifold so $M$ is a real analytic variety in a neighbourhood of $p$

Suppose that $M$ is a real analytic variety and $p \in M$ such that $M \setminus \{p\}$ is a $3-$dimensional (real) manifold. Therefore, $M$ is a real analytic variety in a neighbourhood of the point $...
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How do we derive this formula?! (orthogonal complement involved)

I am studying differential geometry on manifolds at the moment and the following is a part of some notes. Let $\psi:B^2(0,1) \to \mathcal S_+^2$, $\;w \mapsto (\sqrt{1-\vert w \vert^2},w)$ be a ...
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1 vote
2 answers
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Could the linear algebra concepts regarding $n\times n$ matrices be used on any second order tensors?

The metric tensor is actuality is a $(2,0)$ tensor define on a manifold, but in all practical use I've seen, it appears no different than a regular square matrix. In sense that it is just a collection ...
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2 votes
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What is the second derivative of a function on a manifold.

I am a little confused about what the second derivative of a map is. I would like to describe how I understand it and see if I am correct. Let $N,M$ be manifolds and $f:N\to M$ a $C^\infty$ map (for ...
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Does the Differential Topology/Geometry frameworks being able to model solutions to diff. eqs. that are Non-Smooth?

I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to ...
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How to calculate the volume of the image of the manifold

Let $M$ be a $n$ dimensional manifolds, $f:M \rightarrow \mathbb {R}^n$ be a smooth map. Then, how can I calculate $\textrm{vol}(fM)$ ? I'm thinking of calculating it using the area formula as shown ...
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How (if!) can we connect two 2D hyperbolic manifolds by a "wormhole" structure? [closed]

Imagine two 2D hyperbolic manifolds. I connect them by a manifold like one sees pictured in images of a wormhole in 2D. Can we glue them together so the geodesics always diverge? I mean, we can view ...
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Local cross section in smooth manifolds

I have managed to prove that for every field $X$ of class $C^1$ in $\mathbb R^n$ there is a local cross section at a regular point of $X$. I would like to prove that this fact is also true for ...
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manifold criterion with implicit function theorem

I'm studying Spivak's Calculus on manifolds, chapter 5.1. I cannot understand the following theorem. Let $A \subset \mathbb{R^n}$ be an open and let $f : A \rightarrow \mathbb{R^p}$ be a ...
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2 votes
1 answer
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Regarding a proof that $T(M\times N) \cong TM\times TN$

I want to ask about the answer in this link: Tangent Bundle of Product Manifold How is the identification $T_{(x,y)}(M\times N)=T_xM\oplus T_yN$ used in (*) ? And in writing $T(M\times N)$ and $TM\...
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Exercise related to vector fields, map degrees and Poincare-Hopf's Theorem

I got stuck with one exercise from Chapter 3.5 in Guillemin and Pollack's book, which I used to study differential topology by myself: Given a vector field $\overrightarrow{v}$ with isolated zeros in $...
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How to prove that a pullback map is linear

The following question was left as an exercise in my assignment of Manifolds and I am not able to prove this. Question: Define the map $T^{*} : L^{k}(W) \to L^{k} (V)$ , where $\alpha \in L^{k}(W)$ ...
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f-related and smooth map [duplicate]

This question was left as an exercise in my class of manifolds and I am not able to prove this. Question: Let $f : M \to N $ be a smooth map. Suppose that $X_1 , X_2 \in X(M)$ are f-related to $Y_1 , ...
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A doubt on the proof of the isometric invariance of the Levi Civita connection

The Levi Civita connection is isometrically invariant: Note $F_*X$ denotes the pushforward of vector field $X$ Let $M$ and $M'$ be differentiable manifolds having riemmannian metric $g$ and $g'$. Let $...
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The smoothness of distance function on manifolds.

Let $M$ be a manifold. Show that for any closed set $F\subset M$, there exists $\varphi : M\to \mathbb R$ s.t. $\varphi$ is smooth and $F=\varphi^{-1}(\{ 0 \})$. I think the distance function seems to ...
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Quotient of $GL(\mathbb{R}^n)$ by $O(\mathbb{R}^n)$

I suppose the quotient $GL(\mathbb{R}^n)/O(\mathbb{R}^n)$ has manifold structure. Is there a name for this manifold? Google isn't helping find it.
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2 votes
1 answer
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How do I prove $F_*Z=(Z^i\circ F^{-1})\partial_i'$, where Z is a field, and $(F(U),x\circ F^{-1})$ a chart with coordinate fields $\partial_i'$?

I am teaching myself differential geometry on manifolds with some notes a professor gave me. As an initial calculation to prove that the Levi Civita connection is invariant under isometries, the ...
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4 votes
2 answers
216 views

How to see that "two manifolds are diffeomorphic when you can give them each a coordinate atlas with the same transition maps"

This question is about the diffeomorphism of $\mathbb{C}P^1$ and $S^2$. At the end of youler's answer, we read "the general fact that two manifolds are diffeomorphic when you can give them each ...
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1 answer
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Homotopy equivalent to $\mathbb{S}^1$, but not homeomorphic to $\mathbb{R} \times \mathbb{S}^1$ [closed]

What is an intuitive example of a topological object which is homotopy equivalent to $\mathbb{S}^1$, but not homeomorphic to $\mathbb{R} \times \mathbb{S}^1$?
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2 votes
1 answer
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Compressible torus in irreducible 3-manifold bounds a solid torus

In his 3-manifolds notes (page 11, item $(4)$), Hatcher shows that a 2-sided compressible torus $T$ in an irreducible 3-manifold $M$ either bounds a solid torus $S^1 \times D^2$ or is contained in a ...
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Problem on finding Intersection form of compact,orientable $4-$manifolds .

$\mathbf {The \ Problem \ is}:$ Let $M$ be an $\mathbb{F}$-oriented manifold of dimension $2 n$ for a field $\mathbb{F}$. Consider the non-singular bilinear form $H^{n}(M ; \mathbb{F}) \otimes H^{n}(M ...
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1 answer
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Coordinate patches attached to a 2-manifold

In Road to reality page-185, the following photo is shown: I am a bit confused here because I thought the charts of the manifold existed as flat sheets which are subsets of $R^2$ (for a 2- manifold ...
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2 votes
1 answer
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Lee Smooth Manifolds Theorem 6.23's Jacobian Matrix

I am trying to study the tubular neighborhoods and the normal space/bundle section of Lee's Introduction to Smooth Manifolds. I have a minor question about the proof of Theorem 6.23 which states: If $...
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The degree of a foliation $\mathcal{F}$ on the complex projective plane $\mathbb{C}\mathbb{P}^2$

Let $\mathcal{F}$ be a Singular Holomorphic Foliation on the Complex Projective Plane $\mathbb{C}\mathbb{P}^2$. It is well-known that there are too many different equivalent ways to define the Degree ...
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2 votes
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72 views

Free boundary geodesics as a critical point of the energy functional

As a consequence of the formula for the first variation of the energy of a curve, we have the following known characterization of geodesics. A piecewise differentiable curve $c:[0,1]\to M$ is a ...
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3 votes
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Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
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Help understanding smooth covector field on a real closed interval in standard coordinates

In page 138 in John Lee's Introduction to Smooth Manifolds, he stated the following: "We begin with the simplest case: an interval in the real line. Suppose $[a,b] \subset \mathbb{R}$ is a ...
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1 vote
1 answer
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Why is the space of all frames in $\mathbb{R}^3$ the space $\mathbb{R}^3 \times \operatorname{SO}(3)$?

The following regards the textbook "Mathematical Methods of Classical Mechanics" by V.I. Arnold. Definition. The configuration space of a system of $n$ points is the direct product of $n$ ...
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1 answer
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Lee Smooth Manifolds - Lemma (6.14) for Whitney Embedding Theorem

I am studying Lee's Introduction to Smooth Manifolds but I am stuck by a lemma for Whitney's Embedding Theorem. This lemma is trying to prove that: If a smooth n-manifold admits a smooth embedding $\...
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Identity not integrable on surface measure

In my textbook in the definition of center of mass there is a following assumtpion: Let $S \in \mathbb R^m$ be a differentiable manifold such that $l_S(S) < \infty $ and the identity map on $\...
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2 votes
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center manifold and bifurcation: 2D Bifurcation system reduction

I have this system to study $$ \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. $$ I have derived the Jacobian around fixed ...
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Show these topological surfaces are smooth

Consider $\mathcal S^n,$ the space of Schur-convex, simply connected, closed topological $n-$manifolds as subsets of the unit $(n+1)-$cube, which include $p=(0,0,\cdot\cdot\cdot,0)$ and $q=(1,1,\cdot\...
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Good resources that talk about geodesically convex sets for riemannian manifolds?

Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
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Equivalent definitions of Sobolev space on manifold and references

It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$: D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm. D2. All functions ...
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1 vote
1 answer
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Riemannian distance via the length of a curve

I am reading a book where they define the Riemannian distance between two points on a manifold. Naturally it is given as the infimum of the integral of the length of the curves which connect the two ...
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Representation of $L(X)$

Initially posted this, Visualization of L(X), on Cross Validated stack exchange, but since the definition involves measure theory and mathematical statistics I decided to post here as well. $\textbf{...
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An inequality in Hadamard manifolds

Let $M$ be a Hadamard manifold. Is it true that $$\langle v, \exp^{-1}_xy\rangle \le \langle v, \exp^{-1}_xz\rangle + \langle v, P_{x,z}\exp^{-1}_zy\rangle$$ for all $x,y,z\in M$ and $v\in T_x M$, ...
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  • 101
5 votes
1 answer
108 views

Order relation between cohomological dimensions of open orientable manifolds

Let $M$ be an open orientable connected manifold and let $\operatorname{Cohdim}_{\mathbb{Z}_{2}}(M)$ and $\operatorname{Cohdim}_{\mathbb{Z}}(M)$ be the cohomological dimensions of $M$ over $\mathbb{Z}...
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2 answers
66 views

Prerequisites for John M. Lee's Introduction to Topological Manifolds?

I currently have some questions regarding prerequisites, especially prerequisites on analysis before diving into that book. I really liked the way author explained things from some parts I read and I'...
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1 vote
1 answer
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Relation between mod 2 Betti numbers and integral cohomology

Let $M$ be an orientable connected manifold of finite type. My question is that if $H^{i}(M,\mathbb{Z}_{2})$ is non-zero, then can we say that $H^{i}(M,\mathbb{Z})$ is non-zero? I know that this is ...
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  • 1,014
4 votes
2 answers
41 views

Submersion, diffeomorphism, and embedding

Let $\pi:M \rightarrow N$ a submersion and $f:S \rightarrow M$ a smooth map such that $\pi \circ f : S \rightarrow N$ is a diffeomorphism. Show that $f:S \rightarrow M $ is an embedding. I'm having ...
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1 vote
0 answers
74 views

Why are closed manifolds defined as they are?

The standard definition of a closed manifold is the following: A closed manifold is a manifold without boundary that is compact. I am wondering, why we request that the manifold is compact. As I ...
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1 answer
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Quotients of real projective spaces

I am trying to determine whether or not $M := \mathbb{R}P^4/\mathbb{R}P^2$ is a manifold. I believe it is clear that $M$ is second-countable. However, I'm not not sure about Hausdorffness and being ...
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  • 1,572
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Verifying Stokes' Theorem with $\omega = x^2 \text{d}w-2yz \text{d}x$ as a one-form in $\mathbb{R}^4$ over a manifold.

I'm having trouble verifying Stokes' Theorem, and I'm just wondering if someone's able to find my mistake in the calculation. Let $\omega=x^2\text{d}w-2yz \text{d}x$ be a one-form on $\mathbb{R}^4$, ...
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  • 1,892
0 votes
1 answer
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Are these two questions asking the same thing?

Question $(I).$ Show that if the $n$-dimensional manifold $M$ is a product of spheres, then there exists an embedding $M \to \mathbb R^{n+1}.$ Question $(2).$ Show that there exists an embedding $S^{...
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