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
25 views

Homology of manifolds using submanifolds [duplicate]

Motivation I have learned algebraic topology. In simplical homology, we define $C_k(X)$ as an abelian group freely generated by $k$-dimensional skeleton $X^{(k)}$, and boundary operator $\partial_k$ ...
1
vote
1answer
27 views

compact oriented $4k$-manifold-> euler characteristic is congruent to the signature mod $2$

Let $M$ be a compact oriented $4k$-manifold, $\chi_M$ the euler characteristic of $M$ and $sig(M)$ the signature of $M$. Why is $\chi_M \equiv sig(M)$ mod $2$? In the book " a concise course in ...
4
votes
1answer
45 views

Redundancy in the definition of vector bundles?

In John Lee's classic Introduction to Smooth Manifolds, the following definition of vector bundle is given. Definition. Let $M$ be a topological space. A (real) vector bundle of rank $k$ over $M$ ...
3
votes
1answer
44 views

Using 1-forms to integrate along curves

The following is written in my lecture notes: 'Suppose that $\gamma:[a,b]\rightarrow\mathcal{M}$ is a smooth curve and $\omega$ is a 1-form on $\mathcal{M}$. Then we get a smooth function ...
3
votes
1answer
64 views

References request for prerequisites of topology and differential geometry

I am studying differential geometry and topology by myself. Not being a math major person and do not have rigorous background in analysis, manifolds, etc. I have background in intermediate linear ...
0
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0answers
49 views

Defining the pre-image topology for $f:A\rightarrow B$

Let $f:A\rightarrow B$ be a function from a set $A$ to a topology $(B,\tau)$. Then I can define a subbase on $A$ simply by giving $f[A]\subset B$ the subspace topology inherited from $B$ ...
0
votes
0answers
22 views

Differential of right action map on a manifold?

Let $P$ be a smooth manifold, $G$ a Lie group and $$\mu:P\times G\longrightarrow P, p\longmapsto \mu(p, g):=p\cdot g,$$ a right action of $G$ on $P$.Furthermore, suppose $\alpha:I\longrightarrow P$, ...
2
votes
2answers
87 views

Irreducible link complements in $\mathbb S^3$

Let $L$ be an oriented link in the 3-sphere $\mathbb S^3$, consisting of two knot components, $\gamma_1$ and $\gamma_2$. I wonder now if the following is true: If the linking number $N(L)$ of $L$ is ...
2
votes
1answer
35 views

Projective Plane Embedding Ambiguity

I'm working in a problem in do Carmo: let $F: \mathbb{R}^3 \to \mathbb{R}^4$ be given by $$ F(x,y,z) \;\; =\;\; (x^2 - y^2, xy, xz, yz). $$ Let $\varphi: \mathbb{S}^2 \to \mathbb{R}^3$ be the ...
4
votes
1answer
119 views

Tangent vectors in $\mathbb{R}^n$

I am confused with the idea of tangent vector or tangent space. First of all, I learned that there is an isomorphism from $ \mathbb{R}_a^n$ onto $T_a( \mathbb{R} ^n)$ from John M.Lee' book ...
-1
votes
2answers
50 views

Is this smooth manifold compact? [closed]

Let $M$ be a smooth manifold with boundary such that there exists a smooth function $f:M\rightarrow[a,b]\subset\mathbb R$. Can we conclude that $M$ is compact?
0
votes
0answers
15 views

Is there a term to describe the “transformation distance” of a transition map?

A transition map $T$ takes points on a manifold $A$ and maps them to points on a manifold $B$. Suppose I choose N random points $X$ on $A$ and compute the respective points $Y = {T(x_1), T(x_2), ... ...
1
vote
1answer
67 views

Determining dimension of manifold

I have a question concerning differential manifolds. I need to prove that $$M=\{z-x=\sqrt{x+y^2},0<x<z\}$$ is a $2$ dimensional manifold. I define the function $F(x,y,z)=z-x-\sqrt{x+y^2}=0$. ...
1
vote
1answer
46 views

Mapping tangent vectors between differential manifolds

Nigel Hitchen, in his notes on differential manifolds gives a definition of the derivative of a smooth map between two manifolds that appears to assume the the following assertion is self-evident: ...
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. ...
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
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} ...
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?
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
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} ...
5
votes
0answers
91 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 ...
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 ...
0
votes
3answers
37 views

Does $\pi(\partial M)\subset \partial(M/\sim)$?

Let $M$ a set and $\sim$ an equivalence relation. Let $\pi: M\longrightarrow M/_\sim$ the projection. Do we have that $\pi(\partial M)\supset \partial (M/_\sim)$ ? (where $\partial A$ denote the ...
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 $n=2$. ...
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 ...
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 ...
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 ...
0
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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
55 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 ...
4
votes
1answer
35 views

Flow on compact manifold

These questions seem simple, and but I have not found the answer on the web (I have no mathematician in my neighborhood). Does a continuous injective function from $E$ to $E$ have to be surjective, ...
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 ...
4
votes
1answer
88 views

Grassmannian, symmetric, idempotent matrices of trace $n$?

How do I see that $G_n(\mathbb{R}^m)$ is diffeomorphic to the smooth manifold consisting of all $m \times m$ symmetric, idempotent matrices of trace $n$?
5
votes
3answers
94 views

Generalized Gauss map, giving rise to second fundamental form

I know that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\text{Hom}(\gamma^n(\mathbb{R}^{n+k}), \gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of ...
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. ...
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 $m-n$ ...
6
votes
3answers
209 views

Fundamental groups of codimension 1 manifold complements

Let $M$ be a smooth manifold of dimension at most $3$ and $S \subset M$ a smoothly embedded compact connected codimension $1$ manifold, separating $M$ into two components, $M_1$ and $M_2$. I wonder ...
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
44 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 ...
5
votes
2answers
94 views

Smooth manifold $M$ is completely determined by the ring $F$.

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 ...
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 ...
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 ...
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 ...