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
11 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
7 views

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

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 ...
2
votes
1answer
35 views
+100

Reference request: infinite-dimensional manifolds

The following books develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of Global ...
5
votes
0answers
22 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ The following is my solution. Observe that we can ...
1
vote
1answer
22 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 ...
5
votes
1answer
56 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 ...
0
votes
1answer
35 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 ...
8
votes
1answer
50 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 ...
2
votes
2answers
25 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 ...
6
votes
0answers
79 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 $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
1
vote
0answers
20 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 ...
4
votes
2answers
70 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 ...
2
votes
1answer
40 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 $M \subseteq \mathbb{K}^n$ ...
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
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 ...
0
votes
1answer
58 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. ...
15
votes
1answer
375 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
-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
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 ...
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 ...
-1
votes
2answers
34 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 ...
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 ...
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 ...
7
votes
1answer
82 views

Hurewicz map factors through bordism homology

I've read in multiple sources that the hurewicz map $h \colon \pi_n(X) \to H_n(X)$ factors through oriented bordism homology. I'm particularly interested in the injectivity of the map $h \colon ...
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 ...
1
vote
0answers
27 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
27 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 ...
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
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 ...
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 ...
1
vote
2answers
36 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 ...
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" ...
2
votes
1answer
62 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 ...
0
votes
1answer
22 views

The definition of C^r Structural Stability

I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate. Would the definition ...
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 ...
6
votes
1answer
413 views

proof of stokes theorem

I don't understand the "idea" of the following proof, as well as some of the steps. As i'm not sure about its "ways", im not editing it much and as such it might be in the wrong order. My sincere ...
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 ...
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 ...
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 ...
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$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
-3
votes
1answer
2k views

Lee, Introduction to Smooth Manifolds Solutions

Does anybody know where I could find the solutions to the exercises from the book Lee, Introduction to Smooth Manifolds? I searched on the Internet and found only selected solutions but not all of ...
2
votes
1answer
139 views

What does it mean to “calculate in local coordinates” on a manifold?

In differential geometry textbook one sometimes reads "calculating in local coordinates, we obtain..." What does this expression mean? Say, $M$ is a smooth manifold and $h$ is a function on $M$; what ...
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 ...