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

learn more… | top users | synonyms (1)

0
votes
1answer
25 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 ...
2
votes
2answers
73 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 ...
-3
votes
0answers
17 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 ...
5
votes
1answer
52 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 ...
-2
votes
1answer
20 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
votes
1answer
23 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
votes
0answers
16 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
votes
0answers
22 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
votes
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
votes
1answer
27 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 ...
1
vote
0answers
19 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 ...
0
votes
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 ...
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
363 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
votes
0answers
16 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 ...
1
vote
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 ...
2
votes
0answers
21 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 ...
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
votes
1answer
27 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
vote
0answers
42 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 ...
0
votes
1answer
31 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 ...
3
votes
1answer
36 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
54 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
31 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 ...
2
votes
1answer
24 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 ...
3
votes
1answer
121 views
+50

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

Suppose $M$ is a manifold and $X_r^s:M\to T_r^s(M)$ is a section. Let $P$ 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 $\mathcal T(P)$ and $\mathcal ...
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 ...
3
votes
2answers
88 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 ...
1
vote
1answer
36 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
votes
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
vote
2answers
83 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
vote
0answers
27 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 = ...
1
vote
2answers
28 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
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 ...
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 ...
3
votes
0answers
42 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? ...
0
votes
1answer
19 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
votes
0answers
46 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
61 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 ...
2
votes
1answer
25 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
votes
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: ...
0
votes
0answers
23 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 ...
2
votes
0answers
27 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 ...
0
votes
1answer
7 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
28 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
62 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
52 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 ...