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

4
votes
1answer
165 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
90 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
139 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
46 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
97 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
196 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
82 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
167 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
78 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
91 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
242 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
723 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
307 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
59 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
150 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
79 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
65 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
55 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 ...
1
vote
0answers
45 views

show that $g\circ f :S_1 \to S_3$ be also smooth and $d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$

proposition: Let $S_1$, $S_2$, $S_3$ be 3-regular surfaces and $f:S_1 \to S_2$ and $g: S_2 \to S_3$ be smooth maps. Then show that $g\circ f :S_1 \to S_3$ be also smooth and $$d_p(g\circ ...
0
votes
1answer
111 views

Diffeotopy and connectedness on manifolds

Let $M, N$ manifolds without bundary and $f: M\to N$ and $g: M\to N$ embeddings. We say that a differentiable map $h: [0,1]\times M\to N$ is an isotopy between $f$ and $g$ if each of the maps ...
0
votes
1answer
75 views

$d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$

If $f: S_1\to S_2$ is a smooth map betweentwo regular surface $S_1$ and $S_2$, then $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$ I cannot prove this statement. I think that ...
1
vote
0answers
74 views

Transversality and intersection mod.2

I am troubled by the following fact: If $f,g : X\to Y$ are homotopic and both transversal to $Z$ then the mod.$2$ intersection numbers are equal $I_2(f,Z)=I_2(g(Z)$. (the book by Guillemin and ...
0
votes
0answers
58 views

What is the type of $u$ in this definition of knot?

I am going to quote from the second paragraph of the Introduction of Möbius Energy of Knots and Unknots by Michael H. Freedman, Zheng-Xu He and Zhenghan Wang. Let $\gamma = \gamma (u)$ be a ...
17
votes
1answer
212 views

Reconstructing a manifold from critical points

I am teaching theoretical calculus this semester, and on the last discussion section we were discussing critical points of functions. I explained the idea of Morse theory, and a student of mine asked ...
2
votes
1answer
42 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
1
vote
1answer
223 views

Smooth diffeomorphisms preserving common symmetries

Let $A$ and $B$ be two $C^\infty$ orientation-preserving diffeomorphic, connected, bounded, open subsets of ${\mathbb R}^n$, with finitely many ends and $G$ the group of isometries of ${\mathbb R}^n$ ...
0
votes
1answer
150 views

Prove the directional derivative operators at a point on manifold form a vector space

One of the way to define tangent space is to use directional derivative. However, it's not clear at the first glance that the directional derivative operators form a vector space. Let $D$ be the set ...
1
vote
0answers
60 views

Constructing a non-degenerate vector-field from old one

This is an attempt to understand Milnor's proof that it's always possible to compute a vector field's index using a non-degenerate field. The strategy is simple: take $z$ to be an isolated zero of the ...
3
votes
2answers
137 views

Existence of differential form on a manifold

I have a fundamental question about the existence of differential forms on manifolds. A $k$-form on a manifold in local coordinates looks like $f(x_1,...,x_n)dx_{i_1}...dx_{i_k}$, where $f$ is a ...
0
votes
0answers
43 views

Why is $\frac{f}{\|f\|}$ a submersion when this matrix has rank $k$?

A paper I'm reading defines the following $\left(\frac{(k-1)k}{2}+1\right)\times n$ matrix: \begin{align*} \Omega_f(x) := \begin{bmatrix} \omega_{1,2}(x)\\ \vdots \\ \omega_{i,j}(x) \\ \vdots \\ ...
1
vote
2answers
177 views

Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
3
votes
1answer
95 views

Intersection of lines with compact smooth manifolds

everybody. I need a hint on this: I have to prove that a compact smooth submanifold of R^n intersects almost every one dimensional linear subspace (in R^n) in a finite set of points. I know I have to ...
1
vote
2answers
44 views

The space of regular curves deformation retracts onto the space of arclength parameterized curves

Let X denote the space of smooth maps from the circle into R^3 which have no zero derivatives. This is an open submanifold of the Frechet space of smooth maps from the circle into R^3. Let Y denote ...
3
votes
2answers
197 views

How does a smooth structure on a subset of a manifold determine its status as an immersed submanifold?

As titles are limited to 150 characters, allow me to rephrase my question in a way that is hopefully more precise: Given a $d$-dimensional smooth manifold $M$ and some $k$-dimensional subset ...
4
votes
2answers
119 views

Why is the moduli space of gradient flow lines $\widehat{\mathcal M}(p,q) = \mathcal M(p,q) / \mathbb R$ a smooth manifold?

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
0
votes
1answer
56 views

If $f\colon X\to S^k$ smooth, $X$ compact and $Z\subseteq S^k$ closed, then $I_2(f,Z)=0$.

I've been working through an old example sheet I found online at Cambridge geometry 2011, to fill in gaps in my topology. The question #14 asks: Suppose $f\colon X\to S^k$ is smooth where $X$ is ...
1
vote
1answer
69 views

Normal bundle is locally trivial

Could someone tell me how to prove the following result? Let $Z$ be a submanifold of codimension $k$ in $Y$. Prove that the normal bundle $N(Z;Y)=\left\{(z,v):z\in Z, v\in T_{z}(Y)\text{ and } ...
1
vote
1answer
116 views

notation for a torus

I am trying to search for the meaning of this notation but unfortunately it seems that wikipedia even doesn't have it. The book I am following uses the following notation for a torus: $\mathbb{T}^d = ...
5
votes
1answer
154 views

Visualizing Frobenius Theorem

Given a smooth vector field $v$ on a (finite dimensional) manifold $M$, one can find the associated integral curves i.e. integral submanifolds of M such that the tangent space at any point $p\in M$ is ...
2
votes
0answers
63 views

Uniqueness of the “asymptotic limit” of a sequence of gradient flow lines

Let $(M,g)$ be a compact smooth orientable riemannian manifold, and let $f: M \to \mathbb R$ be a Morse function. All functions here are assumed to be smooth. We will be considering the integral ...
0
votes
1answer
104 views

Extending an embedding from a compact submanifold

Suppose $X, Y$ are smooth manifolds, $Z \subset X$ a compact submanifold and $f: X \rightarrow Y$ a smooth map such that the restriction of $f$ to $Z$ is an injective smooth immersion. As $Z$ is ...
0
votes
1answer
130 views

Intuition of a Submanifold

Could someone explain the intuition behind a submanifold. When, for example, is it appropriate to work with immersed submanifolds vs embedded submanifolds? Why is it important for a submanifold to be ...
1
vote
1answer
44 views

Is there a compact complex manifold with trivial $H_2$?

I don't believe that every complex manifold should have nontrivial $H_2$, otherwise we would easily prove the Chern's conjecture... But the problem is I don't have any counterexample. The Kähler ...
1
vote
1answer
66 views

Differential topology question involving cobordism

Prove that if $X$ and $Z$ are cobordant in $Y$, then for every compact manifold $C$ in $Y$ with dimension complementary to $X$ and $Z$, $I_2(X, C) = I_2(Z, C)$. [HINT: Let $f$ be the restriction to ...
2
votes
3answers
111 views

Showing S^n x R is parallelizable

A manifold $M$ is said to be parallelizable if it admits $k$ linearly independent vector fields. I know that this is equivalent to the tangent space $TM$ being trivial. I am trying to show that ...