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

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3
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3answers
37 views

Inverse function of $f(x,y,z) = (xy-z^2, x+z)$?

How do you determine the inverse function $f^{-1}: \mathbb{R}^2 \to \mathbb{R}^3$ of $f: \mathbb{R}^3 \to \mathbb{R}^2 , f(x,y,z) = (xy-z^2, x+z) $ ? Or to put it into a bigger context: ...
1
vote
0answers
49 views

Equivalence of two norms on $L^p(M)$, $M$ compact manifold.

Let $(M,g)$ be a compact Riemannian manifold, $\mu(g)$ the Riemannian Lebesgue measure. Then we can define the usual $L^p$-spaces (lets assume $p<\infty$), $L^p(M,g):=L^p(M,\mu(g))$. For $f\in ...
0
votes
0answers
15 views

characeterization of zero sets of the riemannian measure of a riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold (does not have to be orientiable). Then there exists the Riemannian measure $\nu(g)$ on $M$. Let $(U_i,x_i)$ be a finite covering of $M$ of charts and let ...
5
votes
1answer
57 views

What are the 8 non-compact Euclidean 3-manifolds?

I have found several sources that mention that there are eight non-compact Euclidean 3-manifolds, with four orientable and four non-orientable. There are two standard references for this, but ...
6
votes
3answers
75 views

The set of all matrix with rank $n-1$ is a hypersurface.

Prove that the set $M$ of $n\times n$ matrices with rank $n-1$ is a hypersurface in $\mathbb{R}^{n²}$ and find the tangent space at $A=(a_{ij})$ where $a_{ij}=\begin{cases} \delta_{ij} \ \text{if} ...
2
votes
2answers
49 views

Prove that the antipodal mapping is an isometry on $S^n$. Help understanding the proof.

Prove that the antipodal mapping $A: S^n \to S^n$ given by $A(p)=-p$ is an isometry. I know that in order to prove that a map $f$ is an isometry of a smooth manifold $M$ it must hold true that ...
1
vote
1answer
32 views

Existence of a open set between a compact and an open set

Let $M$ be a compact manifold, $K\subset M$ compact, $U\subset M$ open. Does in this case always exist a open set $V\subset M$ such that $K\subset V\subset\bar{V}\subset U$ ?
1
vote
1answer
24 views

Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...
3
votes
1answer
75 views

Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$

I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$. A non-orientable ...
2
votes
1answer
40 views

If $\phi: M_1 \to M_2$ a diffeomorphism between diff. manifolds, prove that if $M_2$ is oriented then so is $M_1$

Let $\phi: M_1 \to M_2$ a local diffeomorphism between two differentiable manifolds $M_1,M_2$. I want to prove that if $M_2$ is orientable so is $M_1$. Attempt: In order a manifold to be orientable ...
2
votes
1answer
47 views

Normal coordinate parallel along radial geodesics?

A radial geodesic in normal coordinates is given by $\gamma:t \mapsto t(V_1,....,V_n).$ Is it then true that any normal coordinate $\partial_x|_{\gamma}$ is parallel along $\gamma,$ i.e. ...
1
vote
1answer
37 views

parallel vector field

I was wondering about the following: I know that a vector field along a geodesic that is parallel has a constant angle to the tangent vector of the curve and constant length. Now, is the converse ...
1
vote
1answer
32 views

Why should the gradient of an $n-1$-manifold in $\mathbb R^n$ be nonzero?

I am reading this chapter about manifolds here and the author writes (page 2): There is a very important restriction we impose on this situation. It is motivated by our recognition from p. 2–43 that ...
4
votes
1answer
85 views

Images of a familiar object in $\mathbb{R}^3$ mapped to $\mathbb{R}^2$ by Cantor/Peano/Hilbert

The proofs that, e.g., the cardinality of $\mathbb{R}^3$ is the same as the cardinality of $\mathbb{R}^2$, map $\mathbb{R}^3 \to \mathbb{R}^2$ via some scheme: Cantor's interleaving decimals, the ...
1
vote
0answers
15 views

How to determine the pdf for a model in phase space representation?

Consider a univariate discrete linear model : $z(k) = y(k) -(a* z(k-1) + b * z(k-2))$ where $y(k) = x(k) + \eta(k)$ $x(k) = s(k) + p*s(k-1) + q*s(k-2)$ is a Moving Average model of order 2. ...
4
votes
2answers
54 views

a question on topological manifolds and what topology provides

When one talks of a topological manifold being locally homeomorphic to $\mathbb{R}^{n}$ is it meant that the topology of the manifold is locally identical to a Euclidean topology such that we can ...
0
votes
1answer
15 views

Tangent space to noncompact Stiefel manifold

The noncompact Stiefel manifold is the set of $\mathbb{R}^{n \times p}$ matrices ($p \leq n$) that have rank $p$ (full rank). Based on my readings of ...
1
vote
1answer
26 views

If $M\subset \mathbb{R}^d$ is a manifold of dimension $m$ and $U\subset \mathbb{R}^d$ is open, then $M\cap U$ is not a open.

I'm reading notes about M-estimators, and have within these notes been briefly introduced to manifolds, as a way to create what the author call "smooth hypothesis" for statistical models. A basic ...
2
votes
2answers
120 views

Isn't there a better way to put the canonical smooth structure on the $n$-sphere?

My differential geometry notes put a smooth structure on the $n$-sphere (denoted $S_n$) as follows. Firstly, $S_n$ is taken as a subset of $\mathbb{R}^{n+1}.$ Secondly, we define $U_0 = S_n \setminus ...
2
votes
2answers
82 views

A “parallel manifold” is always orientable

I want to solve the following problem from Spivak's Calculus on Manifolds: Let $M$ be an $(n-1)$ dimensional manifold in $\mathbb{R}^n$. Let $M(\varepsilon)$ be the set of end points of normal ...
31
votes
3answers
411 views

Does every bijection of $\mathbb{Z}^2$ extend to a homeomorphism of $\mathbb{R}^2$?

Given a bijection $f\colon \mathbb{Z}^2 \to \mathbb{Z}^2$, does there always exist a homeomorphism $h\colon\mathbb{R}^2\to\mathbb{R}^2$ that agrees with $f$ on $\mathbb{Z}^2$? I don't see any ...
0
votes
1answer
19 views

Compute $(df)_a$ in chart $\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$

Suppose that for a submanifold $H$ of $\mathbb{R}^3$ we have two charts $$\varphi_1:U=\{(x,y,z)\in\mathbb{R}^3:x\neq0\}\rightarrow\varphi_1(U)$$ ...
2
votes
1answer
54 views

Properly discontinuous action on manifold

I am actually not familiar with topology, but since we had a short outlook on these things in our differential geometry lecture today, I would appreciate some general remarks: Let $M$ be a smooth ...
3
votes
1answer
59 views

Coordinate systems on manifolds

I am fairly new to differential geometry and something I can't get my head around is, if an $n$-dimensional manifold is locally homeomorphic to $\mathbb{R}^{n}$, i.e. Euclidean space, then isn't it ...
1
vote
0answers
16 views

Relation beween transition functions of a principal fiber bundle and its dual

What is the relation between transition functions of a principal fiber bundle and its dual? As an example, consider the transition map of the frame bundle on a manifold of dimension $n$ as the ...
2
votes
1answer
35 views

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$

Find an atlas for $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1+z^2\}$ It is easy for me to check that $M$ is a submanifold of $\mathbb{R}^3$ of dimension $2$ using the following theorem: Let $F:U ...
1
vote
0answers
46 views

Certain principle bundle structure on $\mathbb{R}^{n}\setminus \{0\}$

Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a principle fibre bundle. Here $\mathbb{H}^{2}$ is the Poincare upper plane ...
1
vote
1answer
43 views

Find an atlas for $H=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 : x_1+x_2^2=x_3^2+x_4=1\}$

Find an atlas for $H=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 : x_1+x_2^2=x_3^2+x_4=1\}$ Let $F:\mathbb{R}^4 \rightarrow \mathbb{R}^2$ s.t. $(x_1,x_2,x_3,x_4) \mapsto (x_1+x_2^2-1,x_3^2+x_4-1)$. ...
4
votes
0answers
37 views

Manifolds with 'bad metrics' (reference request)

While studying some differential geometry, a thought crossed my mind that I am sure has been considered before, but I cannot find a reference for it. What can be said about spaces for which the ...
1
vote
1answer
73 views

Integration by parts formula on unbounded manifold

Let $M$ be a closed Riemannian manifold and set $X = M \times [0,\infty)$ with the trivial product metric induced. If $u$ and $v$ are functions defined on $X$, how do I know that the formula $$\int_X ...
1
vote
1answer
38 views

$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$. Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$. Check if the obtained $1$-form respects or ...
1
vote
0answers
28 views

Grassmann and Stiefel manifolds

I want to show these two objects live up to their name in the sense that they actually are manifolds. The Grassmann manifold I understand to be a generalization of projective space (everything is done ...
6
votes
1answer
49 views

Making a set into a manifold

Let $n \in \mathbb{N}$, $M$ be a set and let $\mathcal{A} = \{(\varphi_a, U_a)\}_{a \in \mathcal{A}}$ be a system of tuples so that: $U_{a} \subseteq \mathbb{R}^n$ is open for all $a$; $\varphi_a: ...
1
vote
1answer
31 views

Show that $M=\{(x,y,u,v) \in \mathbb{R}^4 : x^2 + y^2 = u^2 + v^2 = 1\}$ is orientable, explaining the induced orientation.

Let $M=\{(x,y,u,v) \in \mathbb{R}^4 : x^2 + y^2 = u^2 + v^2 = 1\}$. Show that $M$ is an orientable subvariety of $\mathbb{R}^4$, explaining the induced orientation. Consider the $2$-form ...
4
votes
2answers
97 views

Why the lens space L(2,1) is homeomorphic to $\mathbb{R}P^3$?

According to one definition of lens space $L(p,q)$, which is gluing two solid tori with a map $h:T^2_1 \rightarrow T^2_2$. And $h(m_1)=pl_2+qm_2$, $l_i$ means longitude and $m_i$ means meridian of the ...
2
votes
1answer
43 views

Quotient manifold theorem provides a fibration?

It's known that if $G$ is a Lie group and $H\subseteq G$ is a closed subgroup, then the quotient map $p\colon G\to G/H$ is in fact a principal $H$-bundle, which follows from the existence of local ...
1
vote
1answer
51 views

Is GL($2$,$\mathbb{Z}$) is lie group?

This is a very dumb question, but is $\mathrm{GL}(2,\mathbb{Z})$ is lie group? I don't think it is, since its underlying set don't form a manifold, but I am just not sure.
4
votes
1answer
49 views

Are there compact manifolds homotopy equivalent to a wedge sum of compact manifolds?

One example given by Hatcher as an application for the cohomology ring is to distinguish $\mathbb{CP}^2$ from $S^2 \vee S^4$ up to homotopy equivalence despite their cohomology groups being the same. ...
0
votes
0answers
35 views

Generalizing Pearson's coefficient to determine properties of embedded manifold

I have the following dilemma: We know that for random vectors we have Pearson's coefficient of skewness. I think you all agree that in some sense it measures the shape properties of the ...
4
votes
1answer
116 views

Homology of a co-h-space manifold

Let $M$ be a compact connected topological manifold of dimension $n>1$. Suppose the corepresented functor $[M,-]\colon Top_{\ast}\rightarrow Set$ lifts to monoids or equivalently that $M$ is a ...
0
votes
0answers
71 views

Measure of inverse image of points by an analytic mapping

How can one prove the following statement: For any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of $M$ ...
0
votes
1answer
36 views

Isomorphism between two manifolds definition.

I want to try and show that a map between two manifold product spaces is an isomorphism. Im just a bit confused as to what 'isomorphism' means in this sense. At first I thought it was equivalent to ...
4
votes
1answer
54 views

Can we live without neighborhood basis but with open neighborhood basis?

I am reading Lee's Introduction to Topological Manifolds, and he declares that neighborhoods always mean open neighborhoods. So, the definition of a open neighborhood basis goes: Def Let $X$ be a ...
6
votes
1answer
98 views

If $F:M\to N$ is a smooth embedding, then so is $dF:TM\to TN$.

Question: I am trying to show that if $M$ and $N$ are smooth manifolds (without boundary), and $$F:M\to N$$ is a smooth embedding, then the differential $$dF:TM\to TN,\quad ...
3
votes
2answers
63 views

Whether or not such a simple CW complex can be made a $C^{\infty}$ manifold?

Problem Let $X$ be the space obtained by attaching two disks to $S^1$, the first disc being attached by the 7 times around,i.e. $z \to z^7$, and the second by the 5 times around. Can $X$ be made ...
2
votes
2answers
73 views

An orientable manifold of codimension 1 is the zero set of a differentiable function

I want to solve the following exercise from M. Spivak's Calculus on Manifolds: If $M \subseteq \mathbb{R}^n$ is an orientable $(n-1)$-dimensional manifold, show that there is an open set $A ...
0
votes
1answer
22 views

Extending a smooth vector field on a manifold

I want to solve the following exercise from M. Spivak's If $F$ is a differentiable vector field on $M \subseteq \mathbb{R}^n$, show that there is an open $A \supseteq M$ and a differentiable ...
1
vote
0answers
50 views

Lagrangian manifolds: basic standard theory

It is the first time that I start to learn about Lagrangian manifolds so I would like some suggestion about book and article to read. Due to the fact that is the first time, I need a book with the ...
0
votes
0answers
25 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 ...
5
votes
1answer
60 views

Characterization of 1-dimensional manifolds. [duplicate]

My intuition tells me that the only connected 1-dimensional topological manifolds are the real line $\mathbb{R}$ and the circle $S^1$. Is this true? If yes, is it possible to prove it from first ...