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

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2
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1answer
44 views

Foliation dense if $G = \textbf{R}$, where $G$ is a subgroup of a Lie group $G'$.

I have the following statement: Let $G$ be a subgroup of a lie group $G'$, and the action is left multiplication. The leaves are then the left cosets of $G$ in $G'$. If for example, we let $G = \...
0
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0answers
20 views

Proving a formula for the coordinate representation of a mapping inbetween smooth manifolds

Let $M, N$ be smooth manifolds, and let $f: M \to N$ be a smooth mapping. I now want to prove: If $(U, \phi = (x_1, ..., x_m))$ and $(V, \psi = (y_1, ..., y_n))$ are charts for $M$ and $N$ ...
2
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1answer
29 views

Questioning about the meaning of “$1$-dimensional circle”

When we talk about the $1$-dimensional circle, is it a one-dimensional object, although one can embed it into a two-dimensional object? More precisely, is it a one-dimensional manifold?
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2answers
86 views

$\operatorname{SU}(n)$ as manifold

I am trying to do this has a while, but I cannot use correctly the regular value theorem to do so! I appreciate any help. The problem is that I cannot choose the function to take $SU_n$ as a regular ...
1
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0answers
33 views

Mobius strip as manifold and as a bundle over $S^1$

I am trying to construct the Mobius strip bundle onver $S^1$. I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved. My attempt was: $$M = [0,...
1
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2answers
34 views

Local defining functions on real hypersurfaces.

I'm currently reading Real Submanifolds in Complex Space and their Mappings by Baouedi, Ebenfelt and Rothschild. I'm currently stumped by what the author's claim to be an easy check (really blowing ...
3
votes
0answers
60 views

What are the essential tools and proof techniques for beginning smooth manifolds and differential topology?

I am an undergraduate currently taking a first course in smooth manifolds. I feel that I understand the material intuitively. But, I'm having trouble turning my intuition into proofs. I was hoping ...
2
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1answer
38 views

How can I prove that interior product obeys a graded Leibniz rule?

I want to prove that $i_{X}(\omega\wedge\phi)=i_{X}\omega\wedge\phi+(-1)^{k}\omega\wedge i_{X}\phi.$ I was thinking I many be able to adapt the proof that the exterior derivative obeys the graded ...
3
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0answers
51 views

Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ dX_s,...
1
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1answer
23 views

If a metric tensor is not conformally equivalent to the flat metric

If on a manifold $M$ we have two metrics $g_{ab}$ and $g'_{ab},$ which are not conformally equivalent, and we say that $(M,g_{ab})$ is a flat manifold, does it follow that $(M,g'_{ab})$ is not flat? ...
0
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1answer
20 views

Finding the expression of a one form in a chart.

Given a one form on a manifold the formula I was given for finding its expression in a given coordinate chart is very strange and I dont understand it. I would appreciate if someone could give me a ...
4
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0answers
53 views

Possible Generalization of a Manifold

A manifold $M$ is a second-countable, Hausdorff, locally Euclidean topological space. Obviously, there are advantages to requiring $M$ to be locally Euclidean, i.e. in some cases this allows $M$ to be ...
5
votes
1answer
71 views

Tangent bundle of sphere as a complex manifold

I'm trying to show that the tangent bundle, $TS^n$ of the n-sphere $S^n$ is diffeomorphic to the set $\sum z_i^2 = 1$ in $\mathbb{C}^{n+1}$. It's relatively straightforward to see that the tangent ...
3
votes
1answer
42 views

Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$?

Can a connected finite-dimensional manifold have cardinality $>2^{\aleph_0}$? I know that if we either impose the condition "Hausdorff" or "second countable", the assertion is false. What if we ...
2
votes
1answer
26 views

Can a connected second countable manifold have cardinality $>2^{\aleph_0}$?

Can a connected second countable manifold have cardinality $>2^{\aleph_0}$? From here, a connected Hausdorff manifold must have cardinality $2^{\aleph_0}$. How about if we change the condition "...
0
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1answer
29 views

Showing that the “abstract” tangent space of a submanifold of the $\mathbb{R}^d$ is isomorphic to the tangent space that's a subset of $\mathbb{R}^n$

Let $M$ be an $n$-dimensional smooth submanifold of the $\mathbb{R}^d$, and $p \in M$. Let $T_p^{A}M$ denote the "abstract" tangent space of $M$ in a point $p$, given by $T_p^AM = \{\gamma: (-\epsilon,...
0
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0answers
25 views

How is “one of the coordinates non negative” for points in the unit circle?

Looking at a problem, Consider $S^1 \subset \mathbb{R}$. Define $U_{a,b}$ where the indexing set $I=\{(a,b):a\in\{1,-1\},b \in \{1,2\}\}$ is given, so that $U_{a,b}=\{(x_1,x_2) \in S^1:ax_b >0\...
2
votes
0answers
31 views

Newton-Raphson method on manifolds

Has anyone explored the notion of the Newton-Raphson method on manifolds? Or to put it another way, on $\mathbb R^n$, is there a natural coordinate free way of defining an iterate of the Newton-...
0
votes
1answer
8 views

Understanding the Chow-Rashevsky Theorem

I'm trying to understand the Chow-Rashevsky Theorem. I unfortunately do not have a formal knowledge of what's going on but have figured out most of the terms. Basically a system $\Sigma$ must ...
2
votes
1answer
29 views

Level set of the hopf map

The Hopf map is given by the projection $\pi: \mathbb{S}^3 \to \mathbb{S}^2$, and: $\pi: z \mapsto zi\bar{z}$, where $z \in \mathbb{S}^3$ and $i \in \mathbb{H}$ is a unit quaternion. Show that ...
1
vote
3answers
70 views

Degree 1 map from torus to sphere

I'm trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the 2-sphere $S^2$. My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual ...
2
votes
0answers
53 views

Exterior derivative for functions with values in a parallelizable manifold

In Sharpe's text on Cartan geometry, he explains in section 1.5 on page 52 how to define an exterior derivative for maps into a parallelizable manifold $N$. Let $f: M \to N$ be a smooth map, and $\...
1
vote
1answer
37 views

Dimension of topology manifold

In the 3 page of Jurgen Jost's Riemannian Geometry and Geometric Analysis .Why it is harder in topology manifold than differentiable manifold ? I think it is easy in differentiable manifold because ...
1
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1answer
36 views

Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
1
vote
1answer
38 views

An orientation on a $(n-1)$-dimensional submanifold.

This question goes on where this question ended. Given is an non-empty $(n-1)$-dimensional ($n\ge2$) differentiable submanifold $X\subset\mathbb{R}^n$ such that there exists an open $U\subset\mathbb{R}...
0
votes
0answers
103 views

Orientability of the level set of a map between abstract oriented manifold

Let M and N be oriented manifold and let $f:M\to N$ be a smooth map between them. Suppose $y \in N$ is a regular value for $f$, how can we show that $f^{-1}(y)$ is orientable? I've seen a solution ...
2
votes
2answers
44 views

Topological boundary of a specific manifold.

Let $\emptyset\neq X\subset\mathbb{R}^n$, $n\ge2$, be an $(n-1)$-dimensional differentiable submanifold, i.e. for every $p\in X$ there is an open $U_p\subset\mathbb{R}^n$ with $p\in U_p$, and a ...
4
votes
2answers
63 views

connected sum of surfaces is well defined proof attempt

Suppose $S_1$ and $S_2$ are compact surfaces (connected 2-dimensional manifolds). If we cut out of them two closed disks, and glue the surfaces along disk boundaries we get new surface, their ...
0
votes
1answer
19 views

Proof Writing: Given 2 two dimensional manifolds $\emptyset \neq N \subseteq M \subseteq \mathbb{R}^3$, $N$ is compact, $M$ is pconnected: $N = M$

Statement: Given 2 two-dimensional manifolds $\emptyset \neq N \subseteq M \subseteq \mathbb{R}^3 $, if $N$ is compact and $M$ is path-connected, then $N = M$. Proof: We know that there is at least ...
0
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0answers
34 views

Brownian motion on a manifold

If I have a manifold $M$ and a chart $\left(x,U\right)$, is it possible to simulate Brownian motion on that manifold by solving an SDE in the chart representation $x\left(U\right)$ and then use the ...
1
vote
2answers
49 views

Why are tangent vectors coordinate-dependent?

Why does the coordinate basis for $T_pM$ depend on the coordinate chart we are using? Any two charts containing $p$ agree on some neighborhood of $p$, so shouldn't we be able to find a basis for $T_pM$...
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0answers
30 views

Why are PDEs with Hamiltonians usually solved on compact manifolds?

The title is self explaining: I see in a lot of literature that PDEs with some Hamiltonian structure in it are solved over a torus or some other compact manifold. Why is that? At least I now that it ...
0
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1answer
44 views

Reference for real analytic manifolds

I'm trying to find a reference for some introduction to real analytic manifolds. I'm especially interested in the fact, that the set of regular points of an analytic function $F \colon M \to \mathbb{R}...
0
votes
1answer
29 views

Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
0
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0answers
14 views

Is there an atlas of Kirby diagrams of 4-manifolds?

We've defined an invariant of 4-manifolds (article) in terms of Kirby diagrams and I'm looking for a lot of manifolds to test it on. Now I'm not a big differential topologist myself, and coming up ...
2
votes
1answer
36 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let $\phi:\...
0
votes
1answer
18 views

Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be $$...
2
votes
1answer
21 views

how to prove that $C^{k}$ map does not depend on choice of the charts

I was reading an article about Manifolds.They have defined a $C^{k} $ function in the following way : Let $M$ and $N$ are two $C^{k}$ manifolds of dimensions $m$ and $n$ respectively.A continuous ...
2
votes
1answer
33 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
3
votes
1answer
68 views

Conditions on characteristic polynomial to define a matrix submanifold.

I'm trying to find conditions on the characteristic polynomial, $p$, of a matrix such that the pre-image of matrices with characteristic polynomial $p$ form a manifold. More precisely, we can write ...
0
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0answers
21 views

Prove that integration of a differential $k$-form is independent of choice of basis

This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book: (Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if $\mathbf{a}_1,\ldots,\mathbf{a}...
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0answers
74 views

Explicit form of the exponential map

I am stuck by the following problem. Let $ g = 1/y^2 (dx \otimes dx + dy \otimes dy)$ be a Riemannian metric on the half-plane $ S = \mathbb{R}^2_{y > 0}$. What is the explicit form of the ...
1
vote
1answer
24 views

Oriented atlas on a circle

I'm trying to find an oriented atlas on the circle $S^1$, i.e., I want to find an atlas for $S^1$ such that for any two overlapping charts $(U,s)$ and $(V,t)$ of the atlas, the derivative $d s/d t$...
0
votes
1answer
25 views

Integration on k-1 form

If $\omega$ is a $k-1$ form on a closed $k$-dimensional manifold $M$ then $\int_M d \omega = 0$. I'm looking for a short proof to this problem, would Stokes be helpful?
2
votes
1answer
29 views

How transversality condition implies that a value is regular?

Currently I am self-learning some manifold theory and just come across concept of functions transverse to submanifolds. It seems that this concept is used a lot for proving regularity of values, but I ...
1
vote
1answer
27 views

Find a nontrivial bundle of $S^1$ with fibre isomorphic to $\mathbb{R}^n$

Show that such a nontrivial bundle exists for every $n\in\mathbb{N}$. I don't really have any useful ideas here. I'm not sure if there is a general approach I should be taking or if there is just a ...
-1
votes
1answer
30 views

analysis on manifods

Let $M$ be a compact oriented $k+l+1$ dimensional manifold without boundary in $\mathbb R^n$. Let $\omega$ be a $k$-form and let $\eta$ be an $l$-form, both defined in an open set of $\mathbb R^n$ ...
0
votes
1answer
29 views

Why is $\int_{X} d(\alpha \wedge *\bar{\beta})$ zero on a compact hermitian manifold?

I am reading the book Complex Geometry - An Introduction by Huybrechts. In proving Lemma 3.2.3 that $\partial$ and $\partial^*$ are formal adjoints to each other, he mention that the following ...
1
vote
2answers
42 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
3
votes
0answers
18 views

Counterexample for the density of smooth functions in Sobolev spaces on a manifold

I'm in desperate need for help understanding a counterexample for the assertion that for (appropriate) manifolds $M,N$ the space $C^\infty (M,N)$ is dense in $L^p (M,N)$ if $\dim(M) > p$. (The ...