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

learn more… | top users | synonyms (1)

4
votes
1answer
39 views

Characterization of 1-dimensional manifolds.

My intuition tells me that the only connected 1-dimensional topological manifolds are the real line $\mathbb{R}$ and the circle $S^1$. Is this true? If yes, is it possible to prove it from first ...
0
votes
0answers
39 views

Is a polyhedron an affine manifold?

I was reading the definition of an affine manifold (https://www.wikiwand.com/en/Affine_manifold) and was wondering if a polyhedron is an affine manifold. Could you also provide any hints to the proof ...
0
votes
0answers
22 views

Normal coordinates

I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates. So the exponential map is a diffemorphism $exp:U \subset T_pM \rightarrow V \subset ...
1
vote
0answers
22 views

Zips and Zippers

I'm currently reading Differential Manifolds by Antoni Kosinski, and the concept of a zip--defined as half of a zipper--is mentioned early on, of course with the intent of connecting manifolds. This ...
-2
votes
0answers
33 views

Almost independent vectors- Where do they live on a manifold [on hold]

I am new to this thing. I am having the next question : Almost independent vectors- Where do they live on a manifold? In a manifold with larger dimmension? Tnks!So don't be tuff with me cause I am ...
0
votes
0answers
10 views

Does the gradient gives a natural orientation in a manifold? [duplicate]

I want to solve the following problem: 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 ...
1
vote
0answers
48 views

Representation of sum of homology classes

Let $X$ be a path-connected topological space, let $x, x' \in H_{k}(X)$ for $k>0$ be represented by two connected manifold i.e. there exist two compact oriented connected manifolds $M$, $N$ and two ...
2
votes
1answer
23 views

The definition of a differentiable vector field on a manifold

I have a question regarding the following section from M. Spivak's Calculus on Manifolds: Let $M$ be a $k$-dimensional manifold in $\mathbb{R}^n$ . . . . . . Suppose that $A$ is an open set ...
4
votes
0answers
20 views

G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
3
votes
1answer
43 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
40 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
63 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
32 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
56 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
41 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
33 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
31 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
41 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
28 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
106 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
80 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
8 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
28 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
31 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 ...
-1
votes
0answers
25 views

how to embed a square into $R^2$?

By Whitney embedding theorem you can embed a smooth 1-manifold in $\mathbb{R}^2$. Now if you give the unit square a smooth structure(for example by inducing the unit circle's smooth structure on it), ...
3
votes
1answer
41 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
67 views
+50

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
36 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
22 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
17 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
29 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
74 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
62 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
64 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 $\mathbb{R}$ ...
1
vote
0answers
36 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
59 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
15 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
27 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
25 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
42 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
31 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
18 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 ...