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

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6
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0answers
48 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 ...
-1
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0answers
15 views

What is the covariant basis around a schwarzschild black hole? [migrated]

First of all, I'm not interested in time for this question. So lets consider a 3-manifold whose metric is the spatial part of the Schwarzschild metric, so it has the event horizon and the singularity ...
1
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1answer
83 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
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0answers
29 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
57 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 ...
1
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0answers
24 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 ...
0
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0answers
26 views

To Prove that The Level Set Of AConstant Rank Map is a Manifold

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of ...
2
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1answer
67 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
22 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
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0answers
52 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
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1answer
52 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
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1answer
32 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
52 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
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2answers
44 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
63 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
81 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 ...
1
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0answers
19 views

Function represented as composition

Question:Prove that if $\vec{g} : \mathbb{R}^n \rightarrow \mathbb{R}^n $ and $ \det(\vec{g}') \neq 0$, then in some open set $V \subset \mathbb{R}^n $ such that $\vec{x} \in V$ we have: $\vec{g} = ...
0
votes
0answers
38 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
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2answers
46 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, ...
4
votes
2answers
37 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
117 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
66 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
35 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
72 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
34 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 ...
2
votes
2answers
68 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
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1answer
47 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!
-2
votes
2answers
52 views

If $f : M\rightarrow N$ be immersion then $f_*$, derivative of $f$ is an immersion. [closed]

Suppose that $f : M\rightarrow N$ be immersion. Prove that $f_*$, derivative of $f$, is immersion too?
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0answers
33 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 $\beth_1=\mathfrak c$, ...
1
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1answer
45 views

A question regarding Jacobi fields and families of geodesics

I'm trying to show that for any one-parameter family of geodesics $\gamma(s,t)$ (where $\gamma(s_0,t)$ is a geodesic for any constant $s_0 \in (-\epsilon, \epsilon)$) defined on a Riemannian manifold ...
2
votes
0answers
38 views

Neighbourhood in a manifold is open

I'm trying to solve a problem in Spivak's A comprehensive introduction to differential geometry. Here, the definition of a manifold is the next A metric space $X$ is said to be a manifold if ...
1
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3answers
35 views

Non-compact manifold with compact boundary

What is an example of a non-compact manifold with compact boundary?
0
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1answer
30 views

Connected manifold with disconnected boundary?

Is there any simple example of a connected manifold with disconnected boundary?
3
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2answers
50 views

Why do a set of continuous transformations form a manifold?

I am reading Sean Caroll's book on GR, and he defines manifolds to be "a space that may be curved and have a complicated topology, but in local regions looks just like R$^n$. Here by "looks like" we ...
0
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0answers
52 views

Vectors in tangent space to a manifold independent of coordinates

In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator $X$ acting on some function ...
2
votes
1answer
73 views

Is every hypersurface in $\mathbb{R}^n$ the boundary of an open domain?

We know if $\Omega \subset \mathbb{R}^{n}$ is a bounded $C^k$ domain, then its boundary $\partial\Omega$ is a $C^k$ compact hypersurface of dimension $n-1$. Is it true that every $m-$dimensional ...
1
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1answer
63 views

Why klein Bottle is 4-D?

I am wondering that Klein Bottle is 4-D. Can any body tell me how it is possible? I can give coordinates for each point of the Klein Bottle with 3 values. Then how it can be 4-D? What is immersion? ...
3
votes
1answer
46 views

Variation on Stokes Theorem for Manifolds (2)

Let $\omega \in \Omega^0(\mathbb{R}^{2}\setminus\{0\})$ be a $0$-form such that $d\omega=0$. Is the following statement true: For any compact, oriented, $0$-dimensional submanifold $M$ of ...
4
votes
1answer
44 views

Bundle metric and connection on trivial vector bundle

I read this: Let $(M,g)$ be a compact Riemannian manifold and let $W$ be a vector bundle (rank $n$) over $M$ with $h_W$ a bundle metric of $W$ and $D$ a bundle connection of $W$. I choose $W$ ...
0
votes
0answers
21 views

Differentiability of a function on a manifold is independent of the coordinate chart

I am currently working through Nakahara's book, "Geometry, Topology and Physics", and have reached the stage at looking at calculus on manifolds. In the book he states that "The differentiability of a ...
0
votes
0answers
13 views

What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap ...
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0answers
28 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
6
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1answer
50 views

If M is a manifold of dimension $ n \neq0$ then M has no isolated points.

I am in doubt whether the following statement is true or false: "If M is a manifold of dimension $ n \neq0$ then M has no isolated points." The idea that made me find the true statement was as ...
0
votes
2answers
58 views

Variation on Stokes Theorem for Manifolds

Let $n >1$ and $\omega \in \Omega^{n-1}(\mathbb{R}^{n+1}\setminus\{0\})$ such that $d\omega = 0$. Is the following statement true: For any compact, oriented, $(n-1)$-dimensional submanifold $M$ ...
0
votes
0answers
25 views

Existence of a fixed-point free map in a manifold.

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map. I know ...
2
votes
1answer
32 views

Is the Riemannian distance function Lipschitz on a hypersurface?

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$ of dimenion $n$. Is it true that there exists a constant $C$ such that $$d(x,y) \leq C|x-y|$$ for all $x, y \in M$? Here $d$ is the Riemannian ...
0
votes
1answer
29 views

What is $T^0_0(M,W)$ where $W$ is trivial vector bundle over a compact manifold $M$?

Let $W=(M \times \mathbb{R}, pr, M)$ be the trivial vector bundle over a compact manifold $M$, and define $$V=T^0_0(M,W) := T^0_0M \otimes W,$$ and $V$ is called "the vector bundle of $W$-valued ...
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0answers
23 views

Time Derivative of the integral over a singular k-cube

I am stuck on this question, and was wondering if someone could provide a hint of where to start? I can't see the first step.
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1answer
60 views

Does the Tangent Space Vary Continuously with The Points On a Manifold?

I recently read about Grassmannian manifolds. The following question naturally comes to mind. Let $GR_k(\mathbf R^n)$ is the grassmannian manifold of $k$ dimensional linear subspaces of $\mathbf ...
3
votes
2answers
70 views

How to visualize the gradient as a one-form?

I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I still visualize gradients as vector fields instead of the ...