For questions about smooth manifolds.

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2
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1answer
20 views

$f:M\to N$ smooth manifold map. $F(x)=(x,f(x))\in M\times N$. For each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$.

Let $M$ and $N$ be to manifolds and $f:M\to N$ smooth map. Define $F:M\to M\times N$ by $F(x)=(x,f(x))$. Show that for each $X\in \mathfrak{X}(M)$ there's a F-related $Y\in \mathfrak{X}(M\times N)$. ...
2
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1answer
19 views

Let $X_i, Y_i$ be vector fields on the manifolds M and N. $X_i\oplus Y_j$ on $M\times N$. $[X_1\oplus Y_1,X_2\oplus Y_2]=[X_1,Y_1]\oplus [X_2,Y_2]$

Let $M$ and $N$ be two differentiable manifolds and $X_1,X_2$ be two vector fields on $M$ and $Y_1, Y_2$ on $N$. Using the fact that $T_p(M)\oplus T_q(N)$ is naturally isomorphic to $T_{(p,q)}(M\times ...
0
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0answers
19 views

A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
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0answers
16 views

1-parametric subgroups of diffeomorphisms induce a complete vector field

I have been working through this book on differential equations and I do not quite understand the justification for one claim. Namely, the author claims that every 1-parameter subgroup $\{\psi_\...
4
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1answer
88 views

About codifying length and energy using a one-form

Let $M^n$ be a smooth manifold and $g$ be a Riemannian metric on $M$. Is there $\omega \in \Omega^1(M)$ such that $\int_c \omega = \ell(c)$ for every curve in $M$? In general the answer seems ...
2
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1answer
32 views

Viewing complex projective space as a Grassmannian manifold

Complex projective space $\mathbb{CP}^n$ carries the structure of a complex manifold of dimension $n$, hence has the underlying structure of a real manifold of dimension $2n$. It is the set of complex ...
2
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1answer
30 views

“Approximate Isometry” in Riemannian Geometry

I apologize if the notion I'm asking about is well known, I'm no expert in geometry (and I did not find an answer via google). Suppose $(X,g_X)$ and $(Y,g_Y)$ are (smooth) Riemannian manifolds. I'm ...
3
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1answer
41 views

Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\...
2
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1answer
33 views

Integration over the Haar measure of a compact Lie group preserves smoothness?

Let $G$ be a compact Lie group. Then there is a unique Haar (probability) measure on $G$. Let $f_g \colon G \to \mathbb{R}$ be a family of smooth functions $(f_g)_{g \in G}$, is the function $$ G \to \...
0
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1answer
19 views

Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis. Let $M$ be a compact and smooth ...
0
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0answers
13 views

Distance in submanifold vs ambient manifold

In Audin-Lanfontaine "Holomorphic curves in symplectic geometry", we need the following condition (Def. 4.7.1, p.182 by Sikorav): Let $L$ be a submanifold in a Riemannian manifold $(W,g)$. We want ...
0
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1answer
26 views

Construct a global smooth vector field

Assume the following lemma: Let $K$ be a compact subset of a smooth n-dimensional $\mathbb{R}$-manifold $M$ and $U$ an open subset of $M$ such that $K\subset U$. Then there exists a differentiable ...
1
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1answer
49 views

Vector fields (on a manifold) and terminology

I read in several books (Do Carmo, Riemannian Geometry or John M. Lee, Smooth manifolds) that a vector field $X$ on a smooth manifold $M$ is a mapping which associates to each point $p \in M$ a ...
2
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1answer
23 views

Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero.

My question is as the title states: Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero. Now, ...
3
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1answer
57 views

How do we call a map $F$ such that $F(g\cdot p)=\varphi(g)\cdot F(p)$?

Let $G$ and $H$ be groups acting on sets $M$ and $N$. Suppose that there is a group homomorphism $\varphi:G\to H$ and a map $F:M\to N$ such that $$F(g\cdot p)=\varphi(g)\cdot F(p)$$ for all $p\in M$ ...
2
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1answer
30 views

Irreducible representations of the fundamental group of a closed surface in $SU(2)$

For a compact Lie group $G$, consider the map $f : G^{2n} \to G$ given by $f(A_1, B_1, \ldots, A_n, B_n) = \displaystyle\prod_{i = 1}^{n} A_i B_i A_i^{-1} B_i^{-1}$ A theorem of Goldman (from the '...
2
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1answer
41 views

Group of deck transformations acts properly discontinuously

Let $M$ be a connected (smooth Riemannian) manifold which admits a universal cover $\tilde{M}$. Let $\Gamma$ be the group of deck transformations on $\tilde{M}$. I want to show that $\Gamma$ acts ...
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2answers
40 views

When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I ...
1
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1answer
32 views

Where does $f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$ have a locally differentiable inversion?

$$f(r,\varphi,\theta)=(r\sin{\theta}\cos{\varphi},r\sin{\theta}\sin{\varphi},r\cos{\theta})$$ $$f:(0,\infty)\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^3$$ How can I find out on which ...
1
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1answer
59 views

Equivalences of the definition of smooth vector fields

Let $M$ be a smooth manifold and $X\colon M \to TM$ a vector field on $M$. I'm having some trouble proving that these assertions are equivalent: (i) $X$ is smooth. (ii) for every chart $(U,\varphi) \...
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1answer
48 views

A Riemannian manifold with constant sectional curvature is Einstein. [closed]

A Riemannian manifold with constant sectional curvature is Einstein. Why? It's true the inverse?
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2answers
59 views

Why a Riemannian manifold minus one point is not complete? [closed]

Could you give me a proof that a Riemannian manifold minus one point is ever complete? Thanks!!
2
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1answer
68 views

Preimage of tubular neighborhood

Let $f:M' \to M$ be a map between smooth manifolds. Let $S \subset M$ be a submanifold, and let $T$ be a tubular neighborhood of $S$, ie. $T$ is diffeomorphic to the normal bundle of $S$ in $M$. If $f$...
25
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2answers
278 views

Origins of Differential Geometry and the Notion of Manifold

The title can potentially lend itself to a very broad discussion, so I'll try to narrow this post down to a few specific questions. I've been studying differential geometry and manifold theory a ...
0
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1answer
41 views

Smooth homotopy between $\Bbb R^2-\{0\}$ and $S^1$

In Tu's book "An Introduction to Manifolds" he defines smooth homotopy as follows. $M,N$ smooth manifolds, two $C^\infty$ maps $f,g:M\to N$ are smoothly homotopic if there is a $C^\infty$ map $F:M\...
3
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2answers
35 views

Suppose $M$ has trivial 1-st de Rham cohomology group. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$?

Let $M$ be a compact oriented smooth $n$-manifold, with $H_{dR}^1(M)=0$. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$? I know that if $M$ is simply-connected, we ...
0
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2answers
57 views

Munkres Problem: Define a $C^{\infty}$ map $f: \mathbb{R}^9 \to \mathbb{R}^6$ such that $O(3)$ is the solution of $f(x)=0$.

On Munkres's book analysis on manifold chap "the boundary of manifold", question 3, says: let $O(3)$ the set of orthogonal matrices, as a subspace of $\mathbb{R}^9$. a) define a $C^{\infty}$ map $f: \...
0
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0answers
15 views

Velocity of a 2-parameter curve

Let $M$ be a manifold and $I,J$ be two intervals on $\mathbb{R}$. Suppose $\alpha:I\times J\longrightarrow M$ a smooth map. It is clear that $s\mapsto\alpha(s,t_0)$ and $t\mapsto \alpha(s_0,t)$ are ...
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1answer
80 views

Topology of the tangent bundle

Let $M$ be a smooth manifold, and let $\pi:TM\to M$ be its tangent bundle. We define the topology of $TM$ by declaring a subset $V$ of $TM$ to be open if and only if $\psi_\phi(V\cap\pi^{-1}(U))$ is ...
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0answers
21 views

Derive geodesic equation

Let $(M,g)$ be a riemannian manifold and $(U,\psi)$ local chart on $M$. If $\alpha=(x^1,\ldots,x^n)$ is a curve on $U$ such that verify: $$\sum_{i,j=1}^n \Big(\frac{1}{2}\frac{\partial g_{ij}}{\...
10
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1answer
97 views

Is a bijective smooth function a diffeomorphism almost everywhere?

Suppose I have $f: M \rightarrow N \in C^{\infty}$ a smooth bijection between $n$-dimensional smooth manifolds. Does it have to be a diffeomorphism except for a set of measure 0? I think the proof ...
0
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0answers
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Regarding Problem 5 on p. 71 of Bredon's Topology and Geometry

Someone else asked this question here long ago. Sadly, no answer was provided. I though the natural approach would be trying to show that $x\rightarrow x^{2}$ is an isomorphism of functionally ...
1
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1answer
50 views

Topology, atlas, smooth manifold

Let $X$ be a set, $n\in\mathbb{N}$ and $((U_i,\phi_i))_i$ a family of subsets $U_i\subseteq X$ with injective functions $\phi_i: U_i\to\mathbb{R}^n$, which hold the following conditions: $\...
2
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1answer
34 views

Is this map smooth?

Let $M$ and $N$ be smooth manifolds and $$f:M\times N\to \mathbb{R}$$ a map. Suppose that the maps $$M\to\mathbb{R},\quad p\mapsto f(p,q_0)$$ $$N\to\mathbb{R},\quad q\mapsto f(p_0,q)$$ are smooth for ...
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0answers
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Relation between nonorientability of the Möbius strip and the Möbius bundle

There are two ways in which the open Möbius strip $M$ is related to orientability: $M$ is nonorientable as a manifold; $M$ is the total space of the nonorientable line bundle $M \to S^1$. Is there ...
3
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0answers
47 views

Does $S_1\subseteq \overline{S}_2-S_2$ $\implies$ $\dim S_1<\dim S_2$?

Question: Let $M$ be a smooth manifold and $S_1,S_2\subseteq M$ two locally-closed submanifolds (i.e. they are open in their closure). If $$S_1\subseteq\overline{S}_2-S_2,$$ is it true that $$\...
3
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1answer
26 views

Obtaining embedding from geodesic

Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
3
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1answer
43 views

Normal bundle of an embedding of a parallelizable manifold into a parallelizable manifold

This question is motivated by an observation of Milnor. Theorem: Let $M^m$, $N^n$ be parallelizable smooth manifolds, and $i:M\to N$ an embedding. If $n>2m$, the normal bundle is trivial. The ...
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0answers
22 views

Difference between critical and singular point

I'm sorry if this has been asked before, it sounds like it should be obvious but is there a difference between critical and singular points of a smooth map between manifolds? In several books and ...
3
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1answer
56 views

What is the definition of the dimension of an algebraic manifold?

I have a very basic question. It says on Wikipedia that an algebraic manifold is an algebraic variety which is also a manifold. So suppose I have an algebraic manifold $V$ which is an affine variety ...
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1answer
41 views

Smooth structure on open subsets of manifolds

Let $X$ be a smooth manifold and $U\subseteq X$ open. Define a canonical smooth structure on $U$, for which the embedding $U\to X$ is smooth. Hello, I want to solve this task. My try was as follows:...
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2answers
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Is $T_pM$ isomorphic to $T_{F(p)}N$? [on hold]

Let $M$ and $N$ be two smooth n-dimensional manifolds and $F:M\to N$ be a diffeomorphism. Is it true that $F_{*p}:T_pM\longrightarrow T_{F(p)}N$ is an isomorphism?
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1answer
31 views

Can we define flat connection on any given smooth manifold?

For example, a sphere $S^2$ in $\mathbf{R}^3$ is apparently not flat with respect to the Euclidean connection, but can we define a flat connection and thus with affine charts on $S^2$?
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Existence of submersions between manifolds

I have a ton of problems, where I need to prove (or disprove) the existence of submersions between given manifolds. I will give you some examples, and hopefully I can learn the techniques to solve all ...
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0answers
25 views

Existence of map between projective planes.

I'm struggling to solve this problem, any indications would be appreciated. Is there an application $f : \mathbb RP^3 \to \mathbb RP^1$ of class $\mathcal C^3$ such that $f^{-1}(p)$ is the union of ...
0
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0answers
26 views

Derivative of projection

Let $\Omega \subset \mathbb{R}^n$ be a limited domain of class $C^\infty$ (open, connected) and $\varepsilon > 0$ small such that $$ \Omega_\varepsilon = \{y \in \overline{\Omega} : d(y, \partial \...
3
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2answers
74 views

Defining a Riemannian metric

Let $M$ be a smooth manifold. I have seen a Riemannian metric be defined in many ways: A smooth choice of an inner product $g_p:T_pM\times T_pM\to\mathbb{R}$ which is symmetric and positive-definite,...
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0answers
29 views

Smooth structure in reconstruction theorem

Let $M,F$ be smooth manifolds, $\{U_i:i\in I\}$ an open cover of $M$ and a cocycle $\{t_{ij}:U_i\cap U_j\to\mathrm{Diff}(F)\}$. In almost any book which discusses fibre bundles, one can find the ...
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1answer
30 views

Do these assumptions on a mapping ensure it is a diffeomorphism?

$A\subset \mathbb{R}^m$ is an open and bounded set, $f:A\longrightarrow \mathbb{R}^n$, $m\leq n$, is injective, continuously differentiable and its Jacobian matrix has full rank on $A$. Does this ...
0
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1answer
30 views

How to prove that a space is not a differential manifold?

Given a box (the surface of a cubic) in R^3 space, can I give a smooth structure on it to make it a differential manifold?