For questions about smooth manifolds.

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2
votes
1answer
12 views

The definition of a differentiable vector field on a manifold

I have a question regarding the following section from M. Spivak's Calculus on Manifolds: Let $M$ be a $k$-dimensional manifold in $\mathbb{R}^n$ . . . . . . Suppose that $A$ is an open set ...
3
votes
0answers
25 views

Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...
4
votes
1answer
48 views

Redundance of the Smoothness of the Inversion Map in the Definiton of a Lie Group.

$\DeclareMathOperator{\inv}{inv}$ I am trying to understand the proof of the following from this document: Let $M$ be a smooth manifold which admits a group structure such that the multiplication ...
5
votes
1answer
47 views

Real Manifold … Complex Coordinates?

I'm working in an earlier edition of John Lee's book on smooth manifolds, and he has a number of problems where he represents a real manifold using complex variables. For instance in chapter 3 ...
1
vote
1answer
27 views

Extension Lemma for Functions on Submanifolds

The following lemma is my question. (cf GTM218, Introduction to Smooth manifold) I can prove (b) using partion of unity as follows: $Proof$ for any $p \in S$ choose a slice chart $W_p$ centered at ...
1
vote
1answer
34 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
1
vote
0answers
36 views

A problem possibly using the technique which has been used to prove the Whitney Embedding Theorem.

Problem. (1) Suppose $F:S^1 \to \mathbb{R}^3$ is a smooth immersion (i.e. $dF$ is nowhere zero). Prove that there is a unit vector $\mathbf v$ in $\mathbb{R}^3$ such that $\pi_{\mathbf v} \circ F : ...
0
votes
1answer
28 views

Confused with The Transversality Theorem when all manifolds are boundaryless

In Guillemin-Pollack's book Differential Topology, the Transversality theorem states that The transversility Theorem. Suppose that $F:X \times S \to Y$ is a smooth map of manifolds, where only $X$ ...
2
votes
1answer
23 views

Expressing a differentiable map and curve in a parametrization

This is a question mainly about notation that I just cannot seem to understand. I'm reading Do Carmo's book "Riemannian Geometry" on page 7. Here is some context: (Here, $\alpha : (-\varepsilon, ...
1
vote
0answers
14 views

What is an “essential 2-sphere” in a 4-manifold?

I am aware what an essential 2-sphere in a 3-manifold is. But in several Articles by Fintushel and Stern essential 2-spheres in 4-manifolds occur. The articles are: "Immersed Spheres in 4-manifolds ...
2
votes
0answers
31 views

Try to use Homogeous space Characterize the space of all lines in the plane .

The problem is just to gives a smooth manifold structure of all straight lines in $\mathbb{R^2}$ ( not just those which pass through the oringin).Moreover, identify it with a well-known manifold! My ...
0
votes
1answer
30 views

Special linear group as a submanifold of $M(n, \mathbb R)$

I have that $SL(n,\mathbb{R})$ is an embedded submanifold of dimension $n^2-1$ in $GL(n,\mathbb{R})$, and I know that $T_XGL(n,\mathbb{R})$ is isomorphic to $M(n,\mathbb{R})$ for all $X \in GL(n , ...
2
votes
0answers
28 views

determinant of general linear group

I know that for the general linear group, the coordinate derivatives of the determinant function $\det:GL(n,\mathbb{R})\to \mathbb{R}$ are \begin{equation*} \frac{\partial}{\partial X^i_j}\det X=(\det ...
-3
votes
0answers
16 views

Submanifold and tangent space [closed]

Suppose $M_1$ and $M_2$ submanifold $N$ such that for all $p \in M_1 \cap M_2$ has $T_ {p} M_1 + T_ {p} M_2 = T_ {p} N$. Show $M_1 \cap M_2$ is a submanifold of $N$.
6
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0answers
68 views
+50

Symplectic reversing diffeomorphisms

Let $(M,\omega)$ be a compact symplectic manifold. Is there a diffeomorphism $f$ on M with $f^{*}\omega =-\omega$?
2
votes
0answers
17 views

How to verify F-relatedness?

This question is from Lee's Introduction to Smooth Manifolds p182. I would like to verify the following vector fields are F-related using two ways, i.e. confirming either $dF_p(X_p)=Y_{F(p)}$ for ...
1
vote
1answer
23 views

Diffeomorphism to tangent space

I had to solve the following problem. Let $M$ be a differenciable $m$-manifold, which admits a global base of differianciable vector fields $\{X_1,\ldots,X_m\}$. This means $\{X_1(p),\ldots,X_m(p)\}$ ...
0
votes
1answer
21 views

Supporting lines of closed Jordan curve

Given a simple closed Jordan (i.e. continuous) curve $\gamma:[a,b]\to\mathbb{R}^2,\ \gamma([a,b])=C_\gamma$, how can I prove that $D_\gamma$ (the set of all interior points of $\gamma$ toghether with ...
4
votes
4answers
74 views

Can someone illustrate the definition of manifold with a simple example?

In my text the definition of a differential manifold is given as follows: A subset $M$ of $\mathbb{R}^n$ is a $k$ dimensional manifold if $\forall x \in M$ there are open subsets $U$ and $V$ of ...
1
vote
0answers
43 views

Are $\mathbb{CP}^{n}$ and $\mathbb{RP}^{2n}$ diffeomorphic?

I understand that they are homeomorphic but couldn't find a proof that they are diffeomorphic. If they are diffeomorphic and if the proof is simple enough, I would imagine it would look like the ...
1
vote
2answers
48 views

Dual Bundles' Local Trivializations Confusion

$$\newcommand{\R}{\mathbb R}$$ Let $\pi:E\to M$ be a rank $k$ smooth vector bundle over a smooth manifold $M$. I will below describe how to form the dual bundle, wherein lies my question. Let ...
1
vote
0answers
38 views

Existence of Immersion of a manifold in Euclidean space?

I am trying to prove the following claim which I saw in some paper: Let $M$ be an $n$-dimensional smooth, oriented, simply connected manifold, which is homeomorphic to a bounded subset of ...
3
votes
1answer
28 views

Smooth action of a lie group on a manifold?

Let $G$ be a Lie group and $M$ be a $C^\infty$ manifold. My textbook defines a differentiable action of $G$ on $M$ as a map $$G\times M\longrightarrow M, (g, p)\longmapsto g\cdot p,$$ such that: (i) ...
0
votes
1answer
59 views

pullback of continuous maps of manifolds

I'm trying to prove the following: (a) If $X, Y$ are smooth manifolds, then the map $\psi:X\to Y$ is smooth $\Leftrightarrow$ $\psi^*(C^\infty(Y))\subseteq C^\infty(X)$ (b) If $\psi:X\to Y$ is a ...
2
votes
0answers
27 views

$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
3
votes
0answers
37 views

Exercise 3.3 Riemannian Manifolds an Introduction to Curvature

STATEMENT: Let $\gamma(t)=(a(t),b(t)),t\in I$(an open interval), be a smooth injective curve in the $xz$-plane, and suppose $a(t)>0$ and $\dot{\gamma}(t)\neq 0$ for all $t\in I$. Let $M\subseteq ...
0
votes
1answer
28 views

Manifold arising from particular proof of Hairy Ball Theorem

Background, aka considerations to find my actual question In Geometry three, at the end of the last lesson, we sketched a proof of the famous Hairy Ball Theorem. The proof goes as follows. Lemma: ...
-1
votes
0answers
86 views

Is a line with all points 'doubled" a differentiable manifold?

The line with two origins is $ X=\mathbb{R}∖\{0\}∪\{0',0''\}$, that is X is the union of the reals minus 0, and two points. Let, $$U_a=(−a,0)∪{0'}∪(0,a)$$ $$V_a=(−a,0)∪{0''}∪(0,a)$$ where $a>0$. ...
2
votes
1answer
31 views

If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
3
votes
1answer
69 views

If a Subset Admits a Smooth Structure Which Makes it into a Submanifold, Then it is a Unique One.

$$ \newcommand{\wh}{\widehat} \newcommand{\R}{\mathbf R} \newcommand{\mr}{\mathscr} \newcommand{\set}[1]{\{#1\}} \newcommand{\inclusion}{\hookrightarrow} \newcommand{\vp}{\varphi} $$ I am trying to ...
3
votes
1answer
54 views

Exercise 2.3 Lee's Riemmanian Manifolds

Statement: Suppose $M\subseteq \tilde{M}$ is an embedded submanifold. a)If $f$ is any smooth function on $M$, show that $f$ can be extended to a smooth function on $\tilde{M}$ whose restriction to ...
1
vote
1answer
27 views

Smooth function from function with singularity

Having an application $f:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{C}$ in $C^{\infty}((t_0-\varepsilon, t_0+\varepsilon))$ with $f(t)=0\Leftrightarrow\ t=t_0$ and knowing that: $\exists\ ...
2
votes
1answer
69 views

An interpretation of $ \frac{\partial^{2}}{\partial x^{2}} $.

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...
1
vote
1answer
31 views

Degree of smooth maps are equal $\Rightarrow$ homotopic

It is an easily proven theorem that if $f,g:M\to N$ are smooth maps that are homotopic maps between compact, connected, oriented, smooth manifolds of dimension $n$, then $\deg f=\deg g$. I was ...
0
votes
0answers
24 views

Prove that orthogonal n x n matrices form a $C^1$ surface of dimension n(n-1)/2 in $\mathbb{R^n}^2$ [duplicate]

Consider the function $F:Mat_n → Sym_n$ defined by the formula $F(A) = A∗A$. $Mat_n$ denotes the vector space of n × n matrices with real entries, while $Sym_n$ denotes the vector space of symmetric ...
0
votes
1answer
22 views

The definition of C^r Structural Stability

I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate. Would the definition ...
2
votes
1answer
36 views

Explaining problem in Gadea's “Analysis and Algebra on Differentiable Manifolds”

I have a lot of trouble trying to explain to myself what the author did in problem 1.102 (the answer is in the link): Let $TM$ be the tangent bundle over a differentiable manifold $M$. Let ...
7
votes
2answers
83 views

Top degree de Rham cohomology determines an orientation

Let $M^n$ be a smooth, compact, orientable, connected manifold. We know then that $H^n_{dR}(M^n)\simeq \mathbb{R}$ by the map $[\omega]\mapsto \int_{M^n} \omega$. I was wondering if, given an ...
-1
votes
2answers
29 views

How can I show that the function is smooth?

I got an assignment which I just can't find the right way to solve. It goes like this: Let $\Omega \in R^n$ be a domain and $b_1,...,b_n:\Omega \to R$ smooth mappings (or functions, don't know the ...
4
votes
1answer
42 views

Nonorientable manifolds being a boundaries

I will say that a manifold (smooth, compact, without boundary) is itself boundary if there is some smooth compact manifold $W$ with boundary, such that $M=\partial W$. I'm interested in nontrivial ...
1
vote
0answers
37 views

De Rham cohomology of $T^n$ using Künneth formula and Chevalley-Eilenberg theorem.

I want to calculate $H^*(T^n)$ with ring structure using both of these methods. Künneth formula gives $$ H^p(T^n)=H^p(S^1\times T^{n-1})=\bigoplus_{i+j=p}H^i(S^1)\otimes H^j(T^{n-1}) $$ for each ...
3
votes
1answer
40 views

Find a surface that has positive constant curvature that is not open subset of sphere

Can some one find a surface that has positive constant curvature that is not open subset of sphere. I know every connected and compact surface with positive constant curvature is sphere. I need ...
1
vote
1answer
65 views

Boundary of the boundary of a manifold with corners

A point of a manifold with corners is a boundary point by definition if one of its coordinates is $0$ by some (hence in all) chart with corners (see here). In the same page one can read: The ...
2
votes
1answer
38 views
+100

Reference request: infinite-dimensional manifolds

The following books develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of Global ...
1
vote
0answers
27 views

Glueing smooth functions give a smooth function if reparametrized

Given $\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$ be a $C^{1}$ application, with $\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and $$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 ...
2
votes
1answer
43 views

Tangent space change of bases

Let $M\subset \Bbb{R}^m$ be a $k$-dimensional differentiable submanifold. Let $(\varphi, U)$ and $(\psi, V)$ be two charts for $p\in M$ with $\varphi(x)=p$ and $\psi(y)=p$. Then we have two bases for ...
0
votes
0answers
14 views

Area between two curves as manifold with boundary

Let $U \subset \mathbb{R}^n$ be open set, $F,G:U \to \mathbb{R}$ smooth function such that $F(x)<G(x)$. We define: $$\Omega=\{(x,y) \in U \times \mathbb{R}:G(x) \leq y \leq F(x)\}$$ I would like ...
1
vote
0answers
12 views

About smooth function whose reciprocal is also smooth

I know that for exponential functions $e^{at}$, or functions like $1/t^m$, t>0, both they and their reciprocals are smooth. Could you please give me more classes of smooth functions, or Analytic class ...
1
vote
0answers
14 views

Any good resources for learning about the moduli space of symplectic structures on a given manifold?

What I can find on the subject are the papers by Fricke and the Habermanns: http://www.researchgate.net/publication/227336993_On_the_geometry_of_moduli_spaces_of_symplectic_structures and ...
-1
votes
0answers
15 views

How to find the limit of this flow $\lim_{t \rightarrow \infty} \phi^i_t(p)$ defined by a vector field?

Could anyone help me with how to begin to solve the following problem? We have: Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ ...