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
31 views

Multiplication in homotopy groups and cobordism

Each homotopy class of a map of an n-sphere into the Thom space of the universal vector bundle determines a cobordism class of embedded smooth manifolds in Euclidean space. How do the cobordism ...
3
votes
2answers
183 views

How does one prove that the Klein bottle cannot be embedded in $R^3$?

How does one prove that the Klein bottle cannot be embedded in $R^3$? I'm talking about smooth embeddings.
5
votes
3answers
264 views

Application of Hodge decomposition

Hodge decomposition states any $p$ form can be decomposed into three orthogonal $L^2$ components: exact form, co-exact form and hamonic form. But actually we don't know how to decompose a general one. ...
1
vote
1answer
89 views

Locally Euclidean can be defined for whole of $\mathbb{R}^n$, or an open set or open ball of $\mathbb{R}^n$?

Topological manifolds are defined to be locally Euclidean (e.g. John Lee). That is, any point is in an open set that is homeomorphic to either $\mathbb{R}^n$, an open ball in $\mathbb{R}^n$ or an open ...
5
votes
5answers
365 views

Trivial tangent bundle of sphere with handles

I am wondering if there is a simple proof of this statement: A sphere with $g$ handles has trivial tangent bundle iff $g=1$ I know that it is a corollary of Poincaré-Hopf theorem, but it seems ...
2
votes
1answer
38 views

isometric imbedding of the projective plane

How to isometrically imbed the projective plane (identifying antipodal points of the unit sphere) in $\mathbb{R^5}$? My textbook indicates that there is an isometric imbedding of the projective ...
1
vote
0answers
65 views

Reference: forms invariant under Lie group action give the de Rham cohomology?

I'm looking for a reference for a proof of the following fact: Let $G$ be a compact, connected Lie group acting on a smooth manifold $M$. Then inclusion of the differential forms invariant under the ...
0
votes
1answer
107 views

Meaning of differentiability

Could anyone give an intuitive idea of the meaning of differentiability in general in any dimension and any space?
2
votes
1answer
69 views

Product of manifolds with boundary

If $M$ and $N$ are manifolds with boundaries and $\{(U_a,f_a)\}$ and $\{(V_a,g_a)\}$ are their respectives $C^r$ atlas, why $\{(U_a \times V_b,f_a \times g_b)\}$ isn't an $C^r$ atlas for $M \times ...
0
votes
2answers
68 views

Isometry of surfaces in $\mathbb{R}^3$

Let $F$ be an isometry of the Euclidean space $\mathbb{R}^3$. Hence $F$ is orthogonoal transform followed by translation by a constant vector. Let M be a surface of $\mathbb{R}^3$ that is connected, ...
2
votes
1answer
147 views

About Stokes' theorem

I noticed that in the proof of Stokes' theorem on manifolds, the condition that the form $\omega$ is compactly supported ensures that the sum is finite so that we can change the order of sum and ...
4
votes
1answer
174 views

differentiable maps between topological spaces without using the idea of manifolds

Is it possible to define differentiable maps between topological spaces without using the idea of manifolds?
1
vote
1answer
73 views

Moves for regular homotopies of immersions of $S^1$ in the plane

What is a set of moves to combinatorially describe regular homotopies of (smooth) immersions $S^1\to \mathbb R^2$?
2
votes
1answer
118 views

Orthonormal frame bundle orthogonal to a curve

Let $M$ be a $n$-dimensional smooth riemannian manifold and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow M$ an embedding. $\varphi$ will denote the image of $\varphi$, too. Consider the bundle ...
2
votes
3answers
285 views

How to prove there exists a solution? Guillemin Pollack

Prove there exists a complex number $z$ such that $$ z^7+\cos(|z|^2)(1+93z^4)=0. $$ (For heaven's sake don't try to compute it!)
5
votes
3answers
926 views

Top homology of an oriented, compact, connected smooth manifold with boundary

Let $M$ be an oriented, compact, connected $n$-dimensional smooth manifold with boundary. Then, is it true that $n$-th singular homology of M, that is $H_n(M)$, is vanish? I can't make counterexamples ...
1
vote
1answer
239 views

Is a curl-free vector field always a gradient?

I tried to prove this problem using the Helmholtz decomposition theorem, but it seems the two are entirely contradictory--thus leaving me with empty hands. Does anyone know how to proceed? Thanks ...
1
vote
1answer
55 views

Maps of $G$-bundles

A vector bundle is a $GL(\mathbb{R}^m)$-bundle with fiber $\mathbb{R}^m$. A principal $G$-bundle is a $G$-bundle with fiber $G$ (where the "$G$" in "$G$-bundle" embeds into $\mathrm{Aut}\, [G]$ by ...
0
votes
1answer
41 views

Computing the coordinate representation of a vector field

Let $V=x\frac{∂}{∂x}+y\frac{∂}{dy}$ be a vector field on the plane. Compute its coordinate representation in polar coordinates on the right half-plane $\{(x,y):x>0\}$. What I got so far: The ...
6
votes
1answer
514 views

Euler characteristic is equal to self-intersection number of zero-section?

As I recall (from Guillemin and Pollack "Differential Topology") the Euler characteristic of a (for my purposes, compact and oriented) smooth manifold X is defined as $\chi(X)=I(\Delta,\Delta)$, ...
1
vote
1answer
161 views

A specific example of $F$-related vector fields

I need to prove the following: Let $F:\Bbb{R}\to\Bbb{R}^2$ be the smooth map $F(t)=(\cos t,\sin t)$. Then $d/dt\in\mathcal{T}(\Bbb{R})$ is $F$-related to the vector field $Z\in\mathcal{T}(\Bbb{R}^2)$ ...
2
votes
1answer
73 views

compact surface with two non-intersecting geodesics

I need to find an example of a compact geometric surface M such that Gaussian curvature $K>=0$ M is diffeomorphic to a sphere M has two simply closed geodesics (smoothly closed loops) that ...
7
votes
2answers
92 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
1
vote
1answer
59 views

If $F$ is smooth then the pushforward $F_*$ is smooth

Question: Suppose $F:M\to N$ is a smooth map. Show that $F_*:TM\to TN$ is a smooth map. We need to find charts $(U,\phi)\subset TM$ and $(V,\psi)\subset TN$ such that $\phi\circ F_*\circ\psi^{-1}$ is ...
4
votes
1answer
164 views

Hopf invariant and the linking number

The Hopf invariant of a map $f:S^{2n-1}\to S^n$ can be defined in various ways, in particular: (1) as the linking number of the preimages of two points and (2) using the cohomology ring of the space ...
1
vote
0answers
46 views

immersions of spheres with handles minus disks

i am at a loss as far how exactly to immerse a sphere with genus g minus a disk into the plane. Thanks in advance
7
votes
0answers
338 views

Trying to Understand Lefschetz Pencils

I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) better, tho I would appreciate insights on condition i), and in general. A Lefschetz pencil on a $4$-manifold $X$ is a ...
4
votes
1answer
169 views

Uniqueness of integral curves.

Suppose we have a smooth compact manifold $M$ with boundary and a vector field $X$ on $M$. The maximal integral curves on $M$ are unique and hence their images give a partition of $M$. Let $p\in M$ ...
3
votes
1answer
91 views

Unknotting number formally?

I am reading Colin C. Adams's very nice but not always rigorous "The Knot Book" right now. How does one formalize the unknotting number? (For example, is some restriction on embeddings ...
4
votes
0answers
88 views

Homotopic maps to $S^n$

I'm working through a proof that, given an oriented compact (connected) $n$-manifold $M$ with boundary, any two continuous maps $f,g:M\to S^n$ are homotopic. The proof uses the double of $M$, which is ...
1
vote
2answers
144 views

Frame bundle of orthonormal frames orthogonal to a submanifold.

Suppose we have a smooth manifold $M$ of dimension $m$ with a Riemannian metric and a connected submanifold $N$ of dimension $n$ in $M$ with $n<m-1$. Let $n\le k<m-1$ and consider the bundle ...
2
votes
1answer
49 views

Every chart is smoothly compatible, what is wrong with my argument?

Let $A_1$ and $A_2$ be two distinct smooth structures on the manifold $M$. For any two charts $(U,\phi)\subset A_1$ and $(V,\psi)\subset A_2$, both $\phi$ and $\psi$ are diffeomorphisms. If we assume ...
0
votes
1answer
102 views

Torus in differential geometry.

I want to write separately parametrizations (surface patches) $\sigma$ for torus when (1) x-axis rotation in the first part of the picture and (2) y-axis rotation in the second part of the picture. ...
3
votes
1answer
201 views

Non-vanishing vector fields on non-compact manifolds

In several papers the following result is invoked: Theorem. Every connected, non-compact, smooth manifold $M$ carries non-vanishing smooth vector fields $v$. (we are assuming $M$ is $2$nd countable ...
2
votes
3answers
83 views

Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style. For (connected, pointed) topological spaces with trivial ...
2
votes
1answer
170 views

Is the map from the circular half cone to the $xy$ plane a local isometry?

This is a text book exercise. And I think that this map is not a local isometry. But, I don't know how to show this question. Please help me explaining this question. Thanks a lot. I posted its ...
0
votes
1answer
83 views

First fundamental form of a surface patch $\sigma$

I have its answer, but I cannot understand again. Please explain its solution clearly. Thanks a lot.
1
vote
1answer
96 views

A surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$

Suppose that a surface patch $\tilde{\sigma}(\tilde u,\tilde v)$ is a reparametrization of a surface patch $\sigma (u,v)$. Let $$\tilde E d\tilde u^2+ 2\tilde F d\tilde ud\tilde v+\tilde G d\tilde ...
2
votes
1answer
245 views

First fundamental form question.

The question I posted; $6.1.2\quad$ Show that and apply an isometry of $\Bbb R^3$ to a surface does not change its first fund. form. What is the effect of a dilation (i.e., a map $\Bbb R^3\to ...
1
vote
1answer
65 views

The question related to a regular surface.

Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$. I know ...
0
votes
3answers
766 views

what's the difference between isomorphism and homeomorphism?

I think that they are similar or same but am not sure. Can anyone explain the differences?
1
vote
1answer
71 views

Euler characteristic of part of the sphere

Let R be the part of the sphere in $R^3$ bounded by two smoothly closed curves that do not intersect. For instance, R is the region bounded by a great circle and a smaller circle paralle to it. How ...
3
votes
2answers
325 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
1
vote
0answers
62 views

prove that the shapes are isometric

I want to prove that the shapes are isometric. How to prove? There is no info except for the picture. First of all I need to write surface patches. Please can someone help me? The definition of ...
1
vote
0answers
155 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
2
votes
1answer
80 views

Differential forms on the torus correspond to periodic forms on $\Bbb{R}^n$?

Let $T^n=\Bbb{R^n/Z^n}$ be the torus. Is it possible say that forms on the torus bijectively correspond to forms on $\Bbb{R}^n$ invariant under translations by integers?
0
votes
1answer
66 views

Verify this is not orientable.

Verify this is not orientable. Möbius transformation: $$U=\{(t,\theta) \mid \frac{-1}{2}\lt t\lt \frac{1}{2}, 0\lt \theta \lt 2\pi \}$$ $\sigma (t, \theta)=<((1-t\sin (\theta/2))\cos (\theta), ...
2
votes
0answers
56 views

Show that the Mod 2 Linking Number is well defined.

If $X$ and $Y$ are compact, boundaryless smooth manifolds, $f\colon X \to \mathbb R^n$ and $g\colon Y \to \mathbb R^n$ with $f(X) \cap g(Y) = \emptyset$, and with $\dim (X) +\dim (Y) =n-1$. Define ...
0
votes
1answer
52 views

find f and $d_pf$

Let $$S_1=\{(x,y,z) | x^2+y^2+z^2=1\}-\{N\}$$ $$S_2= \{(x,y,z,0) | x,y\in \Bbb R\}$$ $f:S_1\to S_2$ $f$ is stereographic projection. ,where $\ell$ is a line passing through $N=(0,0,1)$ and $p$ ...
2
votes
2answers
59 views

$d_pf$ is a linear invertible map.

If $f: S_1\to S_2$ is a diffeomorphism, then $$d_pf: T_p(S_1)\to T_q(S_2)$$ is an invertible linear map and $$(d_pf)^{-1}=d_qf^{-1}$$ for any $p\in S_1$ $q\in S_2$ and $f(p)=q$ I cannot prove this ...