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
1answer
35 views

$\int _c \omega$ is independent of orientation-preserving re-parameterization of c

I'm working on the following problem from Guillemin and Pollack's Differential Topology: Let $c : \left[a, b\right] \rightarrow X$ be a smooth curve, and let $f: \left[a_1, b_1\right] \rightarrow ...
2
votes
1answer
33 views

Divergence free vector field on compact surface

I get a free divergence field $X$ on a compact surface $(\Sigma, g)$ and I would like to integrate it. On the sphere $X=\nabla^\bot f$ since the spher is simply connected.($\nabla^\bot =J\circ ...
1
vote
1answer
20 views

Check that F*(Xp) is a derivation at F(p)

To show linearity is simple but I am stuck on derivation
2
votes
0answers
19 views

Equivariant Poincaré Duality

Let $\Gamma$ be a group acting by smooth orientation-preserving diffeomorphisms on a smooth compact oriented manifold $M$ of dimension $n$. The de Rham complex $\Omega^{\bullet}(M)$ of differential ...
2
votes
1answer
73 views

Cone with deficit angle $2\pi$

I've managed to confuse myself with cones and deficit angles. Let's consider a conical defect in 2 dimensions. So the metric is the usual one in polar coordinates, $$ ds^2 = dr^2 + r^2 d\phi^2,$$ ...
1
vote
1answer
58 views

Collar neighborhood theorem- Cobordism

I am studying cobordism theory and I basically follow Milnor's book, Characteristic classes. I want to prove that cobordism is a transitive relation so I need the collar neighborhood theorem. In ...
0
votes
1answer
11 views

The nature of components in a certain manifold

Let $N$ be a smooth, connected manifold and $f:N \to \mathbb R$ a smooth, proper and surjective map, transverse to some $k \in \mathbb N$. This means that $M:=f^{-1}(k) \subset N$ is a finite ...
0
votes
0answers
42 views

A question about the number of points of the preimage of a regular value.

I would like to have some explanation about a fact which is written in the first chapter of Milnor's book "topology, from the differentiable viewpoint". The fact is the following: Suppose $M,N$ be two ...
1
vote
1answer
43 views

embeddings of projective spaces into Euclidean spaces

Let $\mathbb{R}P^n$, $\mathbb{C}P^n$, $\mathbb{H}P^n$ be the real, complex, quaternionic projective spaces resp. I want to find all $n$ such that $\mathbb{R}P^n$ can be embedded into ...
0
votes
1answer
26 views

Example of germs not involving series

note: moved from mathoverflow, as off topic. I'm currently reading this book: http://www.springer.com/us/book/9781441973993 And when speaking about germs of functions the only example provided of two ...
1
vote
1answer
39 views

General linear group over reals: diffeomorphism [closed]

What does the general linear group over reals look like; i.e., what is it diffeomorphic to? I mean some other manifold I can picture if any?
2
votes
1answer
28 views

Is the set of points where two functions agree a submanifold?

Let $M, N$ be smooth manifolds, $f : M \to N$ a smooth submersion and $g : M \to N$ a smooth function. Is it true that $R = \{x \in M\ |\ f(x) = g(x)\}$ is an embedded submanifold of $M$? How about if ...
3
votes
1answer
59 views

$c_1^3$ of 6 manifolds

For a closed oriented smooth 4 manifold $X$ we have $c_1^2(X) := 2e(X) + 3σ(X)$, $c_1$ is the first Chern class and $σ$ is the signature. For 6 manifolds is there such a relation with $c_1^3$? I'd ...
0
votes
0answers
27 views

How to integrate by parts the symmetric covariant derivative on a manifold with boundary?

Let $(M,g)$ be a compact Riemannian manifold. Also let $\mathcal{S}^2=\Gamma(S^2T^*M)$ and $\mathcal{S}^3=\Gamma(S^3T^*M)$ be respectively the spaces of symmetric $2$-covariant and $3$-covariant ...
4
votes
0answers
72 views

Manifold is cut along hypersurfaces; how to define a connection on this?

May $M$ be a smooth manifold with boundary $\partial M$. Metrics and Connection can be defined everywhere. But now this manifold is cut by a smooth hypersurface $A$ and the cut goes along $M \cap A$ ...
2
votes
0answers
36 views

Brown's theorem and regular values

I would like to review the definition of the degree (Brouwer and mod.2) of an application between smooth manifolds. Let $f:M\to N$ be a smooth map: Is the set $R$ of regular values open and dense in ...
7
votes
1answer
175 views

Visualization of SU(3)

I am trying to visualize the $SU(3)$ group used in quantum field theory. I have a (reasonably) good understanding of $SU(2)$ as the double cover of $SO(3)$ and also that this is homeomorphic to $S^3$. ...
1
vote
0answers
60 views

Is the trivial sheaf fine?

A sheaf is called fine if basically it has partition of unit for endomorphisms. That is, if $P:E \rightarrow M$ is a sheaf, then $P$ is called fine if for for every locally finite open cover ...
1
vote
1answer
31 views

Degree of the map $S^1\to S^1:z\to z^n$

I want to compute the (topological/differential) degree of the map $f:S^1\to S^1:z\mapsto z^n$. I've have shown that the degree of the map $g:\mathbb{C}^*\to \mathbb{C}^*:z\mapsto z^n$ is $n$. Is ...
1
vote
1answer
30 views

dimension of level set

If I have some equation of the form $F(x,y,z)=0$ which defines level set of the function $F$ and I want to know what is the dimension of the set $A:=\{(x,y,z):F(x,y,z)=0\}$ Is there any special ...
2
votes
2answers
231 views

Do there exist manifolds which cannot be smoothly embedded in a compact manifold?

Let $M$ be a smooth manifold. Is it always possible to find a manifold $N$, compact, and a diffeomorphism of $M$ onto an open subset of $N$? I know that if $M$ is given some Riemannian metric, then ...
2
votes
1answer
36 views

A guess about Jacobi

Let $\Omega \subset R^n$ is a open set of $R^n$ , $f:\Omega\rightarrow R^n $ is $C^2$ function.The Jacobi of $f$ is : $$ J_f(x)=\frac{D(f_1,...,f_n)}{D(x_1,...,x_n)}=|\frac{\partial f_i}{\partial ...
0
votes
0answers
38 views

What is dual to relative homology?

Let $(X,A)$ be a pair of manifolds, where $A \subseteq X$. Then we can define the relative cohomology of the pair $H_{\bullet}(X,A)$ to be the homology of the chain complex ...
1
vote
1answer
33 views

Relations between covering map and (co-)homology groups

This occurs to me when considering the homomorphism $p_*$ induced by the covering map/attaching map: $$p:S^n\to \mathbb{R}P^n$$ on its homology group: $H_n(S^n)\to H_n(\mathbb{R}P^n)$ which sends the ...
0
votes
1answer
54 views

How to write differential forms on manifolds?

In the book "Differential Forms and applications" by Manfredo do Carmo, he says that a differential $k$-form on a $n$-dimensional smooth manifold $M$ is determined by a choice, for each ...
3
votes
0answers
26 views

Immersed subgroup of a Lie group is a Lie group?

Let $G$ be a Lie group, $H$ a subgroup of $G$, which is an immersed submanifold of $G$. Kirillov, in his book "Introduction to Lie Groups and Lie Algebras" claims* it's easy to see $H$ will be a Lie ...
1
vote
2answers
35 views

Grammatically confused: $\omega=4dV$ for 3-form $\omega$ and volume in $\Bbb R^4$?

Background: Against the advice I should have been given but wasn't, I'm taking a Lie theory course with no background in differential geometry. We finally made it into the part of the course where we ...
0
votes
1answer
12 views

Given a real matrix $W$, find a complex matrix $V$ such that $df_A(V) = W$ where $df_A: V \mapsto A\bar{V}^T + V\bar{A}^T, A\bar{A}^T = 1_{n\times n}$

I am trying to complete the proof that the unitary group is a smooth manifold using regular values. To complete this proof I need to show that for any real-valued matrix $W$, there exists a ...
1
vote
1answer
46 views

Map isotopic to identity is orientation preserving

Let $M$ be an $n$-dimensional orientable and compact smooth manifold and $f:M\to M$ be a smooth map isotopic to the identity map. Is it true that $f$ is orientation preserving?
1
vote
1answer
35 views

Compact manifold, regular value

I am still trying to convince myself about a fact which I've seen in Milnor's book ''Topology from the differentiable viewpoint'': Let M and N be manifolds of the same dimension. If M is a compact ...
3
votes
1answer
36 views

Flow of a vector field cannot reach a singular point in a finite amount of time?

$\newcommand{\R}{\mathbb{R}}$ Let $M$ be a smooth manifold. Let $X \in \Gamma(TM)$ be a smooth vector field, and $\gamma$ some non-constant integral curve of $X$. Let $p \in M$ be a singular point of ...
2
votes
2answers
78 views

Proving that $I \times I$ is not a manifold

I want to prove that $I \times I$ is not a manifold where $I=[0,1]$. But I believe this is false. Here we will denote closed unit disc as $\overline{D}^2$.Since we can take a homeomorphism $\phi : I ...
2
votes
1answer
38 views

Can every map between manifolds be factored as $p\circ i$

Can every map between topological manifolds $f\colon X\to Y$ be factored as $p\circ i\colon X\to \overline{Y}\to Y$ with $i$ inclusion of open subset in another topological manifold $\overline{Y}$ and ...
0
votes
0answers
34 views

compute compactly supported cohomology of a space

Given space $M=R^2/0$ ($R^2$ excluding origin), how to compute compactly supported de Rham cohomology of M, in notation, $H^*(c(M))$? I am thinking of raising a vector v in $R^2$ with |v| at most ...
3
votes
1answer
42 views

Integral form of Euler characteristic

There is a known formula for Euler characteristic in terms of Ricci scalar: \begin{equation} \chi(M)=\frac{1}{4\pi} \int_M \sqrt{g} \,R\,d^2x\,. \end{equation} I am sure that this formula holds for ...
1
vote
1answer
51 views

How to prove the given sets are equal?

I did not understand the highlighted part(in red). How is the equality obtained? One inclusion is clear namely, the left hand side is a subset of the right hand side. Why is this an open subset ...
2
votes
0answers
70 views

Extending function using a vector field

Let $M$ be a smooth manifold with a smooth submanifold $N$ and vector field $v$ that's never parallel to $N$. Suppose we want to extend some function $f: N \to \mathbb{R}$ to a function $F$ on all of ...
2
votes
1answer
92 views

Change of basis between coordinate charts in $\mathbb RP^2$.

I'm looking at the real projective plane $\mathbb RP^2$, with homogeneous coordinates $(x:y:z)$. I want to find the transition matrix between bases associated with different coordinate charts. I've ...
0
votes
0answers
23 views

Intersection between sphere and ellipsoid - number of connected components

Let $S$ be the unitary sphere of $\mathbb{R}^3$ and for $0<a<b<c$ and $A>0$ let $G(x,y,z)=ax^2+by^2+cz^2 $ and the ellipsoid $E_A$ : $G^{-1}(A)$. How many connected components does ...
2
votes
0answers
50 views

Classification of Torus bundle over $\mathbb{S}^1$ [closed]

$T^2$ has a fixed free isometry group as $D_8$, is this true that $T^2$ bundle over $S^1$ have three type?
0
votes
0answers
40 views

Action of discrete subgroup of Lie Groups

Given $\Gamma$ a discrete subgroup of a lie group $G$ I want to show that the action is wandering: $\forall x\in G\exists U_x\vert \{\gamma\in\Gamma\vert \gamma U_x\cap U_x\neq \emptyset\}$ is finite ...
1
vote
0answers
24 views

Different notions of Submanifold

There are three types of submanifolds discussed in my book. Let $M$ be a smooth manifold. Then 1.) An immersed submanifold of $M$ is a set $S\subseteq M$ such that $S=F(S)$, where $F:N\to M$ is an ...
0
votes
0answers
15 views

Proving that a particular function belongs to a sheaf of functions

I have the following exercise Let $M,N$ be open subsets in $\mathbb{R}^{n}$ and let $\Psi:M\to N$ be a smooth map. Show that $\Psi$ defines a morphism of spaces ringed by smooth functions. The ...
4
votes
1answer
53 views

Reference for Whitney's approximation theorem for manifolds with boundary

I am aware of Whitney's approximation theorem for manifolds without boundary but I was wondering if there is reference which states the above theorem for manifolds with boundary. Thank you.
0
votes
0answers
14 views

Cone of $1$ dimensional manifold.

How I can show that cone on $[0,1]$ is not a smooth manifold whereas cone on $S^1$ it is a smooth manifold? I am unable to prove this explicitly. I need some ideas how to prove this.
6
votes
1answer
72 views

Applications of Whitney's Approximation Theorem

I am reading the book Introduction to Smooth Manifolds by John Lee. In his book he proves a theorem called the Whitney's Approximation theorem which essentially states that any continuous map can be ...
0
votes
1answer
20 views

Integrability and area-preservation property of maps

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
2
votes
1answer
22 views

Smoothness of the projection operator

Let $V$ be a smooth vector field defined on $\mathbb{R}^n$ and let $\Pi$ be a smooth $k$-dimensional subspace. For each $x \in \Pi$ consider the projection of $V(x)$ onto $\Pi$. Call this new vector ...
1
vote
1answer
32 views

Openness of homeomorphism in $C^0_S(M,N)$

I'm studying differential topology on Hirsch book, in particular the part on function spaces. There's the proof (page 38) of the fact that the set of $C^r$ diffeomophisms between two $C^r$ manifolds ...
2
votes
0answers
66 views

Show that $df_x: TM_x\rightarrow TN_{f(x)}$ is well-defined

Let $f\colon M\rightarrow N$ be a $C^\infty$ function between $C^\infty$ manifolds. Show that $$df_x: TM_x\rightarrow TN_{f(x)}$$ defined by $$df_x([c_0]) := [f \circ c_0]$$ is a well-defined map. ...