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

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3
votes
1answer
51 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
0
votes
0answers
13 views

Prove that unitary normal vector to a manifold is $\nabla g / |\nabla g|$ [duplicate]

I want to solve the following exercise: Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable ...
7
votes
1answer
48 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
5
votes
1answer
67 views

Can a $1d$ space never be curved?

I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature $-1.$ I believe ...
2
votes
2answers
37 views

Homeomorphism in the definition of a manifold

Many texts will define a manifold as "a second-countable Hausdorff space that is locally homeomorphic to Euclidean space". By definition of homeomorphism, shouldn't this really and officially read as ...
8
votes
1answer
72 views

Can (singular) homology classes always be represented by images of closed manifolds?

My intuition tells me that if $A \in H_2(M;\mathbf Z)$, then $A$ can be represented by a map $\Sigma \to M$, where $\Sigma$ is a closed (= compact boundaryless) surface, i.e., the connected sum of ...
0
votes
1answer
48 views

Compute the volume element in a differentiable manifold.

Let $A\subset\mathbb{R}^n$ open $g:A\to\mathbb{R}$ of class $C^{1}$ and $g'(x)\not=0$ in each $x\in A$ then I want to compute $dV$ in the differentiable manifold $ M = g^{-1}(0)$. The thing is that ...
1
vote
1answer
41 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
1
vote
1answer
36 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
2
votes
1answer
49 views

Groups $\pi$ with $K(\pi, 1)$ a finite CW-complex

For what (say, finitely generated) groups $\pi$ is $K(\pi, 1)$ a finite CW-complex? Such a group must necessarily be torsion-free, since otherwise $H^*K(\pi, 1) = H^* \pi$ would be nontrivial in ...
4
votes
1answer
34 views

Revolving a $k$-manifold around an axis gives a $(k+1)$-manifold

I want to solve the following problem from M. Spivak's Calculus on Manifolds: Let $\mathbb{K}^n=\{x \in \mathbb{R}^n:x^1=0 \text{ and }x^2>0,\dots,x^{n-1}>0\}$. If $M \subseteq \mathbb{K}^n$ ...
7
votes
0answers
149 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
4
votes
2answers
95 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
0
votes
0answers
9 views

Proof of the categorisation of 1 dimensional connected differential manifolds, using the topological classification?

If we know that every connected, second countable topological 1-manifold is homeomorphic to the circle or the real line, is there a simple way to use it to prove the analogous statement for ...
0
votes
1answer
35 views

Confused with The Transversality Theorem when all manifolds are boundaryless

In Guillemin-Pollack's book Differential Topology, the Transversality theorem states that The transversility Theorem. Suppose that $F:X \times S \to Y$ is a smooth map of manifolds, where only $X$ ...
2
votes
0answers
35 views

Try to use Homogeous space Characterize the space of all lines in the plane .

The problem is just to gives a smooth manifold structure of all straight lines in $\mathbb{R^2}$ ( not just those which pass through the oringin).Moreover, identify it with a well-known manifold! My ...
3
votes
1answer
46 views

Manifold notes in more informal way

When defining the properties of scalar functions that live in manifold $M$ in a less formal way, the following is said: "We no longer refer to a covering by coordinate patches. Instead we conceive of ...
0
votes
1answer
107 views

The closure of an open set in $\mathbb{R}^n$ is a manifold

I want to solve the following exercise from M. Spivak's Calculus on Manifolds (p. 114): (a) Let $A \subseteq \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n-1)$-dimensional manifold. ...
-1
votes
2answers
35 views

How can we show that $S^0$ is a manifold?

Recall $S^n = \{ (x^0, ..., x^n) \in \mathbb{R}^{n+1}: {x^0}^2 + ... + {x^n}^2 = 1 \}$ $S^0$ is a very cute set on $\mathbb{R}$ consisting of points $\{-1, 1\}$. How can we show that it satisfies the ...
5
votes
0answers
41 views

Cartan geometry on manifolds with boundary

I was reading Sharpe's text on Cartan geometry, and I started to wonder: Does the theory change in any significant way if the base manifold for the Cartan principal bundle is allowed to be a smooth ...
0
votes
1answer
24 views

Connection and curve

Let $\nabla$ be a connection on a Riemannian manifold and let the differential of a curve be given by $$c'(t)=c_1'(t)\partial_1 + c_2'(t) \partial_2.$$ Now I was wondering how we define ...
0
votes
0answers
18 views

how many Poincare dodecahedrons fill Poincare dodecahedral space?

I was reading Jeffrey's Weeks "shape of space" and that made me wonder: Every spherical 3d manifold (3d Sphere) has a finite volume, The Poincare dodecahedral space is a 3d Sphere. this manifold ...
1
vote
0answers
33 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
4
votes
4answers
84 views

Can someone illustrate the definition of manifold with a simple example?

In my text the definition of a differential manifold is given as follows: A subset $M$ of $\mathbb{R}^n$ is a $k$ dimensional manifold if $\forall x \in M$ there are open subsets $U$ and $V$ of ...
1
vote
0answers
32 views

Topological properties of regular and critical points and values

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds. Consider the following two statements, the second one under the assumption The set of regular points of $f$ are open in $M$, ...
0
votes
0answers
29 views

The same manifold embedding in different ambient spaces

Suppose we have a manifold embedding in different ambient spaces ($R^{n_1}$,$R^{n_2}$,...,$R^{n_k}$), and we observe some sample data from these ambient spaces, our goal is to acquire the latent ...
4
votes
1answer
70 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
1
vote
1answer
48 views

Sufficient condition for $\mathbb{Z}$-orientability

Let $X$ be a topological $n$-manifold. Let's define a R-orientation on $X$ as a choice of generators $\alpha_{x}\in H_{n}(X,X\setminus\lbrace x\rbrace;R)$ that is consistent. Suppose that $X$ is ...
3
votes
1answer
66 views

Is there an infinite dimensional Lie group associated to the Lie algebra of all vector fields on a manifold?

Since the space $\Gamma(TM)$ of all vector fields on a smooth manifold $M$ is a real Lie algebra with respect to the usual commutator bracket, I was curious if in fact it is the Lie algebra of some ...
1
vote
1answer
31 views

Differential manifold of dimension $m^2$

How could I prove that $\mathrm{GL}(m, \mathbb{R})$ is a differential manifold of dimension $m^2$? $\mathrm{GL}(m, \mathbb{R})$ is the set of all non-singular $m\times m$ matrices in $\mathbb{R}$ ...
2
votes
1answer
50 views

Projection of surfaces in $\mathbb{C}^2, \mathbb{C}P^2, \mathbb{C}H^2$ to $\mathbb{R}^3$

As part of my thesis in Riemannian geometry, I study surfaces in $\mathbb{C}^2$, $\mathbb{C}P^2$ and $\mathbb{C}H^2$. Since visulation is always nice, I was wondering if there existed any "nice" ...
0
votes
1answer
60 views

pullback of continuous maps of manifolds

I'm trying to prove the following: (a) If $X, Y$ are smooth manifolds, then the map $\psi:X\to Y$ is smooth $\Leftrightarrow$ $\psi^*(C^\infty(Y))\subseteq C^\infty(X)$ (b) If $\psi:X\to Y$ is a ...
0
votes
0answers
16 views

About submanifolds, diffeomorphisms, compact subsets and jordan measurable subsets.

I am given the following problem set which left me pretty puzzled. Let $M \subset \mathbb{R}^n$ be a $C^1$ submanifold and $$\psi : \Omega \rightarrow U \cap M$$ a chart of $M$. $\Omega \subset ...
2
votes
0answers
28 views

$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
1
vote
1answer
31 views

Poincaré invariant corresponds to area?

In this paper(equation 34) click me it says that if we have a symplectic egg, then the area of a $(x_i,p_i)$ clone intersected with the egg is given by the poincaré invariant $\int p dx = \int_0^{2 ...
3
votes
1answer
103 views

Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine.

Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a smooth function. The theorem of Sard gives us that ...
0
votes
0answers
26 views

A manifold with boundary in $\mathbb{R}^{n}$.

I want to show that the cylinder $C = \{ (x,y,z)\in\mathbb{R}^3: x^2 + y^2 = 1, 0 \le z \le 1 \}$ is a differentiable manifold with boundary, of dimension 2, this is: A subset $M \subset ...
1
vote
1answer
43 views

Show that this is indeed a differentiable manifold with boundary.

I want to show that the cylinder: $$C = \{ (x,y,z)\in\mathbb{R}^3: x^2 + y^2 = 1, 0 \le z \le 1 \}$$ is indeed a a differentiable manifold with boundary, this means the following: A subset $M ...
0
votes
2answers
41 views

Local representation of a submanifold as a graph over the tangent plane

I'd like to verify the following statement, which intuitively seems quite reasonable, by a rigorous proof: Let $M \subset \mathbb{R}^D$ be a $d$-dimensional $C^1$ submanifold embedded in ...
0
votes
0answers
20 views

Isomorphism of the Clifford bundle of a Riemannain manifold

Let $(M,g)$ be an oriented Riemannian manifold and $Cl(M):=\bigcup_{x\in M}Cl(T_xM,g_x)$ be the clifford bundle of $(M,g)$. (Here $Cl(T_xM,g_x)$ denotes the clifford algebra of the vector space ...
2
votes
1answer
67 views

Sheaf, étalé space with Riemann surfaces.

Let $f:X\rightarrow Y$ be an holomorphic map betwen two Riemann surfaces and let: $\Gamma:=${ $(x,y)\in X\times Y|y=f(x)$ } $\subset X\times Y$ be the graph of $f$. I have to show that ...
4
votes
1answer
49 views

$\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
1
vote
1answer
55 views

For $(n-1)$-form $\omega$ on $M^{n}$ compact, orientable without boundary, then $d\omega$ vanish for some point

Let $M^{n}$ manifold compact, orientable without boundary and $\omega$ $(n-1)$-form then there is $p\in M$ such that $d\omega(p)=0$. This is for my homework of integration on manifolds & Stokes ...
1
vote
2answers
38 views

Explain why this set is not a differentiable manifold

I want to figure out why the set of zeros of the function $g:\mathbb{R}^{2} \to \mathbb{R}$ defined as $g(x,y) = x^2 - y^2$ is not a differentiable manifold. So what I want to use is the following ...
2
votes
1answer
47 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
0
votes
1answer
26 views

Show that the tangent space of a manifold is a certain set.

Let $A\subset \mathbb{R}^n$ an open set, and $g:A\to \mathbb{R}$ continously differentiable such that $g'(x)\not=0 $ for $x\in A$. If $M = g^{-1}(\{0\})\not=\emptyset$, then I want to show that the ...
2
votes
1answer
76 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
1
vote
2answers
78 views

Why the set $g^{-1}(\{0\}) $ is not a differentiable manifold?

Let $g:\mathbb{R}^2 \to \mathbb{R}$ given by $g(x,y) = x^2 - y^2$. Then I am triying to figure out why this function is not a differentiable manifold , I was trying to give an explicit coordinate ...
1
vote
4answers
66 views

Are there compact manifolds without boundary?

Based on this question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the topology of the manifolds has to be the trace ...
1
vote
0answers
19 views

Pipe-fitting conditions in 3D

Let's we have smooth (continuous and infinitely differential) curve $f(x(t), y(t), z(t)) = 0$ in 3D. Now I want to build a pipe of diameter $D$ around it. Questions: What are the set of conditions ...