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

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18
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3answers
337 views

Is the tangent bundle of $S^2 \times S^1$ trivial or not?

As the question title suggests, is the tangent bundle of $S^2 \times S^1$ trivial or not? Progress: I suspect yes. If I could construct three independent vector fields, I would be done. But I'm not ...
2
votes
1answer
30 views

How can uncountably many closed smooth 4-manifolds be presented by an essentially countable alphabet (Kirby diagrams)?

A smooth, closed 4-manifold admits a handle decomposition which is specified completely by its Kirby diagram. A Kirby diagram, up to isotopy, can be seen as a labelled morphism in the tangle category. ...
2
votes
0answers
77 views

Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
2
votes
1answer
113 views

How can a Kirby diagram fail to determine a handle decomposition?

I've read that a handle decomposition for 4-manifold determines a unique smooth structure, and I've also read that every 4-manifold admits a Kirby diagram. So when does a Kirby diagram fail to ...
3
votes
1answer
163 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
1
vote
1answer
43 views

Why is the kernel of the connection one form a connection on a principal bundle?

Let $\pi:P\rightarrow M$ be a principal bundle and let $\omega\in \Omega(P;\mathfrak{g})$ be a one form satisfying $\omega(\sigma(X))=X$ and $R_g^*\omega=\text{Ad}_{g^{-1}}\circ\omega$ Then ...
5
votes
1answer
63 views

Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type \begin{equation}(1)\quad \ddot x + f(x)=0 \end{equation} ...
0
votes
1answer
25 views

Show that $M\#\mathbb S^n\cong M$.

I recall that $M_1\#M_2$ is the connexe sum of two manifolds and it's defined as following: Let $B_1\subset M_1\backslash \partial M_1$ and $B_2\subset M_2\backslash \partial M_2$ where $M_i$ have ...
1
vote
1answer
38 views

differential forms of 2 sphere

Assume that $w$ is a 1-form on the 2-sphere $S^{2}$ so that $A^{*}w = w$ for all $A \in SO(3)$. Show that $w = 0$ I have tried to apply the definition of pullback and special orthogonal group, but I ...
0
votes
1answer
26 views

The basis of a tangent space [closed]

I don't know how to start the second part. Because I can't find a chart for this submanifold, is there anyway to do this problem without an explicit chart?
0
votes
1answer
25 views

Existence of a chart with given properties.

I am trying to prove that for a smooth manifold $M$ there is a chart $( U, \phi = (u_1, \dots, u_m))$ such that $\phi(U)=\mathbb{R}^m$, $\phi(p)=0$ and $\xi= \left. \frac{\partial}{\partial u_1} ...
1
vote
1answer
17 views

Prove that the map $\pi$ is a submersion

I'm trying to solve the following problem: Let $\tilde M$ be the set of real matrices $3 \times 2$ of rank 2. $$ (u,v) = \begin{pmatrix} u_1 & v_1 \\ u_2 & v_2 \\ u_3 & v_3 ...
5
votes
0answers
92 views

Cell-structure for Grassmann manifolds, is restriction homomorphism an isomorphism for $p < k$? [closed]

Is the restriction homomorphism$$i^*: H^p(G_n(\mathbb{R}^\infty)) \to H^p(G_n(\mathbb{R}^{n+k}))$$an isomorphism for $p < k$? Here, any coefficient group may be used.
0
votes
1answer
33 views

Prove that $Q=\{(x_1,…,x_n)\in \mathbb R^n\mid \forall i, x_i\geq 0\}$ is a topological manifold with boundary.

I have to prove that $Q=\{(x_1,...,x_n)\in \mathbb R^n\mid \forall i=1,...,n,\ x_i\geq 0\}$ is a topological manifold with boundary. The fact that the topology is second countable and hausdorff is a ...
2
votes
2answers
38 views

If $f:\overline{U}\longrightarrow \overline{V}$ is a homeomorphism, then $f|_{Bd(U)}$ is a homemorphism on $Bd(V)$

Let $U,V\subset \mathbb R^n$ open and $f:\overline{U}\longrightarrow \overline{V}$ a homeomorphism. Let $U=int(\overline{U})$ and $V=int(\overline{V})$. Show that $f|_{Bd(U)}$ is a homeomorphism in ...
1
vote
1answer
31 views

Need clarification to understand an example of manifold (the $n$-sphere)

This is one of the first examples of manifolds, from John M. Lee - Introduction to Smooth Manifolds. I'm having trouble to understand two things, which I'm gonna show in the simple case of $n=2$. ...
0
votes
0answers
21 views

Annulus 1-1 mapping onto a single coordinate patch.

In the book in Geometrical methods of mathematical physics by Bernard Schutz he says in the end of section 2.2 [...], the two-dimensional interior of the annulus bounded by two concentric circles ...
1
vote
1answer
56 views

The exterior derivate and pullback commute

The above question is from a past exam. I am having trouble with the fine details, ie what $F*dw$ and $dF*w$ actually look like. Can anybody show me how this question is solved? I have solved it ...
2
votes
2answers
71 views

Use of partial derivatives as basis vector

I am trying to understand use of partial derivatives as basis functions from differential geometry In tangent space $\mathbb{R^n}$ at point $p$, the basis vectors $e_1, e_2,...,e_n$ can be written ...
3
votes
2answers
104 views

Introductory book on differential geometry for engineering major

I am an engineering major and looking for a straightforward, easy to understand basic book on differential geometry to get started. At starting point, I am not looking for a comprehensive book (may be ...
0
votes
0answers
41 views

Understanding manifolds (asking just for confirmation)

In lecture we used the following definition of manifolds: A subset $ M \subset \mathbb{R}^n $ is called a k-dimensional manifold of the class $C^\alpha$, if $ \forall a \in M $ there is an open ...
0
votes
1answer
30 views

The $f^{-1}(y)$ is locally constant where $ y$ ranges through regular values

Let $f: M \longrightarrow N$ be a smooth map between two manifolds of the same dimension, with M compact, and a regular value $y \in N$. Then the number of points in $f^{-1}(y)$ is locally constant as ...
1
vote
0answers
34 views

Some details on the tangent space of a manifold

Let $x$ be a point of some smooth manifold $(M,\mathcal U)$ of dimension $n$. Let $I_x$ be the set of all $U\in \mathcal U$ containing $x$. Define the relation "$\sim_x$" on $I_x\times \mathbb R^n$ by ...
1
vote
1answer
96 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 ...
0
votes
0answers
20 views

How does one see, if a set is a manifold or not ? Understanding sharp corners/edges

I am still having troubles to understand, when sets are manifolds and not. So I stumbled across the following posts: Deciding whether a given set is a manifold In the lecture we used the following ...
0
votes
2answers
58 views

how to show that $S^2/\Gamma$ is not a manifold

Let $\Gamma$ be the cyclic group generated by the matrix $$\begin{pmatrix} \cos(2\pi/3) & \sin(2\pi/3) & 0 \\ -\sin(2\pi/3) & \cos(2\pi/3) & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ Show ...
9
votes
1answer
102 views

Can $\mathbb C P^4$ be smoothly embedded in $\mathbb R^{12}$?

In Bott and Tu's Differential Forms in Algebraic Topology, the authors show using Pontrjagin classes that $\mathbb CP^4$ cannot be smoothly embedded in $\mathbb R^k$ when $k\le 11$. The obvious ...
3
votes
1answer
53 views

Set consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into additive group? [closed]

How do I see that the set $\mathfrak{N}_n$ consisting of all unoriented cobordism classes of smooth closed $n$-manifolds can be made into an additive group?
6
votes
0answers
33 views

Sobolev space and integration by parts on non-orientable manifolds

Let $M$ be a compact manifold without boundary which is not orientable. Do all the standard facts that apply to oriented manifolds and Sobolev spaces also apply here? Like Green's formula for example. ...
1
vote
1answer
67 views

Showing something is homeomorphic to $S^2$.

Suppose $X,Y$ are compact surface such that $X\#Y \approx X$ for any compact $X$. Show that $Y$ is topologically equivalent to the sphere. I was thinking for a while about this. It seems pretty ...
0
votes
0answers
21 views

Height function on the torus and regular values

I have the theorem that if $f:\mathcal{M}\rightarrow\mathcal{N}$ is smooth, and $y\in{f(\mathcal{M})}\subset\mathcal{N}$ is a regular value, then $f^{-1}(y)$ is a proper submanifold of dimension $m-n$ ...
10
votes
2answers
227 views

What are examples of parallelizable complex projective varieties?

A smooth complex projective variety is the zero-locus, inside some $\mathbb{CP}^n$, of some family of homogeneous polynomials in $n+1$ variables satisfying a certain number of conditions that I won't ...
1
vote
0answers
34 views

Polynomial functions on a smooth manifold

If one views $\Bbb R ^{2n}$ as the cotangent bundle of $\Bbb R ^n$, with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$, then in order to do classical Hamiltonian mechanics on it one considers ...
4
votes
1answer
53 views

Any $\mathbb{R}$-linear mapping $X: C^\infty(M, \mathbb{R}) \to \mathbb{R}$ with $X(fg) = X(f)g(x) + f(x)X(g)$ given by $X(f) = Df_x(v)$?

Let $M$ be a smooth manifold, and let $C^\infty(M, \mathbb{R})$ denote the collection of smooth real valued functions on $M$. For $x \in M$, how do I see that any $\mathbb{R}$-linear mapping $X: ...
1
vote
1answer
45 views

Why is an embedding an injective immersion?

My course on manifolds defines an embedding as follows: 'A smooth map $f:\mathcal{M}\rightarrow\mathcal{N}$ between manifolds $\mathcal{M}$ of dimension $m$ and $\mathcal{N}$ of dimension $n$ is an ...
7
votes
2answers
63 views

If $M$ is compact, every maximal ideal in $F$ arises in this way as a point of $M$?

For any smooth manifold $M$, the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and every point $x \in M$ determines a ring homomorphism $F ...
8
votes
2answers
229 views

Do the singular matrices form a topological manifold

So the definition of manifold I'm using is that of a topological manifold (a topological space with an atlas of homeomorphisms to $\mathbb{R}^n$). I have two related questions: Is the set of ...
3
votes
1answer
49 views

Homology of a subspace of a $2$-manifold

In some homework solutions someone posted online, one of the problems was from Hatcher's Algebraic Topology: computing the homology groups of the space which is the union of the boundary of $I \times ...
3
votes
1answer
42 views

Collection of smooth real valued functions on smooth manifold has ring structure.

For any smooth manifold $M$, how do I see that the collection $F = C^\infty(M, \mathbb{R})$ of smooth real valued functions on $M$ can be made into a ring, and that every point $x \in M$ determines a ...
7
votes
1answer
67 views

How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
1
vote
1answer
40 views

What is the relationship between the unit simplex and the nonnegative orthant?

I came upon this figure while reading something online. Pictured above is the intersection of the unit sphere in the nonnegative orthant and the unit simplex. Question: What is the relationship ...
4
votes
1answer
53 views

Does every torus $T\subset S^3$ bounds a solid tours $S^1\times D^2\subset S^3$?

I want to show this by using Alexander's Theorem's proof method. So here's what I thought. As I surger $T$, I have 2 $S^2$. So one bounds $S^1$ and the other bounds $D^2$. By reversing the surgery, ...
3
votes
1answer
58 views
3
votes
1answer
123 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 ...
2
votes
1answer
37 views

Optimization on manifold via Lagrange multipliers

Let the manifold $S$ in $\mathbb R^n$ be defined by $g(x)=0$. If $p$ is a point not on $S$, and $q$ is the point of $S$ which is closest to $p$, show that the line from $p$ to $q$ is perpendicular to ...
4
votes
1answer
31 views

If $M$ is Riemannian, then $\kappa_f \oplus f^*TN \cong TM$, where $\kappa_f$ is built out of kernels of the $Df_x$?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. I know how to construct a vector bundle $\kappa_f$ built out of the ...
2
votes
0answers
73 views

Correct definition of gradient (and divergence) on smooth manifolds (for engineers)

I am very sorry if this is a very trivial question or my formulation is inadequate, I am only an engineer, and I am not familiar with the topic. I am looking for the definition of the gradient of a ...
6
votes
2answers
246 views

Unique factorization of manifolds?

I wonder if there is a result on the unique factorization of manifolds. Call a topological manifold to be indecomposable if it is not homeomorphic to a product of manifolds of positive dimension. Is ...
1
vote
0answers
30 views

Why do those terms vanish if the metric is Hermitian?

On this [page][1], the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor ...
4
votes
1answer
38 views

Constructing a vector bundle built out of kernels of the Jacobian?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. How do I construct a vector bundle $\kappa_f$ built out of the kernels of ...