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

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5
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2answers
115 views

Interior of a compact 3-manifold

I have an orientable 3-manifold $X$, such that $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ How to find a compact 3-manifold $M$ such that $X= ...
0
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0answers
25 views

What geometric information can be recovered from $L^2(X)$ for a manifold $X$?

It is well known that a compact Hausdorff topological space can be fully reconstructed from its $C^\ast$-algebra of complex valued continuous functions with the sup norm. Are there similar (partial) ...
1
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1answer
56 views

Changing $[0,2\pi)$ with $S^1$ such that a map defined on $[0,2\pi)$ stays unchanged

* Consider the following procedure of changing the domain of a map, but the map remaining essentially the same - illustrated, for concreteness, in case of the polar-coordinates map.* Let ...
0
votes
1answer
33 views

Covariant Stared Vector Spaces?

I reading john Lee's book entitled "Introduction to Smooth Manifolds" and on page 311 $$T^{k}\!\!\left(V^{*}\right)=V^{*}\!\!\times\!\!V^{*}...V^{*}\;\;k\text{-times}$$ is defined as the vector ...
2
votes
1answer
70 views

$G$ is an $(n-1)$-manifold without boundary and is the topological boundary to an open $K\subset \mathbb{R}^n$. Prove $G \cup K$ is an $n$-manifold.

All manifolds are smooth. Let $M = G \cup K$. The interior of $M$ is an open set in $\mathbb{R}^n$ and can be given a global coordinate by the identity map. The points in $M$ not on the interior of ...
7
votes
2answers
110 views

What's the “easiest” closed 3-manifold with a nonabelian fundamental group?

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group. It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one ...
2
votes
1answer
64 views

Lie Subgroup Example - Explanation?

I'm currently working through Jeff Lee's 'Manifolds and Differential Geometry'. He defines a Lie Subgroup, $H$, to be an abstract subgroup of a Lie Group $G$, such that the inclusion map ...
6
votes
2answers
798 views

Proof of whitney's embedding theorem?

While learning about the rigorous definition of manifolds, my text mentions that any $n$-dimensional manifold can be embedded in $\Bbb{R}^{2n}$, which is called Whitney's embedding theorem. I have ...
1
vote
1answer
59 views

Polynomials of a fixed degree have a nonzero partial at all points of their vanishing set?

I'm having difficulty with the second half of a question from an old homework assignment (for a differential geometry class I am currently taking). The first half of the question asked me to assume ...
3
votes
0answers
84 views

Generalizations of the Hairy Ball Theorem to wider classes of manifolds

In 2 dimensions the hairy ball theorem generalizes from spheres to all orientable closed manifolds with nonzero Euler characteristic. The hairy ball theorem holds for all even dimensional ...
4
votes
1answer
137 views

Using Alexander's Theorem to show that the sphere $S^3$ is a prime manifold

I'm completely aware of the triviality of this question, but for some reason, I can't visualize the argument. In Hatcher's 3-manifold notes, the form of Alexander's theorem stating that Every ...
2
votes
1answer
37 views

Embedding of classical Lie groups

This is somehow very natural question so I'm sure that the answer should be well known: Whitney theorem states that each (say paracompact) $n$-dimensional manifold could be embedded in ...
1
vote
0answers
54 views

Smooth sections of smooth vector bundle

Suppose that $E \to B$ is a (real for example) smooth vector bundle ($B$ is assumed to be a smooth manifold). There is a important notion of the smooth section $s:B \to E$: is has to satisfy $s(x) \in ...
1
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0answers
41 views

Question about Nehari manifold

I have two questions about the Nehari manifold, the first one is where i can found the properties of this manifold ? and the second one is in a theorem it says "let M a Banach-finsler manifold" can ...
2
votes
1answer
87 views

Evaluating contractions of a tensor product

Consider $T = \delta \otimes \gamma$ where $\delta$ is the $(1,1)$ Kronecker delta tensor and $\gamma \in T_p^*(M)$, the co-tangent space over some manifold $M$. Evaluate all possible contractions of ...
1
vote
1answer
41 views

Vector Fields on Real Numbers

I'm looking at vector fields on the manifold $\mathbb{R}$, in the sense that a vector field $v$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}\times T_p\mathbb{R}$. These seem so simple that ...
0
votes
1answer
35 views

Differential: Smoothness

This is a lemma for another thread. Given smooth manifolds. The differential is a smooth map: $$F:M\to N:\quad F\text{ smooth}\implies\mathrm{d}F\text{ smooth}$$ How to check this?
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0answers
133 views

Question about Milnor's proof of Sard's Theorem

We've just covered Sard's theorem and have just started to look at transversality in my differential geometry class and I'm trying to understand a proof of Sard's theorem (based on Milnor's proof): If ...
1
vote
0answers
58 views

Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
0
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2answers
32 views

What kind of manifold is a configuration manifold?

I have recently been learning about the basic properties of topological, smooth, and Riemannian manifolds. But I frequently hear the term configuration manifold referenced in relation to Lagrangian ...
3
votes
0answers
68 views

When did the notion of the tangent space emerge?

When did the modern conception of/notation for the tangent space to a manifold come into use?
2
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0answers
22 views

Charts/transition charts of the $\mathbb{CP}^3$ tangent bundle

I would like to explicitly compute the charts and transition charts for the tangent bundle of $\mathbb{CP}^3$. I know the charts of $\mathbb{CP}^3$ are $\phi_i: U_i=\{[z_0,z_1,z_2,z_3]; z_i \neq ...
8
votes
2answers
252 views

Equivalence of two distance function on a Riemannian manifold

Let $(M,g)$ be a closed connected $m$ dimensional smooth Riemannian manifold and assume that it is isometrically embedded in a Euclidean space $\mathbb{R}^q$ by $\iota:M\to\mathbb{R}^q$. $|\ast|$ ...
3
votes
0answers
65 views

Smooth manifolds having an analytic structure.

I am reading about Manifolds and Lie groups and I have certain questions related to them. Let me first explain why I am asking these questions. We know that not all $C^{\infty}$ functions are ...
6
votes
0answers
102 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
2
votes
2answers
436 views

Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space $$[0, 1] \times [0, 1]/{\sim}, \text{ where the ...
0
votes
0answers
75 views

Does isometry preserve volume on open sets?

Suppose there are two open sets $A,B$. $h$ is an isometry. And the function $h$ maps $A$ to $B$; $h(A)=B$. I need to show that isometry is volume preserving. Any hint would be appreciated! Thanks ...
0
votes
2answers
99 views

The quotient map and isomorphism of cohomology groups

Let $X$ be a closed $n$-manifold, $B$ an open $n$-disc in $M$. Suppose $p:X\rightarrow X/(X-B)$ is a quotient map. Notice that $X/(X-B)$ is homeomorphic to the sphere $\mathbb{S}^n$. My question is ...
0
votes
0answers
47 views

Tangent space of a Product of two manifolds

Suppose $M$ and $N$ are two $C^\infty$ manifolds. Take $p\in M$ and $q\in N$. We have the following maps between these: $\iota_1 : M\to M\times N$, $\iota_2:N\to M\times N$, $\pi_1:M\times N\to M$ and ...
2
votes
1answer
143 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
1
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0answers
57 views

What does matrix decomposition really mean?

Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & ...
4
votes
2answers
169 views

Invertibility theorem on the boundary for a function between two closed 2D manifolds

Assume a function $f:\mathbb{R}^2\to\mathbb{R}^2$ on a simply connected, closed domain $D\subset\mathbb{R}^2$ including its boundary $\partial D$. I am interested in the local invertibility of $f$ ...
5
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0answers
63 views

Is there a way to define the concept of manifolds so it looks more like “generalised affine spaces”?

What I have in mind is along the lines of this: Let $M$ a topological space, $V$ a normed vector space, and $$ \boxminus \colon M\times M \to V, $$ $$ \boxplus \colon M\times V \to M. $$ Then ...
3
votes
1answer
203 views

Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of non-singular projective varieties over $\mathbb C$. (Where a variety is understood as in ...
1
vote
1answer
46 views

How to prove that volume forms agree on $U_\alpha \cap U_\beta$?

I am familiarizing myself with Riemannian manifolds. Let $M$ be an orientable smooth $n$-manifold with atlas $(U_\alpha, (x_1^\alpha, \dots, x_n^\alpha))$ and let $g$ be a Riemannian metric on $M$. ...
1
vote
2answers
82 views

Stokes or homotopy?

The problem states Show that if $X$ is a simply connected manifold, then $\oint_{\gamma}\omega=0$ for all closed 1-forms $\omega$ on X and all closed curves $\gamma$ in $X$. However I have ...
1
vote
2answers
61 views

Question concerning tensors

As some of you may have seen from my previous question(s), I am working through Spivak's Calc on Manifolds and this happens to be the first time I've been introduced to tensors formally, though I have ...
0
votes
1answer
93 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
0
votes
1answer
413 views

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
0
votes
0answers
49 views

Is $x:\emptyset\to\emptyset$ a chart?

In the definition of a manifold, one defines, in particular, a chart as a homeomorphism $x:U\to O$ where $U\subseteq M$ is an open set of the topological space $M$ and $O\subseteq \mathbb{R}^n$. ...
1
vote
2answers
104 views

The top singular cohomology group with coefficient $\mathbb{Z}_2$ for non-compact, non-orientable manifolds without boundary

I want to know the top singular cohomology group with coefficient $\mathbb{Z}_2$ for non-compact, non-orientable manifolds without boundary. I think Poincare and Lefschetz duality may help. However, ...
5
votes
2answers
336 views

Compute a parallel transport

Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ be the $2$-sphere ($\mathbb{S}^{2} = \left\{ (x,y,z) \in \mathbb{R}^3, \; x^2+y^2+z^2 = 1 \right\}$). Let $p \in \mathbb{S}^{2}$ and $\xi \in T_{p}S^{2} = ...
2
votes
2answers
130 views

The space $x^3-y^2=0$

Consider $\{(x,y)\in\mathbf{R}^2 \ | \ x^3-y^2=0\}$ as a subspace of $\mathbf{R}^2$. Intuitvely I understand that this is not supposed to be a differentiable manifold because it has a cusp at $0$. But ...
0
votes
1answer
180 views

Prove the manifold of SU(2)/U(1) is the 2-sphere.

I want to demonstrate that the manifold of $SU(2)/U(1)$ is a 2-sphere. In a text-book I've found this way of solution, where there are some unclear points. Let to be $g= a\mathbb{1} + i b_j\sigma_j$ ...
3
votes
1answer
77 views

Proving that something is a manifold from the definition

Consider a set $$M = \{ (s\cos t, s\sin t, t) \colon s,t\in \mathbb{R}\}\subset \mathbb{R}^3.$$ I am asked to show from the definition that $M$ is a 2-dimensional submanifold of $\mathbb{R}^3$ ...
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0answers
79 views

Prove that if $T$ is one-to-one on $D$, then the set $T(D)$ is open

Let $f$ and $g$ have continuous first-order partial derivatives on an open set $ D\subseteq\mathbf{R}^2 $ and let $T :D \to \mathbf{R}^2 $ be defined by $ T(u,v)=(f(u,v),g(u,v)). $ ...
0
votes
1answer
40 views

Topology on the tensor Bundle $T^{r, s}(M)$?

Let $M$ be a smooth manifold and for $r, s\geq 0$ define the tensor bundle: $$T^{r, s}(M):=\bigcup_{p\in M} T_pM.$$ I'm trying to understand its topology. I'm following Homology and Curvature written ...
3
votes
2answers
173 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
0
votes
1answer
80 views

How to prove 2x2 rotation matrix is a manifold [duplicate]

How can I prove that this matrix is a manifold? $\begin{pmatrix} \cos(a) & -\sin(a) \\ \sin(a) & \cos(a) \end{pmatrix}$ Thanks!
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0answers
39 views

Topology of $L^2$ space

Cardinality of space of all funcions $f: \mathbb R \rightarrow \mathbb R$ is $\beth_2$. However, cardinality of space of all such square-integrable functions, space $L^2$, is $\beth_1=\mathfrak c$, ...