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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0answers
35 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
44 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
28 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
49 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
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
62 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
65 views
+50

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
vote
2answers
41 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
112 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
63 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 $\mathbb{R}^3$ ...
1
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0answers
66 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
31 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
66 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
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!
-3
votes
2answers
49 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?
-2
votes
0answers
25 views

Question about continuous and injective map between two same dimensional smooth manifold. [closed]

Suppose that $M$ and $N$ be two same dimensional smooth manifold. Prove that $f(M)$ is open in $N$ where the map $f : M\rightarrow N$ is continuose and injective.
1
vote
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
vote
1answer
42 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
37 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
34 views

Non-compact manifold with compact boundary

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

Connected manifold with disconnected boundary?

Is there any simple example of a connected manifold with disconnected boundary?
2
votes
2answers
49 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
votes
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
vote
1answer
61 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
45 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 ...
1
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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
votes
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
57 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 ...
0
votes
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.
8
votes
1answer
59 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 ...
2
votes
2answers
66 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 ...
2
votes
0answers
35 views

How to visualize cotangent spaces.

I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
1
vote
1answer
34 views

Why is the image of a smooth embedding f: N \rightarrow M an embedded submanifold?

I'm reading An Introduction to Manifolds (Tu) and got confused on p.123 Theorem 11.13. Let me briefly explain what was done before that. The author defines an embedding between two manifolds $f: ...
0
votes
1answer
23 views

Negative Gauss Curvature

Let S be a manifold of dimension 2, compact and orientable. Suppose its border is made of k geodesic circumferences, with $k \geq 3$. Show that there exists a point in S with negative Gauss ...
1
vote
1answer
27 views

Find an explicit atlas for this submanifold of $\mathbb{R}^4$

I'm having a hard time coming up with atlases for manifolds. I can prove using the implicit function theorem that $M = \{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=x_3^2+x_4^2=1\}$ is a ...
1
vote
2answers
40 views

Give an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping.

I'd like to know if there is an explicit atlas for the manifold $\mathbb{R}P^3$ which is defined as the quotient of the three-sphere by the antipodal mapping. Thanks.
0
votes
0answers
42 views

Integral Curve of the vector field

If we have a 2-sphere with coordinates $x=r \cos \theta \cos \phi$, $y= r \sin \theta \sin \phi$ and $z=r \sin \theta$ and the vector field $X= (-r\sin \theta \cos\phi, r \cos \theta \sin \phi, r \cos ...
0
votes
1answer
20 views

What is the significance of incompatible coordinate charts for a manifold?

For reference, here is my definition of a "manifold". A $\,C^\infty$ manifold is a topological manifold together with all the admissible charts of some $C^\infty$ atlas. When considering the ...
0
votes
1answer
37 views

Locally exact vs globally exact

Why the volume form in Sphere is locally exact but not globally exact? here the integral is integral $$\int_{S^n}w$$ with $$w = \frac{1}{r} \sum_{i=1}^{n+1} (-1)^{i-1} x_i dx_1 \cdots\hat{dx_i},\cdots ...