For questions about smooth manifolds.

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Differential geometry

If we have integrable distribution D of rank k on a manifold, and we have k functions which are constants on the associated integral manifolds. can we glue together these functions to obtain global ...
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1answer
30 views

How to compute $[\dot c, X]$ on a manifold?

Consider a smooth curve $c : [0,1] \to M$ and $X \in \mathcal X (M)$. How can one obtain an explicit formula for $[\dot c, X]$? I know the theoretical approach: for every $t \in [0,1]$ there exist a ...
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0answers
17 views

Definition of ruled manifold

What is the definition of a ruled manifold? I'm searching in internet but I can't find the definition of such manifold.
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32 views

How to find the tangent space of a general submanifold?

Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general ...
4
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2answers
32 views

Is every diffeomorphism an element of a one parameter group of diffeomorphims?

I understand that a smooth vector field on a manifold $M$, generates a "flow"/one parameter group action, lets say $\sigma(t,s): \mathbb{R} \times M \rightarrow M$, and $\sigma_t: M \rightarrow M$ ...
0
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1answer
33 views

Proof that the special linear group $\mathrm{SL}(n,\mathbb{R})$ is a smooth manifold

This is my definition of a smooth manifold I am supposed to work with: Let $\mathcal{M} \subseteq \mathbb{R}^{n}$. The set $\mathcal{M}$ is a $k$-dimensional smooth submanifold of $\mathbb{R}^{n}$ ...
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0answers
25 views

Change of variables for path integral.

Let $G=C^\infty([0,1];\mathbb{R}^d)$ be smooth paths, then for the path $A\in G$, consider the translation operator from $G$ to itself $T_A:G\to G$ $$T_A(g)(t):=g(t)+A(t).$$ Does there exist a ...
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1answer
28 views

Darboux coordinates on projective spaces

I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of ...
0
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2answers
29 views

Does the forgetful functor from smooth manifolds to sets preserve colimits?

It is easily shown that the forgetful functor $F: \mathbf{Man} \to \mathbf{Set}$ preserves limits ($F$ is representable), but does it preserve colimits? It certainly preserves all examples of ...
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0answers
27 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
0
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0answers
23 views

Value preservation along geodesics.

Given a differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$, where $n>m$. We define the graph of $f$ $ W_f = \{ (x,y) | x \in \mathbb{R}^n, y \in \mathbb{R}^m,y=f(x) \}. $ Given two ...
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35 views

$C^k$-maps between manifolds is a sheaf?

I know that the functor from the category of open subsets of a manifold $M$ to the Sets, taking an open set $U$ and associating to it the collection of $C^k$ maps to $\mathbb{R}$ is a sheaf. My ...
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1answer
21 views

Are the symplectic leaves of a Poisson manifold submanifolds?

In "Introduction to Mechanics and Symmetry" by Marsden and Raţiu it is written, in chapter 10, page 347, example b, that "[s]ymplectic leaves need not be submanifolds". In "Lectures on Poisson ...
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24 views

Show that the following two collections of charts are equivalent atlases on $S^1$. [closed]

Let $S^1 \subset \Bbb R^2$ denote the unit circle. Let $\mathcal A = \{(U_1, \varphi_1), (U_2, \varphi_2)\}$ denote the following collection of ($1$-dimensional) charts on $S^1$: $U_1 = \{(\cos ...
3
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1answer
20 views

Finding a domain of an integral curve of a vector field

Studying Morse theory, I am stuck on some problem. Let $M$ be a compact smooth manifold, and $f$ is a smooth real-valued function on $M$. Choose a Riemannian metric $g$ on $M$, let $X$ be the vector ...
2
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0answers
26 views

Wedge product of $k$-forms

I'm studying smooth manifolds with Lee's book. He defines a $k$-form on a manifold $M$ as a section $M \to \Lambda^k M$ (where $\Lambda^k M = \bigsqcup_{p\in M} \Lambda^k T_pM$ is the smooth vector ...
4
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0answers
28 views

Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, ...
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0answers
12 views

What is the name of these elliptic surfaces E(n)?

I am referring to the elliptic surfaces $E(n)$, with fibration over $\mathrm{C}\mathbb{P}^1$. They are common in 4-manifold theory and complex geometry. See for example Chapter 7 in Akbulut`s ...
2
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0answers
22 views

Connection on $T\mathbb{R}^n$

Let $\nabla$ be a connection on the tangent bundle $T\mathbb{R}^n$. Now, I need to show that there exist smooth function $C_i: \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$, $i=1,\dots ,n$ such ...
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0answers
24 views

Proving smooth map between smooth manifolds is constant based on push forward being zero

I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following: Let M, N be smooth manifolds, ...
4
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1answer
25 views

Study of algebraic structures analogous to the ring of smooth functions and module of vector fields

$\newcommand{\Ga}{\Gamma}$ Let $M$ be a smooth manifold. $\Ga(TM)$ is a module over the ring of smooth (real) functions (which is also an algebra, and denoted by $C^{\infty}(M)$). Also, each $X \in ...
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1answer
35 views

Differential of a smooth function on a manifold

Let $S^2$ be the sphere in $\mathbb{R}^3$, let's consider the (inverse) chart $\varphi$ $$x=\sin v\cos u, y=\sin v \sin u, z=\cos v$$ now let $f$ be the restriction of the linear aplication of ...
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1answer
16 views

Smooth actions and stabilizer

Let $G$ be a Lie group acting smoothly on a smooth manifold $M$. We consider a point $x$ of $M$ and $g$ an element of its stabilizer $G_x$. The smooth diffeomorphism $\theta_g$ of $M$ defined by ...
0
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1answer
50 views

Guillmin & Pollack's Definition of a Manifold

In Guillemin and Pollack's Differential Topology, they (roughly speaking) define a manifold to be a space which is locally diffeomorphic to Euclidean space. Now this is obviously not the full ...
5
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1answer
66 views

Lie groups where $x \mapsto x^2$ is a diffeomorphism?

In every Lie group $G$ the function $x \mapsto x^2$ is a local diffeomorphism in a neighbourhood of the identiy. (This is because its differential is: $v \mapsto 2v$ when considered as a map from ...
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0answers
45 views

Well definedness of Lie bracket (coordinates approach)

For practice I wanted to define Lie bracet in terms of coordinates and show that this definition is independet of the choice of coordinates. So I started with the definition ...
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0answers
17 views

Criterion for semi-simplicity of Lie algebra generated by vector fields

Suppose I have a finite collection of smooth vector fields $V:=\{V_1,...,V_k\}$ on a smooth manifold $M$. Moreover suppose that the Lie algebra $g$, generated by $V$ (where the Lie bracket is defined ...
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0answers
46 views

The limit of a uniform convergent sequence of isometries is an isometry (problem 6-3 of Lee's “Riemannian manifolds”)

I'm trying to prove the following theorem: let $f_n : M \to N $ a sequence of isometries of Riemannian manifolds that converges uniformly to a function $f:M \to N$: prove that $f$ is an isometry too. ...
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1answer
20 views

Can we always choose an isometric slice chart for a submanifold of $\mathbb{R}^n$

Let $S$ be a submanifold of $\mathbb{R}^n$. Let $p \in S$. Is there an isometric slice chart for $S$ in $\mathbb{R}^n$, around $p$? i.e, I am asking whether there is a diffeomorphism $\phi$ from some ...
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Show a set is $k$-submanifold

Suppose you have $n-k$ differentiable functions $F_i, \ i=1, \dots, n-k$ on the space $\Bbb R^k \times \Bbb R^{n-k}$ and the system $$F_1(v_1, \dots, v_k, x_{k+1}, \dots, x_n)=0 \\ F_2(\dots)=0 \\ ...
0
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1answer
27 views

Defining submanifolds without charts

In his elegant 2012 introduction to smooth manifolds, Nigel Hitchen minimizes his reliance on charts. In stating what it means for a manifold $M$ to be a smooth submanifold of $N$, for example, he ...
0
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1answer
28 views

Quotient of a manifold

Suppose we have a manifold $M$, and a connected submanifold $N$. We can make the quotient $\frac{M}{N}$, which send $N$ to a single point. Now, there are known restrictions on $N$ such that ...
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1answer
45 views

Differential of Sum of Two Functions is Sum of Differentials

Let $M$ be a smooth $n$-manifold and $f, g:M\to \mathbf R^n$ be smooth functions on $M$. Let $p$ be a point on $M$. I want to show that $d(f+g)_p=df_p+dg_p$ without passing to a chart about $p$. ...
0
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1answer
33 views

Normal coordinates and the metric matrix

While trying to follow and check the proof of Theorem 1 in this work on manifold averaging I reached the notion of normal coordinates. An important property is that the metric tensor at a point ...
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0answers
30 views

convergence in the $C^r$-topology on $C^r(M,N)$ for $M$, $N$ compact manifolds

Let $M$ and $N$ be smooth (finite dimensional) manifolds without boundary. On the set $C^r(M,N)$ we choose the compact-open $C^r$-topology. This topology is defined as follows (I take the definition ...
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1answer
18 views

Quotient by a distribution/foliation?

Suppose we have a (smooth) manifold $M$, and an integrable smooth distribution $\Delta$ on $M$. Somewhere, I read that we can define a natural map $\pi:M \rightarrow \frac{M}{\Delta}$. First of all, ...
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0answers
39 views

Connection with zero torsion and curvature

I remember encountering the following theorem many, many time ago: if a connection $\nabla$ has $R = 0$ and $T = 0$ then... The problem is that I cannot remember the conclusion. Was it that the ...
1
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1answer
29 views

Approximation of piecewise smooth curves with same-lenght smooth curves in Riemannian manifolds

Let $M$ be a Riemannian manifold, and let $\gamma : [a,b]\to M $ be a piecewise smooth curve. Then, using Whitney's theorems, it can be proved that $\gamma$ is homotopic (by a homotopy relative to $a$ ...
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1answer
116 views

Is a general smooth rescaling of a complete vector field itself complete?

$\newcommand{\Ga}{\Gamma}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\til}{\tilde}$ $\newcommand{\M}{M}$ $\newcommand{\ep}{\epsilon}$ $\newcommand{\brk}[1]{\left(#1\right)} $ ...
0
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1answer
43 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
0
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1answer
34 views

The identity map from $\mathbb{\bar B}^3$(as a subset of $\mathbb{R}^3)$ into $\mathbb{\bar B}^3$(as a smooth manifold with boundary) is not smooth?

Let $U$ be the open rectangle $(0, \pi) \times (0,2 \pi) \subset \mathbb{R}^2 $ and let $X : U \rightarrow \mathbb{R}^3$ be the following map: $$X(\varphi , \theta)=(\sin \varphi \cos \theta , ...
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1answer
56 views

Prove S is a manifold.

At the moment the definition of a manifold I'm working with is that of a set $X$ equipped with a smooth atlas $A$. I want to prove that $\{(a,b)\in \mathbb{R}^n\times\mathbb{R}^n \mid a\cdot a=b \cdot ...
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1answer
29 views

Show that a submersion is open

Let $M,N$ smooth manifold of dimension $m$ and $n$ respectively. $f:M\longrightarrow N$ a submersion. Show that $f$ is open. My proof Let $W\subset M$ an open. Let $p\in W$. By the theorem of ...
5
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1answer
67 views

Smooth manifold which is a group, but not a Lie Group

Are there (preferably non-pathological) examples of smooth manifolds, which are groups, but not Lie groups? In books one can see plenty of examples of Lie groups, but I haven't seen an example where ...
2
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0answers
58 views

When are differential forms related by a base space automorphism?

Let $w$ and $u$ be nowhere-vanishing smooth differential forms fields of degree $n$ on a smooth manifold $M$ (aka smooth sections of $\Omega^n(M)$). When does there exist an automorphism $f: M \to M$ ...
0
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1answer
21 views

Example of a diffeomorphism that preserves orientation but not homology orientation

I wondered if there is someone who knows an example as described in the title. If $M$ is a closed oriented smooth 4-manifold, then a homology orientation on $M$ is a choice of orientations on ...
2
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1answer
29 views

Construct a smooth function on $T^2$ that has exactly three critical points

By the results in Morse theory, a smooth function on $T^2$ has at least three critical points, and at least one of them is degenerate. I'm asked to construct a smooth function that has exactly three ...
4
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0answers
268 views

Integral over “infinitesimal” transformed manifold

Suppose I have a $d$ dimensional manifold $\mathcal{M}$ on which I want to perform the integral of a certain function $\mathcal{f}: \mathcal{M} \longrightarrow \mathbb{R}$ I will have then ...
0
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0answers
20 views

regular submanifold

I have some problem to understand this . if this means of regular submanifold is a subset $S$ of a manifold $N$ of dimension $n$is regular submanifold of dimension $k$ if for every $p\in S$ there ...
4
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1answer
45 views

Redundancy in the definition of vector bundles?

In John Lee's classic Introduction to Smooth Manifolds, the following definition of vector bundle is given. Definition. Let $M$ be a topological space. A (real) vector bundle of rank $k$ over $M$ ...