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
17 views

Frobenius theorem

I came across the following conclusion in a textbook, but can't really understand it. I would be grateful if anyone could elaborate: Assume that we have three linearly independent vector fields ...
3
votes
1answer
180 views

How can we define $\partial x_{i_r}^p(X_p^r)$?

Let $M$ be a smooth manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P\subseteq M$ be an open set and $X_1,\ldots,X_s\in\mathcal T(P)$ and $X^1,\ldots,X^r\in\mathcal T^*(P)$ where $\mathcal T(P)$ ...
1
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0answers
10 views

Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
0
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1answer
39 views

Show $\mathbb{C}P^n$ is a $2n-$manifold [in singular homology theory]

There is a Theorem in the book that says: The space $\mathbb{C}P^n$ is CW complex of dimension $2n$. I wonder some questions: Is there any Theorem or result that if a space has CW complex ...
6
votes
1answer
610 views

Local homology group: a homeomorphism takes the boundary to the boundary

Let $X \subset \mathbb{R}^{n}$ be the subspace $\{ x_{1},...,x_{n} \mid x_{n}\geq 0 \}$, and let $Y$ is the subspace with $x_{n}=0$. Let $x\in Y$, calculate the local homology of $X$ at ...
1
vote
2answers
32 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 ...
1
vote
2answers
43 views

Is the Lie derivative $L_{X}(\omega \wedge \mu)$ an exact form?

Let $\omega$ be an $n$-form and $\mu$ be an $m$-form where both are acting on a manifold $M$. Is the Lie derivative $L_{X}(\omega \wedge \mu)$ where $X$ is a smooth vector field acting on $M$ an ...
1
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1answer
33 views

Low torsion in orientable manifolds?

The final sentence on page 170 of Stillwell's Classical Topology an Combinatorial Group Theory is: Poincaré justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only ...
0
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1answer
29 views

Transversality: what is wrong with this counter example to persistence for small perturbations?

Let $M$ and $N$ be differentiable manifolds in $\mathbb{R}^{n}$, and let $p \in \mathbb{R}^{n}$. We say that $M$ and $N$ are transversal at $p$ if $$T_{p} M + T_{p}N = \mathbb{R}^{n}.$$ By dimension ...
4
votes
1answer
61 views

Apparent violation of fundamental theorem of ODEs, how to resolve?

Consider, in the $(x, y)$-plane, the family of curves given by $y = (x - c)^3$, for the various possible values of the number $c$. Denote by $v$ the unit vector field everywhere tangent to this family ...
4
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2answers
83 views

Concept of Manifold

The concept of manifolds is freaking me out. For me it seems like a manifold is just a subspace embedded in a higher dimension. In order to clear out my confuision I have created a list and I would ...
5
votes
1answer
69 views

Is Whitehead's manifold with a point removed homotopy equivalent to a sphere?

A contractible open subset of $\mathbb{R}^n$ need not be homeomorphic to $\mathbb{R}^n$. The Whitehead manifold is an open subset of $\mathbb{R}^3$ which is contractible but not homeomorphic to ...
-3
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0answers
18 views

constructing a manifold structure for a cylinder [on hold]

Any help on this problem would be greatly appreciated. thanks! let M be the cylinder {$(x,y,z)\in \mathbb{R}^3:x^2+y^2=1$} in $\mathbb{R}^3$. Construct a manifold structure each topological space ...
-2
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1answer
21 views

constructing a manifold structure for a plane in $\mathbb{R}^3$ [on hold]

Any help on this problem would be greatly appreciated. thanks! Let M be the plane in $\mathbb{R}^3$ with normal vector (a,b,c)$\neq$0. Construct a manifold structure each topological space (M,$\tau$) ...
0
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0answers
77 views

torus parametrization inverse

I've been looking at the torus parametrization \begin{equation} \phi(u,v)=((r\cos u+a)\cos v, (r\cos u +a)\sin v, r\sin u) \end{equation} with $a>0, r\in(0,a)$. I want to invert this map to get a ...
0
votes
1answer
25 views

What is the tangent space o SO(n) [on hold]

It should be the kernel of the map $H\mapsto A^TH+H^TA$ at some $A$ such that $A^TA=I$ . But I cant find this Kernel can someone help me?
0
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0answers
20 views

Real Analitic Manifolds, Tubular Neighborhood, Radius of Convergence

Given a Real Analytic Manifold isometrically embedded into an Euclidean Space. Gicven the maximum value of the radius of a Tubular Neighborhood "around" the manifold: what relation does it have with ...
0
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0answers
26 views

Injectivity Radius vs. Radius of Convergence in Analytic Manifolds

I would like to ask the following: How does the Injectivity Radius relate to the Radius of Convergence (of the analytic function to its power series) of any local (parametrization) map in the ...
0
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0answers
20 views

Immersions-possible error in Dieudonné III?

Below I refer to [D] Dieudonné Treatise on analysis III [B] Bourbaki VARIETES DIFFÉRENTIELLES ET ANALYTIQUES [M] Michor Topics in differential geometry In [D,16.7.7], we can read: Let $f ...
0
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1answer
32 views

Prove there exists a smooth unit normal at the boundary of the following manifold

Let $M$ be a compact subset of $\mathbb{R}^3$ with smooth boundary $S=\partial M$. Consider M with the standard orientation $\mu=\mu_{0}$ from $\mathbb{R}^3$ and $S$ with the boundary orientation ...
1
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0answers
20 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
1
vote
1answer
30 views

Determining a derivation on the unit sphere of the $\mathbb{R}^3$

Let $S^2 \subseteq \mathbb{R}^3$ be the unit sphere in $\mathbb{R}^3$ (which is a smooth manifold of dimension $2$). Let $\phi = (x_1, x_2): S^2 \backslash \{N\} \to \mathbb{R}^2 $ be the ...
8
votes
2answers
373 views

The reason behind the definition of manifold

I was going thorough the definition of a manifold and needless to say it wasn't something that I could digest at one go. Then I saw the following Quora link and Qiaochu's illustrative answer. It was ...
0
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0answers
22 views

Converting a vector $v \in \mathbb{R}^2$ given in Polar coordinates to Cartesian coordinates

I know that switching inbetween Polar coordinates and Cartesian coordinates in $\mathbb{R}^2$ can, on suitable open subsets of $\mathbb{R}^2$, be done via $(x, y) = (r cos \theta, r sin \theta)$. Let ...
0
votes
1answer
33 views

Orientation on the boundary

If $M$ is an oriented without boundary manifold, and $\mu$ is it volume form, is true that the boundary of $M\times [0,1]$ is $ M \cup M$, right? It is true also that the orientantion on the boundary ...
0
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0answers
17 views

For manifolds $M,N$ show that $W^{1,p}(M,N)$ is path-connected iff $C^0(M,N)$ ist path-connected.

I'm asked to show that for compact, smooth Riemmanian manifolds $M,N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from ...
2
votes
0answers
22 views

Simplification of Levi-Civita in an orthonormal frame

I have been struggling to understand how picking an orthonormal frame for the tangent space of a Riemann surface with local coordinates ${x_1,x_2}$ simplifies the matrix of one forms associated to its ...
1
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0answers
19 views

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$?

Divergence theorem not consistent when calculating the flux of $F= (x^k,y^k,z^k)$ on $S^2$? ($ k > -2$). Using the divergence theorem, I got that the flux is: $\frac{3\pi}{k}(1-(-1)^k)$ and ...
0
votes
1answer
28 views

push forward of the levi civita connection

Let $M$, $M'$ be riemann manifolds with levi-civita connection $\nabla$,$\nabla'$. If $\phi$ is an isometry (global so diffeomorphism too) I want to show: $ \nabla'_{X'} Y'=D\phi (\nabla_X Y) $ where ...
1
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0answers
44 views

Fundamental class for tangent bundle

For closed orientable manifolds $X$ we can define the fundamental class $[X]$ which is just a choice of generator of the top homology group $H_n(X; \mathbb{Z})$. However, in the context of the ...
3
votes
2answers
92 views

Trying to prove that $TM$ is a manifold: Is this function an homeomorphism?

I am trying to prove that if $M$ is a $k$-manifold in $\mathbb R^n$, then $TM=\{(p, v): p \in M, v \in T_pM\}$ is a manifold. Here, $T_pM$ is defined as a subset of $\mathbb R^n$. I know that ...
14
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3answers
445 views

Is every self-homeomorphism homotopic to a diffeomorphism?

Given a smooth manifold $M$, is every homeomorphism $M \to M$ homotopic to a diffeomorphism? Hirsch's "Differential Topology" has a proof that every $C^1$ diffeomorphism of smooth manifolds is ...
4
votes
1answer
37 views

Why is a diffeomorphism preserving a parallelism locally uniquely determined by its value at $1$ point?

According to the title, why is a diffeomorphism preserving a parallelism locally uniquely determined by its value at $1$ point?
5
votes
1answer
57 views

Can every Riemmanian Manifold be completed?

I had two trails of though.. is either of them fruitful? I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's ...
4
votes
2answers
32 views

Maps between manifolds with boundary and homeomorphism

Assume we have $f:(M,\partial M)\rightarrow (N,\partial N)$ connected 3-manifolds, not compact, such that $f$ is an homeomorphism onto its image and $f(\partial M)=\partial N$. Can say that $f$ has to ...
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 ...
2
votes
1answer
25 views

How to tell if a manifold can be embedded as the interior of a compact manifold with boundary?

Some (topological) manifolds can be embedded as the interior of a compact manifold with boundary. Any closed manifold, for example, or any closed manifold with some points removed, and so on. On the ...
2
votes
1answer
26 views

Is an open subset $U \subset \mathbf{R}^{n}$ diffeomorphic to the product $U' \times \mathbf{R}$ with $U' \subset \mathbf{R}^{n - 1}$ open?

I'm trying to prove that $U$ is diffeomorphic to the product of some open subset $U' \subset \mathbf{R}^{n}$ with $\mathbf{R}$, $U' \times \mathbf{R}$. I received the hint that this set admits a ...
1
vote
1answer
39 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
votes
0answers
19 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 $N$ ...
4
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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?
1
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2answers
85 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
30 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 = ...
2
votes
1answer
33 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 ...
23
votes
1answer
498 views

How much of an $n$-dimensional manifold can we embed into $\mathbb{R}^n$?

I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded ...
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: ...
3
votes
0answers
58 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 ...
3
votes
3answers
63 views

What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...
3
votes
0answers
44 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 ...
1
vote
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? ...