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

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4
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1answer
55 views

Parametrizing the time an element stays in an open subset

Let $X$ be a topological space (If it helps anything, we can assume $X\subseteq\mathbb{R}^n$ or $X$ being a smooth manifold.) and $U\subseteq [0,1]\times X$ an open subset. Does there exist a ...
1
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0answers
15 views

Quickest way to restrict a homeomorphism

Let $\phi: U \to V \subset \mathbb{R}^n$ homeomorphism. My desire is: I want to say the restriction $\phi|_{\phi^{-1}(B_{r'}(x))}:\phi^{-1}(B_{r'}(x)) \to B_{r'}(x) $ is a homeomorphism in the ...
2
votes
1answer
27 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
2
votes
3answers
43 views

2-Sphere surface coordinate dimension

Ordinary sphere in $\mathbb{R}^3$ is two-dimensional object (2-sphere), i.e. it requires at least two coordinates to define point on a surface. As I notice, however, there is a catch. If we use ...
2
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1answer
49 views

Cut out characteristic Submanifold N ($w_1(M)=w_1$(Normalbundle of N in M)). Remainder M-N is orientable? Orientation Character or CW-Structure?

So I try to understand the following (which is taken from Dold, "Structure of the cobordism ring", Page 3/274, in the paragraph "1. La suite exacte de Wall."): https://eudml.org/doc/109581 ): Giving a ...
1
vote
1answer
39 views

Homeotopy of Shrinking Manifolds

Let $M$ be a $n$-dimensional open manifold in $\mathbb{R}^n$. Let $B^n_k$ be the closed $n$-dimensional ball of radius $k$. Let $$N_k = (M^c \oplus B^n_k)^c$$ where $X^c$ denotes the complement of ...
6
votes
1answer
157 views

Definition of a Cartesian coordinate system

Apologies if this is a basic question, but I'd really like to clarify the exact meaning of what a Cartesian coordinate system is. Heuristically, is it correct to say that a Cartesian coordinate system ...
0
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1answer
53 views

Smooth function, which separates between a closed and a open set.

Let $M$ be a smooth manifold, $O\subseteq M$ an open subset and $B\subseteq M$ a closed subset, such that $closure(O)\subseteq interior(B)$ I think there must exist a smooth function $f\colon ...
5
votes
1answer
95 views

Theorema egregium violated in dimension $n \ge 4$?

Gauß showed that for surfaces in $\mathbb{R}^3$ the Gaussian curvature ( = sectional curvature) is invariant under local isometries. This is known as the thema egregium. Now in another question ...
2
votes
1answer
37 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
1
vote
1answer
66 views

Meaning of “locally homeomorphic to $\mathbb{R}^{n}$”

I am fairly new to differential geometry and approaching it with a physics background (in the study of general relativity), as a result I'm having a few struggles with terminology etc, so please bear ...
1
vote
1answer
37 views

(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
0
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0answers
13 views

Are variables in embedded space Statistically independent variables?

Performing Taken's phase space delay embedding on the observations $\mathbf{z}$ of a univariate random variable, with an embedding dimension $d$, we get a realization of $n$ points such as: ...
0
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0answers
49 views

Constructing coordinate maps on manifolds

I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on ...
3
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3answers
38 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
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0answers
55 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 ...
8
votes
1answer
88 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
77 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
58 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
26 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
80 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
43 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
48 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
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1answer
44 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
35 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 ...
5
votes
1answer
129 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 ...
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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
61 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
19 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
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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
124 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
90 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
434 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
64 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
82 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
19 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
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0answers
47 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
44 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
41 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
91 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
42 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
51 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
33 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
119 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
54 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 ...