Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

learn more… | top users | synonyms

1
vote
0answers
41 views

Existence of Immersion of a manifold in Euclidean space?

I am trying to prove the following claim which I saw in some paper: Let $M$ be an $n$-dimensional smooth, oriented, simply connected manifold, which is homeomorphic to a bounded subset of ...
3
votes
1answer
72 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 ...
5
votes
1answer
69 views

Does every even-dimensional sphere admit an almost complex structure?

We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?
0
votes
0answers
14 views

If $U$ is an open subset of the manifold $X$, check that $T_x(U)=T_x(X)$ for $x \in U$

Know that I'm already aware that there is a similar question on the forum. However, comments do not allow to clarify my ideas. Let $ϕ:W∈R^k→X$ be a parametrization of $X$ around $x$ so that $ϕ(W)∈X$ ...
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 ...
2
votes
0answers
38 views

Proving Sard's theorem

Theorem: (Sard) Let $f:U\to \mathbb R^p$ be a smooth map with $U$ open in $\mathbb R^n$, and let $C$ be the set of all critical points of $f$. Then $m([f(C)]=0$ where $m$ is the Lebesgue measure of ...
3
votes
1answer
108 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
39 views

Is level set near a maximum value a circle?

Let $f$ be a $C^2$ function defined on $[0,1] \times [0,1]$. Let $0 \leq f(x) < 1$ on $[0,1] \times [0,1] \setminus \left(\frac{1}{2},\frac{1}{2}\right)$ and $f(\frac{1}{2},\frac{1}{2})=1$. It has ...
0
votes
2answers
68 views

What is the inverse limit?

In his book of Differential Topology, Hirsch starts a little detour in his theory in order to present a way to see things in a general perspective. The precise fragment I'm referring to is the ...
0
votes
1answer
65 views

About the Chern class of the determinant line bundle

Is it true that the first Chern class of a rank $k$ complex line bundle is equal to the first Chern class of its determinant line bundle?
3
votes
0answers
37 views

Error in Hirsch?

Consider the following lemma: In the fragment: Write $\displaystyle X= \bigcup_{1}^{\infty} X_j$ where each $X_j$ is a compact subset of a ball $B$ as above. why is he allowed to do that? If ...
1
vote
1answer
28 views

Obstruction to the existence of constant-rank sections of $T^*M\odot T^*M$

If $\alpha$ is a section of $T^*M\odot T^*M$, where $M$ is a smooth manifold, the rank of $\alpha$ at $m\in M$ is the codimension of the kernel of $\alpha_m$, i.e. the subspace of vectors $v_m\in ...
0
votes
0answers
27 views

The pullback bundle

Consider a smooth fibre bundle $P:E\to B$ over a base $B$ with fiber $F$, and a trivializing family $\{(\phi_a:P^{-1}(U_a)\tilde =U_a\times F\}_{a\in A}$. Let the clutch functions of this family be ...
3
votes
1answer
79 views

If a Subset Admits a Smooth Structure Which Makes it into a Submanifold, Then it is a Unique One.

$$ \newcommand{\wh}{\widehat} \newcommand{\R}{\mathbf R} \newcommand{\mr}{\mathscr} \newcommand{\set}[1]{\{#1\}} \newcommand{\inclusion}{\hookrightarrow} \newcommand{\vp}{\varphi} $$ I am trying to ...
0
votes
1answer
48 views

Covariant derivative (or connection) of and along a curve

Let $c:[a,b] \rightarrow M$ be a curve parametrized by arc-length. Then $c': [a,b] \rightarrow TM$ such that $c'(t) \in T_{c(t)}(M).$ So essentially, I want to understand how $\nabla_{c'}c'$ is ...
0
votes
1answer
15 views

Express covariant transformation conveniently

Let $\omega = \sum_i \omega_i dx^i = \sum_{i} \nu_i dy^i$ be a 1-form in two different bases. Now, $(\omega_1,...,\omega_n)$ transform covariantly to $(\nu_1,...,\nu_n).$ My question is, can we ...
5
votes
1answer
80 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
3
votes
2answers
66 views

Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
1
vote
0answers
50 views

Difficulties on proof of $\epsilon $-Neighborhood Theorem.

I'm trying to proof the $\epsilon$-Neighborhood Theorem from Guillemin and Pollack's book. I'm not good at topology, and I'm having some difficulties to completely understand the theorem. For the ...
4
votes
1answer
46 views

Nonorientable manifolds being a boundaries

I will say that a manifold (smooth, compact, without boundary) is itself boundary if there is some smooth compact manifold $W$ with boundary, such that $M=\partial W$. I'm interested in nontrivial ...
1
vote
0answers
53 views

De Rham cohomology of $T^n$ using Künneth formula and Chevalley-Eilenberg theorem.

I want to calculate $H^*(T^n)$ with ring structure using both of these methods. Künneth formula gives $$ H^p(T^n)=H^p(S^1\times T^{n-1})=\bigoplus_{i+j=p}H^i(S^1)\otimes H^j(T^{n-1}) $$ for each ...
7
votes
1answer
44 views

Grassmanians and boudaries of manifolds

Let $M$ be a smooth, compact manifold without boundary. I will say that $M$ is a boundary when there is a smooth, compact manifold with boundary $W$ such that $\partial W=M$. After some lectures I ...
1
vote
1answer
31 views

Understanding Result on Non-Degenerate Critical Points

I read a result in a collected works of Steven Smale and one result leapt out at me which I'm clearly not understanding. Stated: Theorem 1.1 (a): Suppose $J: M \to \mathbb{R}$ is a $C^2$ ...
3
votes
1answer
32 views

A differentiable fiber bundle exists for any family of clutching functions.

Lemma: Let $F$ be a smooth manifold and let $\{U_a\}_{a\in A}$ be a covering of a manifold $B$, and let $\{g_{ab}\}_{a,b\in A}$ be a family of clutching functions. That is, $g_{ab}:U_a\cap U_b\to ...
3
votes
1answer
93 views

Is a mapping a homeomorphism

I'm considering the mapping $\Psi: C^2([0,1])$ to $C^1([0,1])$ via: $f(x) \mapsto f(x)+x\cdot f'(x)$. Is this mapping a homeomorphism? It should be continuous given that, for any sequence $(f_n) \in ...
1
vote
0answers
28 views

solubility of nonlinear overdetermined PDE for holonomic scaling of a frame

This question is essentially a tweak of under what conditions can orthogonal vector fields make curvilinear coordinate system? . Suppose we have a frame of vector fields $\nu_i$ on $\mathbb{R}^n$. As ...
2
votes
1answer
39 views

The inverse of a smooth bundle map is smooth.

Consider smooth fiber bundles $P_i:E_i\to B_i,\quad i=1,2\quad$ with fiber $F$. Let $\tilde f:E_1\to E_2$ be a smooth bundle map. That is a smooth map which preserves the fiber $F$. Let $f:B_1\to B_2$ ...
2
votes
1answer
49 views

Tangent space change of bases

Let $M\subset \Bbb{R}^m$ be a $k$-dimensional differentiable submanifold. Let $(\varphi, U)$ and $(\psi, V)$ be two charts for $p\in M$ with $\varphi(x)=p$ and $\psi(y)=p$. Then we have two bases for ...
0
votes
1answer
21 views

Subset of a smooth manifold

I am actually in the resolution of the problem Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - "Differential topology" of Guillemin and Pollack (my own ...
1
vote
0answers
32 views

Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - “Differential topology” of Guillemin and Pollack

I know that we can refered to the question How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?. I have the same question with an answer, and I needed that someone tell me if it is ...
3
votes
0answers
43 views

Proof of Reeb's theorem without using Morse Lemma

I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory. That is, suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both ...
1
vote
0answers
24 views

Show that the projection map $X \times Y \rightarrow X $, carrying $(x,y) \rightarrow x$, is smooth.

How do I show that the projection map from $X \times Y \rightarrow X $, carrying $(x,y) \rightarrow x$, is smooth? A map $f:X \rightarrow R^m$ defined on an arbitrary subset $X \subset R^n$ is called ...
2
votes
1answer
87 views

Prove or disprove any continous map $f$ from $T^2$ to $RP^2$ is null-homotopic.

I tried to solve the following question: Prove or disprove any continous map f from $T^2$ to $RP^2$ is null-homotopic. We know the universal cover of $RP^2$ is $S^2.$ I want to construct a map $g$ ...
107
votes
5answers
3k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
2
votes
0answers
40 views

What can I say of an $m$-dimensional submanifold $S$ of an $m$-dimensional manifold $M$?

I consider a differentiable manifold $M$ of dimension $m$. Let be $S$ a submanifold of $M$ of the same dimension $m$. What can I say about $S$? I have tried to prove that $S$ is open but I get ...
2
votes
1answer
70 views

Diffeomorphism between $T$ (torus) and the cover $S^1 \times S^1$ - Question of the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : ''The'' torus is the set of points in $\mathbb{R^3}$ at distance $b$ from the circle of radius a in the xy plane, where $0<b<a$. Prove that these tori are all diffeomorphic ...
0
votes
1answer
46 views

Closed geodesic minimizing properties

Considering closed geodesics on a compact manifold M of even dimension, what does it mean to say that a curve (any closed geodesic) is locally energy minimizing but not globally ? For simplicity, say ...
0
votes
0answers
40 views

Why gradient-like dynamical systems are special case of Morse-Smale systems?

I'm studying Morse Theory and my question is exactly as stated in the above title. I can't see how a gradient-like dynamical system could be considered as a Morse-Smale system? Thanks in advance for ...
1
vote
0answers
51 views

The inverse image of any regular value is a submanifold

Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear ...
1
vote
0answers
22 views

Applications of $C^\ast$ algebras in differential topology

I was wondering if there were any useful ways $C^\ast$ algebras come into play within differential topology. I know this is a fairly broad question,so any type of input is applicable.
2
votes
3answers
51 views

Union of the two coordinate axes in $\mathbb{R^2}$ - From the book '' Differential Topology '' of Guillemin and Pollack

I have tried to solve the problem : Prove that the union of the two coordinate axes in $\mathbb{R^2}$ is not a manifold. Let $X = \{(x,y) \in \mathbb{R^2} : x=0~or~ y=0\}$ be the union of the two ...
1
vote
1answer
57 views

Topological question from the book '' Differential Topology '' of Guillemin and Pollack

Here's the question : A smooth bijective map of manifolds need not be a diffeomorphism. In fact, show that $$f:\mathbb{R^1}\rightarrow {R^1}$$ $$x\rightarrow f(x)=x^3,$$ is an example. I would like ...
0
votes
0answers
32 views

On the following of the question: every $k$-dimensional vector subspace $V$ of … [duplicate]

This is the continuity of this question I've created this question recently, but I didn't receive all the answers I hoped. Someone could explain me why is it that the problem of approach work? In the ...
3
votes
1answer
55 views

Area form a differential form?

I just read that $\omega_x( \eta, \zeta) := \langle x, \eta \times \zeta \rangle $ is the area form on the sphere, where $x \in \mathbb{S}^2$ and $\eta,\zeta \in T_xM.$ All I see is that this is ...
4
votes
0answers
33 views

Immersion except at the origin. [closed]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
4
votes
1answer
78 views

Is this statement about manifold true? [duplicate]

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
6
votes
1answer
79 views

is there a diffeomorphism with only finite orbits but of infinite order?

Note: after not receiving any answer for some time, I asked this in mathoverflow, and got an answer there. The Question: Is it possible for a diffeomorphism $\phi$ (of a smooth manifold $M$) to have ...
4
votes
2answers
208 views

How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible ...
2
votes
1answer
70 views

Working with homomorphisms and de Rham cohomology.

Here’s my question: Let M be a connected, compact, orientable, smooth n-manifold ($ n \in \mathbb{N}_{\geq 2} $). Let V be a neighborhood of p diffeomorphic to $\mathbb{R}^n$ and let U = M \ {p}. ...
0
votes
0answers
39 views

Tangent space chart dependent or not?

I was wondering whether the definition of a tangent space is chart dependent or not? Cause to me it seems that they depend on the charts: Let $p \in M$ where $M$ is a k-dim smooth manifold and $\phi: ...