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

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2
votes
1answer
19 views

Quick question about covariant derivative

Let $f$ be a function and define $\nabla_X f = X(f)\,\,(1)$, where $\nabla$ is the connection on a manifold and as far as I understand the r.h.s is a function and $X$ is a vector field. I am just a ...
4
votes
1answer
51 views

Is this statement about manifold true? [duplicate]

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
1
vote
0answers
10 views

Extending a function from set without limit points

Problem: Let $D$ be a subset of (smooth) manifold $M$ without limit points in $M$. Let $f \colon D \to \mathbb R$ be any real-valued function. Can $f$ be extended to smooth real-valued function $g ...
1
vote
0answers
13 views

inverse function theorem on manifolds

suppose there are two 3-manifolds(consider them as orthogonal matrices $SL(2,\mathbb R)$), and there is $F:SL(2,\mathbb R)\to SL(2,\mathbb R)$, given by $F(A)=A^3$. Can we apply inverse function ...
4
votes
0answers
25 views

Transversality and homotopic maps

I'm trying to solve some problems in differential topology, and I came across the following: suppose $f:M\times [0,1]\rightarrow N$ is a homotopy, where $M$ is a compact manifold, such that $f_0$ and ...
3
votes
2answers
129 views

Is there a well-defined notion of measure zero on topological manifolds?

We extend the concept of measure zero on manifolds by local parameterization. but in this definition we have to check if it is true for every parametrization. In Guillemin's Differential Topology this ...
1
vote
0answers
10 views

Discrete singularities of $C^k$-functions.

I'm stacked in the following problem: suppose $f:M\rightarrow N$ is a $C^k$ map between $C^k$-manifolds, such that $\dim M=\dim N=n>1$; if the singularities of $f$ are isolated, then the map is ...
0
votes
0answers
18 views

Integrating 2 form over torus

Let $\Bbb M^2 ⊂ \Bbb R^3$ be the torus of revolution obtained by rotating the circle $(x−2)^2 +z^2 = 1$ in the $xz$ plane around the $z$ axis. Consider the orientation on $M$ induced by the ...
1
vote
1answer
26 views

what is the manifold associated with general linear group? [on hold]

It has dimension n^2 but I want to know the exact manifold structure of general linear group.
4
votes
2answers
167 views

How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
1
vote
1answer
27 views

A partition of the unit square such that the quotient space is the Klein bottle

Write down a partition $X^*$ of the unit square $X=[0,1]\times[0,1]$ such that the quotient space is the Klein bottle. I understand the definition of Quotient topology and Partitions, however, ...
1
vote
1answer
32 views

Smooth mapping between manifold such that $\text{Im}(f) \subset \partial N$

Let $f:M \to N$ be smooth such that $\text{Im}(f) \subset \partial N$. Prove that $f$ as mapping $f:M \to \partial N$ is smooth. I've tried to write down $f:M \to \partial N$ as composition of two ...
1
vote
1answer
20 views

Smooth manifold

$A=M\cap N$, $M={(x,y,z\in\Bbb R^3)| x^2+y^2=1}$, $N=(x,y,z)\in \Bbb R^3|x^2-xy+y^2-z=1$. $1$. Is $A$ is smooth manifold? $2$. Find the points of $A$ that are farthest from the origin. This is ...
0
votes
0answers
24 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...
1
vote
1answer
48 views

Using stokes' theorem

$B=\{(x,y), x^2+y^2\le1\} $ is a closed ball and $S=\{(x,y,z), z=x^2+y^2, (x,y)\in B\} $ oriented so that $f:B\to S$ defined by $$f(x,y)=(x,y,x^2+y^2)$$ is orientation preserving. Compute ...
0
votes
1answer
18 views

Diffeomorphism preserves compact support of functions?

Let $M$ and $N$ be two Riemannian manifolds which are diffeomorphic via a $C^k$ map $F:M \to N$. Let $\phi \in C^0_c(M)$ be a continuous function with compact support in $M$. Is it true that its ...
4
votes
1answer
40 views

Can $\mathbb{R}\mathbb{P}^2$ be embedded into an orientable 3-manifold?

We know that $\mathbb{R}\mathbb{P}^2$ cannot be embedded into $\mathbb{R}^3$, but is there an orientable 3-manifold where it is possible?
0
votes
0answers
24 views

Sp(2n) as manifold

How to prove that $Sp(2n)$ is a manifold? We know that $Sp(2n)\subset Gl(2n)$ and $Gl(2n)$ is a manifold. Furthermore $Sp(2n)$ can be described as zeros of $A\mapsto A^TJA-J $, where $J$ is a ...
0
votes
0answers
61 views

Integration over a manifold with boundary (Check).

Assume that $ f: \Bbb{R}^{3} \to \Bbb{R} $ is a smooth function such that $ M \stackrel{\text{df}}{=} \left\{ \mathbf{x} \in \Bbb{R}^{3} ~ \middle| ~ f(\mathbf{x}) \ge 0 \right\} $ is a non-empty ...
1
vote
1answer
25 views

Every submanifold of $\mathbb R^n$ is locally a level set

Is it true a very submanifold $M$ of $\mathbb R^n$ is locally a level set? Given a chart $\phi$ about $p \in M$, how can we construct a smooth function $f$ s.t. $f^{-1}(0)= M \cap U$ for some open ...
1
vote
1answer
40 views

Is the form closed?

$S$ is an n dimensional unit sphere such that $S^n=(x\in \Bbb R^{n+1}: |x|=1)$ with some fixed orientation and $\omega$ is a volume form on $S$. Prove that $\omega$ is closed. Prove that $\omega$ ...
0
votes
1answer
30 views

Homeomorphism between the 1-sphere and a semi-open real interval

I need help with a problem that's troubling me. In Lee's "Introduction to Topological Manifolds" I found this exercise: being given the exponential map $\ a:[0,1[\to\mathbb{S}^{1}$, $\ a(s)=e^{2\pi i ...
2
votes
1answer
59 views

All derivations are directional derivatives [duplicate]

Let $X : C^{\infty}(\mathbb{R}^n) \rightarrow \mathbb{R}$ be a derivation, so i.e. linear and satisfying the Leibniz Rule $$X(fg)=X(f) \cdot g(a)+X(g) \cdot f(a)$$ for some fixed $a \in ...
1
vote
1answer
28 views

Composition of smooth maps between manifolds is smooth

This is a continuation of the problem : Composition of smooth maps. At the moment, I am on the same problem. I am not quite sure of the continuation of the comment '' The point here is another. Are ...
3
votes
1answer
27 views

Vector space operations on fibres of associated bundles.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\text{ad}:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let ...
1
vote
1answer
44 views

Show this is not a manifold with boundary

Consider a curve $\alpha: \mathbb R \to \mathbb R^2$ defined by $t \mapsto (e^t \cos(t), e^t \sin(t))$. Show the closure of $\alpha(\mathbb R )$ is not a manifold with boundary. Denote ...
2
votes
1answer
44 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
3
votes
1answer
56 views

projective space and torus

we defined the projective space as $\mathbb{S}^2$ with opposie side identification and the torus as $\mathbb{R}^2 / \mathbb{Z}^2.$ And now I am concerned with their manifold structure- In fact, I ...
5
votes
1answer
57 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
3
votes
2answers
69 views

Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
1
vote
0answers
12 views

spheres are not simpletic?

Reading some books on diferential geometry, a found that S^2n (with n>1) are not simpletic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not understand this ...
0
votes
1answer
25 views

Unsolved regular value problem in $\mathbb{R}^n$

I want to show that if $F : \mathbb{R}^n \rightarrow \mathbb{R}^{n-k}$ is a $C^1$ function and $rank(DF) = n-k$ then $M:=F^{-1}(\{0\})$ defines a manifold. My idea: Without loss of generality I ...
0
votes
1answer
19 views

Closure of $\{ ( e^t\cos t,e^t\sin t) : t \in \Bbb R \}$

Suppose $\alpha: \Bbb R\to \Bbb R^2$ given by $\alpha (t)=(e^t \cos t,e^t \sin t)$, $A=\alpha(t)$ is a smooth manifold. What is the closure of $A$? I know that the closure of the set is the set ...
1
vote
1answer
23 views

Prove that this application $f:S^n\rightarrow \mathbb{RP}^n$ is a local diffeomorphism, alternative approach using curves

I consider $f:S^n\rightarrow \mathbb{RP}^n$, the restriction to $n$-sphere $S^n$ of the canonical projection $\pi :\mathbb{R}^{n+1}\setminus \{0\} \rightarrow\mathbb{RP}^n $. I have to prove that $f$ ...
0
votes
0answers
30 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
1
vote
1answer
21 views

showing pushfarward

Let $M,N$ be two differentiable manifolds and $f:M \rightarrow N$ be a smooth map. Define a new map $F:M\rightarrow M\times N$ by $F(p)=(p,f(p))$ I can prove first part which is F is smooth but I can ...
1
vote
1answer
26 views

Given $f : P\rightarrow N$ $C^\infty$ and $\pi : M\rightarrow N$ local diffeomorphism show that $\tilde f$ s.t.$f= \tilde f \circ \pi$ is $C^\infty$

Let be $M$, $N$ and $P$ three differentiable manifolds. I consider $\pi: M \rightarrow N$ a local diffeomorphism and $f:P \rightarrow N$ differentiable. I have to prove that the application $$ ...
2
votes
1answer
23 views

Let $G=GL(n,\mathbb R)$, show that this application $ (A,B) \in G \times G \rightarrow AB \in G$ is $C^{\infty}$

Let be $G=GL(n,\mathbb R)$. I consider the application $$a: G \times G \rightarrow G$$ such that $$ (A,B) \rightarrow AB .$$ I have to prove that this application is $C^\infty$. I know the ...
1
vote
1answer
38 views

Induced Connection on $\Sigma\subset M$

Let $(M,g)$ be a Riemannian manifold, $\Sigma$ a manifold and $F:\Sigma \rightarrow M$ a smooth map. For $X,Y \in \Gamma(T\Sigma)$ vector fields and $\tilde{\nabla}$ the pull back connection on ...
2
votes
0answers
46 views

A compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary

Under what conditions is it true that a compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary (or more generally, when a manifold is embedded in some topological space)? For ...
0
votes
1answer
21 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
0
votes
1answer
63 views

What does it mean for a manifold to be oriented?

I'm currently working through Spivak's Calculus on Manifolds. I've got to Stokes' Theorem, which is stated thus (the bold is my emphasis): Stokes' Theorem If $M$ is a compact oriented ...
1
vote
0answers
13 views

Associated bundle - valued 1 forms.

Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$. Fact: (as can be found in lecture notes here) Functions ...
1
vote
1answer
30 views

How to show that a subset of $\mathbb{R}^2$ is a closed one-dimensional submanifold?

I'm trying to solve the following problem: For $c \in \mathbb{R} \setminus \{ 0 \}$, let $$C = \{(x,y) \mid x^3 + xy + y^3 = c \} \subset \mathbb{R}^2.$$ Show that for $c \neq 1/27$ the ...
3
votes
1answer
45 views

How to show that $f : \mathbb{R}^n → \mathbb{R}^n$, $f(x) = \frac{h(\Vert x \Vert)}{\Vert x \Vert} x$, is a diffeomorphism onto the open unit ball?

Could anyone help me with the following problem? The problem Fix $\varepsilon \in (0, 1)$ and choose a smooth function $h$ on $[0,\infty)$ such that $h'(t) > 0$ for all $t ≥ 0$, $h(t) = t$ for ...
0
votes
0answers
28 views

Green's theorem via Stokes's theorem

I am considering the following form of Stokes's theorem: Let $\omega$ be an $n-1$ differential form with compact support on an oriented manifold of dimension $n$. Let us consider the boundary ...
2
votes
1answer
38 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
-1
votes
0answers
47 views

What is the slow manifolds? and how to calculate?

I'm a newbie in slow manifolds and dynamical system. I cannot understand the concept of slow manifolds and how to calculate that. Please explain the concept of slow manifolds intuitively and ...
1
vote
1answer
27 views

Cartan's structural equation

I am reading through a proof of Cartan's Structural equation: $$\Omega=d\omega + \frac{1}{2}[\omega\wedge\omega]$$ In the case when the input is two vertical vectors $V_1$ and $V_2$, we can ...
1
vote
1answer
31 views

Gluing together holomorphic functions on $\mathbb{P}^n$

The problem Let $U_j$ for $0\leq j\leq n$ denote the standard coordinate charts of the complex manifold $\mathbb{P}^n$. Fix $d\geq 1$ and assume we are given holomorphic functions $f_j:U_j\to ...