# Tagged Questions

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

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### Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
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### Higher-Order Differential Operators as Vector Fields

On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$. When realised this way, the tangent ...
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### If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?

I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant ...
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### Involutive distributions?

How do we check exactly that a distribution is involutive? I have the following definition in my book: A $k-$dimensional distribution $\Delta$ on a manifold $M$ is a smooth choice of a k-dimensional ...
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### Defining a metric in the tangent spaces $T_xM$

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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### Transformation of a subset of compact Jordan sets to manifolds

Let $T$(for e.g. $[0,1]^2$) be a Jordan compact sets and $\tau$ be a "smooth enough one-to-one" transformation, i.e.($\tau: [0,1]^2 \rightarrow [0,1]^2$). Lets take a subset of Lebesgue measure ...
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### p-simplex spanned by elements in the boundary of the unit ball of Thurston norm is in the the boundary of this unit ball.

in completing my thesis I have reached a momentary impass. I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to ...
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### Understanding the “shape” of a singular Riemann surface

Consider the singular Riemann surface given by the following expression: $$z^d w^d-z^d-w^d+t=0\ ,$$ where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2. For $t\neq0$ the ...
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### real coordinates of a complex manifold

I have a naive question about real coordinates of a complex manifold. Let's consider 1-dimensional case for simplicity. Let $X$ be a Riemann surface and $z$ be a local complex coordinate. Then one ...
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### How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
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### Basic examples topological manifolds with boundary

I've just started to study differential geometry and I've some problems with the first definitions. We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...
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### n-dimensional lattice as a collection of lower dimensional spaces.

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...
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### Group actions on manifolds - exponential map

Let $M$ be a smooth manifold. Suppose $K$ is a Lie group (with Lie algebra $\mathfrak{k}$) acting EDIT: TRANSITIVELY on $M$ from the left and $G$ is a Lie group (with Lie algebra $\mathfrak{g}$) ...
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### Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
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### Heegaard splitting via a Morse function - twisted union or not?

Let $M$ be a smooth, closed, connected, oriented 3-manifold and let $f: M \rightarrow \mathbb{R}$ be a self-indexing Morse function. Since $\frac{3}{2}$ is a regular value of $f$ it follows from Morse ...
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### Critical and regular values of height functions on a closed hypersurface

Let $M$ be a closed connected hypersurface of $n$-dimensional in $\mathbb{R}^{n+1}=\{(x^1,\cdots,x^{n+1})\}$ and let $\nu$ be a smooth unit normal vector field of $M$ at $\mathbb{R}^{n+1}$, $H$ be the ...
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### equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
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### Tangent space manifold

Let M be a differentiable manifold of dimension m and also let $\{\xi_1,\dots,\xi_m\}\subset \text{T}_pM$ be an linearly independent set of the tangent bundle of M at a certain point $p\in M$. I have ...
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### Push-forward of vector fields by local isometries

I am studying Riemannian Manifolds by John Lee, and there is this lemma: Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if ...
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### What's the geometrical meaning of immersion?

Does that just mean to different tangent vectors, their images are different tangent vectors?
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### Does a proper map have to be continuous?

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...
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### Looking for a good alternative to 'An introduction to manifolds' by Loring W. Tu

I'm currently studying some basic theory about manifolds from the book 'An introduction to manifolds' by Loring W. Tu. The problem I have with this book is that there are very little exercises, and ...
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### Area form and surface area

I know how one can define the surface area via the charts of a surface in $\mathbb{R}^3.$ click here for instance Now, I read that the canonical surface area form for such a surface with surface ...
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### Volume of Manifold with zero Lebesgue measure

Let $M$ be a smooth manifold in $\mathbb{R}^n$. If Lebesgue measure of $M$ is zero i.e $l(M)=0$, does it mean that volume of manifold is also zero i.e $Vol(M)=0$? Are they the same thing (volume and ...
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### Local isometries preserve geodesics?

Question: It is well known that if $\varphi:M\to \tilde{M}$ is an isometry between Riemannian manifolds, then $\varphi$ maps geodesics of $M$ to geodesics of $\tilde{M}$. I am wondering if it is ...
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### Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...
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### Is every compact manifold with boundary a manifold of bounded geometry?

If $M$ is a compact Riemann manifold with boundary, does it have bounded geometry, which means that the injectivity radius of the manifold is positive and every covariant derivative of the Riemannian ...
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### Action of the fundamental group

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined? EDIT: Of course I want my action to be nontrivial, say ...
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### Are $X=M \times [0,T]$ and $\partial X$ smooth compact manifolds when $M$ is smooth compact Riemannian manifold?

Let $X=M \times [0,T]$, where $M$ is a smooth and closed compact Riemannian manifold. I want to know if: $X$ is smooth compact manifold, and if $\partial X$ is smooth compact manifold? I am not ...
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### Smooth map $S^1 \to S^2$ can not be surjective

Why cannot a smooth (or piecewise linear) map $S^1 \to S^2$ be surjective? There are space-filling curves, but the usual examples have very "twisty" definitions. UPD A bit of background for this ...
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### Why do we require differential manifolds to be Hausdorff? [duplicate]

Among the requirements for a differential manifold $M$ is that it be connected and Hausdorff. What fails if a manifold is not Hausdorff?
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### Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
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### Confused (disoriented?) by questions about orientation

I believe I have a reasonable basic understanding of orientation. Yet, i'm finding myself utterly confused when facing a specific question. Here are several examples: Exhibit an ordered basis ...
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### Determining if an equation represents (?) a Riemann surface

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...
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### What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...
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### Maximally symmetric manifold with boundary and non-vanishing extrinsic curvature?

I was wondering if the following requirements are compatible: Given a $d$-dimensional manifold with boundary $M$ with $\partial M\neq \emptyset$ endowed with a metric $g$. The following conditions ...
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### Homeomorphic manifolds have the same dimension

So I want to prove: If two manifolds $M$ and $N$ are homeomorphic then $dim(M) = m = n = dim(N)$. My idea was to use the property of the manifolds that they are locally homeomorphic to the ...
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### Hermitian metric on $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space

Consider the line bundle $L=\mathcal{O}_{\mathbb{P^n}}(1)$ over the projective space. Locally is descriebd by $\{U_a,g_{ab}\}$ where $U_a=\{z_a\neq0\}$ is the standard covering of the projective ...
### What is the dimension of the space of planes in $\Bbb R^3$?
What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...