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

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Proving $O(1,1)$ is diffeomorphic to the disjoint union of 4 copies of the real line.

I'm trying to prove that the pseudo-orthogonal group $O(1,1)$ is diffeomorphic to the disjoint union of four copies of $\mathbb{R}$. Firstly I tried to solve the equation $AI_{p,q}A^T = I_{p,q}$ where ...
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40 views

Which maps are cellular between triangulable manifolds?

Let $X$ and $Y$ be triangulate manifolds. It is clear that all triangulable manifolds can be given a regular CW decomposition. For which continuous functions $f:X \to Y$ can we find regular ...
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2answers
64 views

Understanding Vector Fields

I want to construct a smooth vector field on $\Bbb S^1 \subset \Bbb R^2$ with the standard smooth structure. Can I just take a smooth vector field on $\Bbb R^2$, say $$V(x,y) = x \frac{\partial}{\...
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42 views

Is the complement of the singular locus of an algebraic variety a topological manifold?

Could someone explain to me clearly, please, why is the complement of the singular locus of an algebraic variety a topological manifold ? A lot of thanks for your help.
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21 views

Projection of upper hemisphere is orientation preserving iff n is even

How do I show that the projection map $π : U → R^n , π (x_1 , . . . , x_n , x_{n+1} ) = (x_1 , . . . , x_n )$, is orientation-preserving if and only if $n$ is even My idea is to calculate the ...
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1answer
21 views

To define Dehn surgery, should one allow orientation reversing diffeomorphisms or arbitrary ones?

So for the definition of Dehn surgery (also called rational/integer surgery), what is correct: Definition Let $K$ be a knot in an oriented $3$-manifold with a regular neighbourhood $N(K)\simeq S^1\...
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19 views

schauder estimate for the heat equation on compact manifolds

Let $M$ be a compact manifold without boundary. Consider $Lu:=\partial_tu-\Delta u$. Let $f\in C^{0,0,\alpha}((0,T)\times M,\mathbb{R})$, $u_0\in C^{2,\alpha}(M,\mathbb{R})$ and let $u\in C^{1,2,\...
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38 views

Diffeomorphism between $TS^1$ and $S^1\times\mathbb{R}$

I want to show that there exists a diffeomorphic $\phi$ such that the following diagram commutes: $$ \require{AMScd} \begin{CD} TS^1 @>{\phi}>> S^1\times\mathbb{R}\\ @V{\pi}VV @V{\pi_1}VV \\ ...
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142 views

Eilenberg-Maclane space, when can $K(\pi, 1)$ be constructed as a compact manifold?

See here. Let $\pi$ be any group. Construct a connected CW complex $K(\pi, 1)$ such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. It is rarely the case that $K(\pi, 1)$...
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2answers
51 views

If $f : X \to Y$ is a continuous map between manifolds with boundary and $f(\partial X) \subseteq \partial Y$, is $f$ surjective?

Let $X$ and $Y$ be $n$-dimensional manifolds with boundary and $f:X \to Y$ be a continuous function. Suppose that $f(\partial X) \subseteq \partial Y$. Does this imply surjectivity and that $f(\...
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Interpreting infinite integer lattice as a manifold of negative dimension

Various fractal dimensions coincide on self-similar fractals as the logarithm of self-copies the fractal includes divided by the logarithm of the factor by which the copies are smaller than the ...
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1answer
57 views

Triangulation for a 1-manifold

I'm taking a course on Algebraic Topology and I had a question while studying. I know the answer is affirmative but I don't know why. What I want to prove is There exist a triangulation for every ...
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1answer
54 views

Reconciling definition of tangent vector with intuition

I'm having trouble getting to grips with what a tangent vector really is. Let $M$ be an $n$-dimensional manifold, and let $\alpha:(-\epsilon, \epsilon) \to M$ be a curve in $M$ through $p$ (i.e. $\...
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1answer
103 views

Boundary of a connected manifold is connected?

I assume the answer to this question is simple, but I can't find any references: Let $X$ be a topological space, and $M$ be a (path-)connected component of a manifold in $X$ with the same dimension ...
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1answer
43 views

Convexity over a manifold

First post on math.stackexchange! I have a question which probably requires a bit of Differential Geometry knowledge which I'm lacking. By definition a convex function over a vector space $V$ is such ...
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1answer
118 views

Quotient space $\mathbb{R}/\mathbb{Z}^{2}$ is not a manifold

I need to prove that $\mathbb{R}/\mathbb{Z}^{2}$ is not a manifold when $\mathbb{Z}^{2}$ acts (continuously) on $\mathbb{R}$ by $t\mapsto t+m+n\alpha$ where $\alpha$ is a fixed irrational for all $t\...
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67 views

Show that $df_x: TM_x\rightarrow TN_{f(x)}$ is well-defined

Let $f\colon M\rightarrow N$ be a $C^\infty$ function between $C^\infty$ manifolds. Show that $$df_x: TM_x\rightarrow TN_{f(x)}$$ defined by $$df_x([c_0]) := [f \circ c_0]$$ is a well-defined map. ...
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70 views

How to embed a square as a submanifold in the Euclidean plane?

One can give a square a differential structure, namely by transferring the smooth structure of a circle to it. But then a square in not a sub-manifold of $\mathbb{R}^2$. But the strong Whitney ...
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1answer
79 views

Einstein manifolds and topology

Given a Riemannian manifold $(M,g)$ with Ricci tensor $ R_{mn} = k g_{mn} $. Suppose the Ricci scalar you get is $$ R > 0 $$ What can you tell about the manifold $globally$ ? In particular, can ...
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1answer
28 views

How to prove this equation? Is $dx_i=e^*_i$?

Suppose U$\subseteq R^m$, F:U$\to R^m$ is $C^{\infty}$, f$\in C^{\infty}(R^m)$. And $x_1,x_2...x_m$:U$\to$R are coordinates on U.{$e^*_1,e^*_2...e^*_m$} is the basis of $(R^m)^*$ dual to the basis {$...
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Prove that $|$ det $g| v(U)$ is the volume of $g(U)$ for any linear transformation $g: \Bbb R^n \to \Bbb R^n$.

This is question of spivak's Calculus of Manifolds; (a) Let $g: \Bbb R^n \to \Bbb R^n$ be a linear transformation of one of the fol­lowing types : $$ \left\{ \begin{array}\\ g(e_i) = e_i, i\neq j\\...
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62 views

Composite function between manifolds

I have this claim left as an exercise in my course: Let $f:M\to N$ be some function between two smooth manifolds $M$ and $N$ (respectively of dimensions $m$ and $n$). Prove that, if for any smooth ...
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1answer
72 views

Clarification about Definition of Immersed Submanifold

I am a little confused about if I am understanding the definition of an immersed submanifold correctly, and characterizing which immersed submanifolds are embedded submanifolds. From Lee's ...
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1answer
72 views

An example of a compact topological manifold which has one cover?

I have been studying compact topological manifolds lately, in particular the $n$-sphere, $S^n$. The reason $S^2$ cannot be covered by one chart is because it is closed and bounded (and hence, by Heine-...
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1answer
71 views

Submanifold, Tangent Space.

Show that (1) M= $ \{(x,y,z) \in \mathbb{R}^{3}$ :$ x^{2}+y^{2}+z^{2} - 3xyz = 1 \}$ is a submanifold of $\mathbb{R}^{3}$. (2) Compute the Tangent Space of M at $a$, for $a \in M$. Attempt(1) If I ...
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165 views

Prove the tangent space at a point $x$ of the $n$-sphere is the space $\{v \in \mathbb{R}^{n+1} : v\cdot x=0\}$

I can see why this is true but I'm not sure how to prove it, any help would be appreciated. Prove that the tangent space $TS^{n}_{x}$ at a point $x$ on the $n$-sphere $S^{n}:=\{x \in \mathbb{...
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1answer
119 views

Prove that $df_{x}: TM_{x} \rightarrow TN_{f(x)}$ is a well-defined map

I was wondering if someone could help me with the following problem, any help would be greatly appreciated. Let $f:M \rightarrow N$ be a $C^{\infty}$ map between smooth manifolds. Given $x \in M$, ...
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1answer
23 views

Prove that if (M, U) is a smooth n-dimensional manifold and p ∈ M, then there is a chart x: U → Rn such that x(p) = 0.

This is a textbook problem from the book Introduction to Manifolds by Tu(Pg-41). I have no idea how to go about this problem. Any hints/suggestions would be greatly appreciated.
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1answer
72 views

integrate over a cube given some differential form

What is process of integrating a differential form given some cube (hyperdimensional obejcts)? I read a lot qualitative problems on this, but seem to find rare examples on how to compute such ...
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1answer
276 views

Prove that the Cartesian product of two topological manifolds is a topological manifold.

I need help on the following problem, any responses would be greatly appreciated: Let $M$ be a topological $m$-manifold and $N$ be a topological $n$-manifold. Prove that $M \times N$ is a topological ...
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1answer
34 views

Gradient flow under varying riemannian metric

Suppose I have a smooth flow $\varphi(t)$ on some Riemannian manifold $(M,g)$ and I know that $\dot{\varphi}(t) = \textrm{grad }F$ for some smooth function $F$. If I smoothly modify the metric $g$, ...
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18 views

property of sum of coefs of a chain

Suppose c is a k+1 chain in U(open set in space R^n), then boundary of c (a k chain) can be expressed as a linear combination of k-cubes, using boundary operator: $$∂c=∑_ia_ic_i$$, where a_i are the ...
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24 views

What is the role of the Laplace-Beltrami operator in providing an optimal embedding for a manifold?

I ve read in "Laplacian eigenmaps for dimensionality reduction and data representation" of Belkin and Niyogi that the Laplace-Beltrami operator has a role in providing an optimal embedding for the ...
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Octonionic Hopf bundle is a bundle.

We have the octonionic Hopf map $f: S^{15} \to S^8$, where $f$ is a bundle of fiber $S^7$. To me this is not obvious. How do I see that the octonionic Hopf bundle is indeed a bundle? Thanks.
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A question about Gauss's Lemma in Riemannian Geometry.

I was reading the proof of Gauss Lemma from do Carmo's book on Riemannian geometry. We need to prove $\langle (d\exp_p)_v(v), (d\exp_p)_v(w) \rangle = \langle v,w\rangle$. He decomposes $w=w_T+w_N$ ...
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1answer
45 views

Question About Definition of Lyapunov Exponents

I just have a quick question about the definition of Lyapunov exponents. My textbook defines them for a smooth map $f:M\to M$, where $M$ is a smooth manifold. For $x\in M$ and $v\in T_xM$: $\lambda(x,...
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35 views

About poisson manifolds?

A Poisson manifold is a pair $(M, \{\cdot, \cdot\})$ where $M$ is a smooth manifold and $\{\cdot, \cdot\}$ is a Lie bracket on the $\mathbb R$-algebra $C^\infty(M)$ satisfying $$\{f, gh\}=\{f, g\}h+g\{...
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1answer
58 views

Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$? If the result is not true for ...
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1answer
28 views

Is $T_{(r,0)}(TM)\cong T_rM$?

Where(r,0)$\in T_rM$, $M$ is an manifold.Is $T_{(r,0)}(TM)\cong T_rM$?
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Show that $(V,\|\cdot \|)$ is a smooth manifold.

Let $(V,\|\cdot \|)$ a normed space of dimension $n$. I want to prove rigorously that it's a smooth manifold. This is how I do: $(V,\|\cdot \|)$ is a topological manifold : Let $x\in V$, $(v_1,...,...
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32 views

Problem defining a smooth m-manifold via a smooth atlas

The definition of a manifold in my course notes is given as follows: Let $\mathcal{M}\subseteq\mathbb{R}^n$. A smooth chart on $\mathcal{M}$ consists of a subset $U\subseteq{\mathcal{M}}$, open in $\...
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Prove that the map $F:\mathbb{R}\to S^1$, $F(t)=(\cos t,\sin t)$ is $C^\infty$.

Why isn't smoothness of $(\cos t,\sin t)$ as a map from $\mathbb{R}$ to $\mathbb{R}^2$ enough? IS it because when we have $S^1$, the codomain is changed? But that shouldn't affect smoothness, right? ...
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An equivalent definition of properly discontinuous of a discrete group

A discrete group $\Gamma$ is said to act properly discontinuously on a smooth manifold $M$ if the action is smooth and satisfies the following two conditions: (i) Each $x \in M$ has a neighborhood $U$...
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18 views

How did the smoothness of map into R^m transfer to the smoothness of map into R

The following is a proof from the book Introduction to manifolds by Tu ( Page 63) I did not understand the highlighted step. Please explain.
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109 views

Is the total space of the tautological line bundle over $\mathbb{R}P^{n}$ a non orientable manifold?

Is it true to say that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is a non orientable manifold? Perhaps the question can be indirectly related to the following question:...
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20 views

Why is the composite map defined on the given domain in the definition of a smooth map between manifolds?

This is a definition from the book Introduction to manifolds by Tu (Page 61 definition 6.5) My question is related to the highlighted part. Why is the highlighted portion the domain of the ...
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1answer
45 views

Interpretations of the derivative

In class we were told to think of the derivative of a function in higher dimensions as a linear transformation between the tangent spaces of $\mathbb{R^n}$, that is $$Df_p:T_p\mathbb{R^n} \to T_p\...
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97 views

Smooth manifold has infinite dimensional space of smooth vector field

I want to show smooth manifold has infinite dimensional space of smooth vector field. I know the space of smooth function of a smooth manifold is infinite dimensional. Then can I just choose a local ...
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47 views

Does diffeomorphism act transitively on a $\mathbb B^n$?

I want to find a diffeomorphism F : $\mathbb B^n$ → $\mathbb B^n$ such that F(p) = q for all p,q$\in$ $\mathbb B^n$. Here $\mathbb B^n$ is the just unit open ball in $\mathbb R^n$. My idea is let r(t)...
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55 views

A sphere with a hair in R^3 is not locally euclidean at the root of the hair.

This is a problem from the book Introduction to manifolds by Tu (Pg-57 problem 5.2) The idea being referred to here is about connectedness i.e if we remove the root of the hair then it would ...