For questions about smooth manifolds.

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-1
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0answers
25 views

a local maximum of a $C^∞$ function $f:M→R$ is a critical point of $f$ [on hold]

A real-valued function $f:M→R$ on a manifold is said to have a local maximum at $p∈M$ if there is a neighborhood $U$ of $p$ such that $f(p)≥f(q)$ for all $q∈U. $ a) We know if a differentiable ...
15
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4answers
67 views

group operations are smooth in $\text{SL}(n, \mathbb{R})$

I am told the following reason as to why group operations of multiplication and inversion are smooth on $\text{SL}(n, \mathbb{R})$. Multiplication is smooth because the matrix entries of a product ...
2
votes
1answer
44 views

Differentiable manifolds that allow isometric transition maps.

What is the class of differentiable n-dimensional manifolds that allow a differential structure, in which all transition maps are isometric? Note that isometric must be overlapping pieces of charts ...
7
votes
1answer
32 views

defining $C^\infty$ structure on finite-dim vector space, homeomeomorphism to tangent bundle, such that independent of choice of bases

If $V$ is a finite dimensional vector space over $\mathbb{R}$, how would I go about defining a $C^\infty$ structure on $V$ and a homeomorphism from $V \times V$ to $TV$ which is independent of bases? ...
1
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0answers
11 views

An example of regular singular points

I was reading a book on differential geometry and after the intro to the concepts of regular singular points I came across an example under it: The set $M:=\{(x^2,y^2,z^2,yz,zx,xy)|x,y,z\in ...
2
votes
1answer
24 views

Quick question: Map being smooth vs Graph being submanifold of the product space [on hold]

Is $f:X\rightarrow Y$ smooth if and only if the graph $\Gamma_f$ is a closed submanifold of $X\times Y$? Thank you very much.
12
votes
3answers
247 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
2
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0answers
36 views

Help needed in understanding a question to show that $M$ is a smooth manifold

Let $\rho : \mathbb{Z} \hookrightarrow GL(\mathbb{R}^r)$ be a representation. Consider $\mathbb{Z}$ as a subgroup of $(\mathbb{R},+)$ in the usual way. Define $M$ as the quotient of $\mathbb{R} \times ...
1
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1answer
50 views

Representation of $n$ form and $n-1$ form in local coordinates

Let $M$ denote a smooth $n$-dimensional manifold. (a) Let $\phi$ denote a smooth $n$ form which is nowhere zero. Show that every $x_{0} \in M$ has a neighborhood on which we can find smooth local ...
5
votes
1answer
41 views

identity map is not diffeomorphism, $x^3$ is a diffeomorphism [closed]

Consider the real line $\mathbb{R}$ the two following differentiable structures: 1) $(\mathbb{R}, f_1)$ where $f_1(x) = x$. 2) $(\mathbb{R}, f_2)$, where $f_2(x) = x^3$. How do I demonstrate that: ...
7
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3answers
192 views

“Drawable” Examples of Vector Bundles

I'm looking for examples of vector bundles that can be easily drawn or "illustrated" on a whiteboard for a talk I am giving. I know of a couple simple examples that I could use: When our base ...
2
votes
1answer
39 views

Smooth function on intersection is the difference of two smooth functions

I am trying to understand a proof from Loring W. Tu's An introduction to Manifolds. In order to prove Proposition 26.2, The author must show that if $\{U, V\}$ is a open cover of a manifold $M$ and ...
2
votes
1answer
18 views

How to show $(d\pi^{-1})_{\pi(y)}\circ (d\pi)_x:T_xS^n\longrightarrow T_y S^n$ reverses orientation for $n$ even?

Let $\mathbb R\mathbb P^n$ be the quotient manifold $S^{n}/R$ where $R$ is the equivalence relation given by: $$xRy\Leftrightarrow y=x\ \textrm{or}\ y=-x.$$ We know the canonical map ...
0
votes
1answer
35 views

Prove $O(n)$ is compact

I have to prove $O(n)$ is compact, I know if I can prove it bounded and closed in $\mathbb{R^{n\times n}}$, I will be done. But how to check boundedness and closed ness. For closedness I would like to ...
5
votes
1answer
46 views

$H_2(M)$ is free abelian for any simply connected $4$-manifold

In Naber's book "Topology, Geometry and Gauge Fields. Foundations", it is stated that for each $4$-manifold $M$ which is smooth, closed, connected and simply connected we have $H_0(M) = H_4(M)= ...
0
votes
0answers
14 views

Surjective $\gamma \colon I \to M^1$, $\gamma (t_1)=\gamma (t_2)$ can be extended to a periodic parametrization of $M^1$

Suppose that $\gamma \colon I \to M^1$ is a smooth surjective curve in a Riemannian connected 1-dimensional manifold. Furthermore, suppose that it is parametrized via arc lenght i.e.:$$||\dot ...
1
vote
1answer
29 views

Problem to proof that $SO(3)$ is a differentiable manifold

I and my friend were trying to proof that $SO(3)$ is a Lie Group, and in particular a differentiable manifold. His attempt was to consider a closed ball in $\mathbb{R}^3$, centered in origin and ...
0
votes
0answers
18 views

Poisson bivector on the product of two manifolds

Let $X, Y$ be two manifolds. Let $(U, x_1, \ldots, x_n)$ and $(V, y_1, \ldots, y_m)$ local coordinates of $X, Y$ respectively. A Poisson bivector on $X$ is defined by \begin{align} \pi_X = \sum_{i,j} ...
1
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1answer
27 views

A Smooth map homotopic to a constant map

Q: Let $M^{k}$ be a smooth compact $k$-manifold and let $F:M \rightarrow S^{n}$ be a smooth map, where $n>k$. Prove that $F$ is homotopic to a constant map. Proof: Since $n>k$, by Sard's ...
7
votes
3answers
155 views

Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?

Is there any answer of this question around basic theory of differentiable manifolds?
2
votes
1answer
49 views

Extenting a homeomorphism on subsurface to the entire surface

Suppose you have surface and a subsurface. The complement of subsurface is union of open discs and once punctured open discs. Can all homeomorphisms of the subsurface be extended to homeomorphisms of ...
0
votes
2answers
42 views

How to orient a manifold in the Euclidean space?

I learned that the orientation of a smooth manifold is a smooth choice of an orientation for bases of tangent space. Also I sometimes read that an embedded manifold in $\mathbb{R^3}$ inherites an ...
1
vote
1answer
44 views

Complex structure on a real vector bundle

Let $M$ be a smooth manifold and $\pi:E \rightarrow M$ a real vector bundle and note $E_x:=\pi^{-1}(x), \forall x\in M$. We set a bundle $\text{End}(E)=E\otimes E^*$. Now suppose there exists a smooth ...
2
votes
0answers
19 views

Real vector bundles over $S^1$

Prove that for every real vector bundle over $S^1$ there exist open connected subsets $U_1,U_2 \subset S^1$ with $U_1 \cup U_2=S^1$ such that $E$ is trivial over $U_1$ and over $U_2$. I need an ...
0
votes
1answer
40 views

Smooth structure on $M\cup_f N$?

Let $M$ and $N$ be two smooth manifolds with $$\textrm{dim}(M)=\textrm{dim}(N)=n.$$ Let $U\subseteq M$ and $V\subseteq M$ be two open sets and $f:U\longrightarrow V$ a smooth diffeomorphism. Consider ...
0
votes
0answers
35 views

Space of generalized functions for product manifold

I have a the following definition of a generalized function at a point $m\in M, M$ - a smooth manifold, at hand: it is a linear functional $F: C^{\infty}(M) \to \mathbb{R}$, such that for some $k \in ...
1
vote
1answer
31 views

Atlas on product manifold

If {$(U_\alpha ,f_\alpha )$} and {$(V_i,y_i)$} are $C^\infty$ atlases for the manifolds $M$ and $N$ of dimensions $m$ and $n$, respectively, then the collection {$(U_\alpha \times V_i,f_\alpha \times ...
0
votes
0answers
43 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
3
votes
1answer
181 views

Rank Theorem proof

Let $\phi: M \to N$ be an immersion from smooth manifold $M^m$ into $N^n$ ($\dim M = m$ and $\dim N = n$). Prove there exists smooth charts $(U,h)$ in $M$ with $p \in U$, $h(p) = 0$, and $(V,g)$ in ...
0
votes
1answer
27 views

How do we get from $\mathrm{ker}(Df)$ to $(\nabla f)^\perp$?

A tutorial sheet has the following problem. Find a unit normal vector and a basis for the tangent space of the following smooth manifold $M \subseteq \mathbb{R}^2$ at a point $(a,b) \in M$. ...
2
votes
1answer
55 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
3
votes
1answer
50 views

Is a diffeomorphism's image automatically open?

Sorry if this question is trivial, I am new to smooth manifold theory. Let $\varphi : I \times \mathcal S^{n-1} \to X$ be a diffeomorphism. $I=(0,1)$, $\mathcal S^{n-1}$ is the unit sphere in ...
1
vote
1answer
29 views

Prove without coordinates that covariant derivatives are “almost” related under isometric immersion?

I'm trying to solve this problem: Let $F : (M,g) \to (N,h)$ be an isometric immersion. For any $p \in M$, let $\pi_p$ be the orthogonal projection from $T_{F(p)}N$ to the image of $dF_p : T_pM \to ...
1
vote
1answer
22 views

Confusions on Pullback

I am reading Lee's book Introduction to Smooth Manifold. I am confused about the conception of pullback in this book. Assume $F:M\to N$ is a smooth map. We can define a pullback $F^*$ at $p\in M$ ...
2
votes
1answer
29 views

A function from a smooth manifold with boundary to $[0,\infty)$

Suppose $M$ is a smooth manifold with boundary, show that there exists a smooth function $f: M \rightarrow [0, \infty)$ such that $\partial M = f^{-1}(0)$. My attempt is that given a chart ...
1
vote
2answers
36 views

How to show $X=\{p\in M: \textrm{ker}(df(p))=\{0\}\}$ is open in $M$?

Let $M$ and $N$ be two smooth manifolds and $f:M\longrightarrow N$ a $C^\infty$ map. We say $f$ is an immersion at $p\in M$ if $df(p):T_pM\longrightarrow T_{f(p)}N$ is injective. How can I show the ...
2
votes
3answers
80 views

Constructing a vector bundle using Vector bundle construction lemma

Given are: an open cover of $\{U_\alpha\}_{\alpha\in A}$ of a smooth manifold $M$. smooth maps $\tau_{\alpha\beta}\colon U_{\alpha}\cap U_{\beta}\rightarrow \text{GL}(k,\mathbb{R})$ with ...
1
vote
1answer
23 views

Smoothness of a particular function in two variables

I cannot understand why this doesnt work for $v \neq 0$ and $u=0$? I think it may be to do with my lack of understanding of why $v=0$ and $u \neq 0$ works which I believe has come from the fact that ...
5
votes
1answer
91 views

Does the compact manifold $f=0$ resist small perturbations?

Suppose we have a compact manifold of the form $\left\{f=0\right\}$ where $f:\mathbb R^n\to\mathbb R$ is a smooth Morse function. I am interested in showing that the manifold is topologically ...
1
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0answers
18 views

Reference request: foliations

I am looking for a gentle introduction to foliations for smooth manifolds, but I have a hard time finding a textbook explaining this notion. Wikipedia's links are also to articles. Is there any ...
1
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0answers
52 views

How special are the polynomials amongst the smooth functions?

This is a naive question, so perhaps the answer will be made obvious by the right remark. On a smooth manifold, there is no notion of polynomial (apart from constants). I would like to know if, ...
3
votes
1answer
112 views

How to calculate differential forms on $S^2$ parameterized by stereographic projection?

Suppose that we have the stereographic mapping $\varphi: \mathbb{R}^2\to M$ where $M=S^2-\{(0,0,1)\}$. I've already found that the stereographic parametrization of $S^2-\{(0,0,1)\}$ is given by: ...
2
votes
0answers
44 views

Compute the Jacobian of the following

The question is this: Consider the parabaloid $\{(x,y,z)|z=1-x^2-y^2\}$, let $A$ be the subset satisfying $z>0$. Consider the plane $\pi$ given by $z=1$. The functions $x$ and $y$ act as ...
0
votes
0answers
34 views

$C^{\infty}$-homotopy type of the Moebius band

The Moebius band $N$ has the same $C^{\infty}$-homotopy type of $S^1 \times \mathbb{R}$. What is the explicit expression of the $2$ $C^{\infty}$-homotopies involved ?
0
votes
2answers
45 views

Uniqueness of smooth/symplectic/etc structure

It is well-known that every topological manifold $M$ of dimension $\le 3$ admits a unique smooth structure. That is to say for any choice of atlas on $M$ like $A$ and $B$, the smooth manifolds $(M, ...
0
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0answers
30 views

A surface in $\mathbb{R}^{4}$ diffeomorphic to $\mathbb{S}^{2}$

This is homework so no answers please. The problem is: Show that the surface S given by \begin{matrix}x^{2}=-y\\ 1=y^{2}+s^{2}+t^{2}\end{matrix} is diffeomorphic to $\mathbb{S}^{2}$ My attempt: ...
0
votes
1answer
20 views

Free and proper action

I don't know how to solve this problem. Let G be a Lie group and H a closed Lie subgroup ,that is, a subgroup of G which is also a closed submanifold of G. Show that the action of H in G defined by ...
0
votes
0answers
18 views

Cutting a surface at critical levels produces cylinders

Let $F$ a closed surface with isolated critical points and a homeomorphism $g: F \rightarrow F$ that maps critical levels to critical levels. Let us cut the surface $F$ by the critical levels. Do we ...
2
votes
0answers
50 views

$x^{4}+y^{2}+z^{2}=1$ diffeomorphic to 2-sphere $\mathbb{S}^{2}$

This is homework so no answers please The problem is: $A=\{(x,y,z)\in \mathbb{R}^{3}: x^{4}+y^{2}+z^{2}=1\}$ is diffeomorphic to 2-sphere $\mathbb{S}^{2}$. Any mistakes: Consider ...
3
votes
0answers
33 views

The unit tangent bundle for submanifold $M^{m}\subset \mathbb{R}^{n}$ is a (2m-1)-dim submanifold

This is homework so no answers please Here is the problem: Show that $UM:=\{(x,v)\in T\mathbb{R}^{n}:x\in M^{m}, v\in T_{x}M^{m},|v|=1\}$ is a (2m-1)-dim submanifold of $T\mathbb{R}^{n}$. My ...