Tagged Questions

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.

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1answer
30 views

How can we prove that $GL(2,\mathbb{R})$ is a topological group in $R^{4}$

A group is called topological group if it satisfies three properties 1) G is a Hausdorff space in K (Here we want to prove that if $ A,B \in GL(2,\mathbb{R})$ then we can find two disjoint open sets ...
0
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1answer
25 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
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0answers
23 views

Orientation of the intersection of manifolds

From Guillemin and Pollack Differential Topology: Compute the orientation of $\mathbf{X}\cap\mathbf{Z}$ in the following examples by exhibiting positively oriented bases at every point: a) ...
3
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1answer
88 views

Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
1
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1answer
20 views

Is the image under a homeomorphism of the cut locus $C_p$ a null-set?

Let $M$ be a complete Riemannian manifold with a point $p  \in M$ and let $U \subset T_pM$ be an open disk containing $0_p$ in the tangent space to $p$. By $C_p$ we denote the image of the boundary of ...
1
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1answer
26 views

Taylor development on manifolds and Manifolds of differentiable Mappings?

I'm trying to study the book "Manifolds of Differentiable Mappings" written by Michor and I came across the following: He considers two smooth manifolds $M$ and $N$ and define an equivalence relation ...
0
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1answer
19 views

The space of collars of a manifold is contractible

Theorem: Let $M$ be a smooth manifold with boundary $\partial M$. Let $e_0,e_1 : \partial M\times [0,1]\rightarrow M$ be collars of $M$, i.e. $e_i$ are embeddings such that $e_i(x,0)=x$ for each ...
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0answers
15 views

How to get a Kirby diagram of $S^1 \times M^3$ if $M^3$ is given by a surgery diagram?

In "4-manifolds and Kirby Calculus" by Gompf and Stipsicz, there is a nice description of how to get the Kirby diagram of $S^1 \times M^3$, given a Kirby diagram of $M^3$. Basically, one thickens the ...
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0answers
20 views

Transversality, mod 2 degree, Winding numbers in differential topology

From Chapter 2 Section 5 of Guillemin and Pollack, Differential Topology, $\mathbf{X}$ is a compact connected manifold, and $f:\mathbf{X}\rightarrow \mathbb{R}^n$ a smooth map and $\dim{X}=n-1$. ...
1
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1answer
43 views

Minkowski space is locally Euclidean?

The Minkowski spacetime $\mathbb{R}^{1,3}$ is said to be a manifold (isomorphic to $SO^{1,3}$. But according to the definition of a manifold it should be locally euclidean. However, this seems to be ...
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0answers
44 views

Analytic approximations of the step function

Consider the Heaviside step function: $$H:\mathbb{R}\to \mathbb{R} $$ defined by $$H(x)=\begin{cases} 0 & \mbox{if } x<0 \\ 1 & \mbox{if } x\geq 0\end{cases}$$ Fix any $\delta>0$. Given ...
10
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0answers
123 views

$C^{k}$-manifolds: how and why?

First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the ...
2
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1answer
31 views

In uniform circular motion in R^2, is acceleration in the normal bundle?

In physics we learn that accleration is a vector quantity parallel to the radius and orthogonal to the velocity. With the embedding $\mathbb{S}^1 \hookrightarrow \mathbb{R}^2$ and the induced ...
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31 views

What 's conditions on open set related to connected neighborhood of boundary

I have a question: Suppose $D$ is an open set in $\mathbb{R^n}$ and topological boundary $bD$ is an embedded submanifold of $\mathbb{R^n}$. For each $p\in bD$, we want to have an open neighborhood ...
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0answers
32 views

General expression of smooth sections of tensor bundles.

On page 317 of John Lee's Smooth Manifolds it's said that if $(x^i)$ are local coordinates on a smooth manifold $M$, then sections of the tensor bundle $T^kT^*M=\bigsqcup_{p\in M}T^k(T^*_pM)$ over a ...
1
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1answer
36 views

Homeomorphism of a Genus-2 Surface

Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, $S^1$, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ...
2
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1answer
22 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
3
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3answers
60 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
6
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4answers
608 views

Why is there no natural metric on manifolds?

One of the things that always bothered me after learning introductory differential geometry (as a physics student) and then delving deeper into this field on my own is that, the usual construction of ...
4
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1answer
145 views

Why the whole theory about differentiable manifolds is based on open sets?

I only have studied basic topology, which means i haven't studied any about differentiable manifolds. I just skimmed pages on wikipedia. Here is a simple illustration on a basic situation. Let $E$ ...
4
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2answers
107 views

How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...
3
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1answer
43 views

Why does every noncompact orientable surface have a complex structure?

There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ...
0
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1answer
75 views

Are definitions for smooth map between manifolds are equivalent?

There are two ways to define a smooth map between manifolds. The 1st way (for example, Lee): $f:M\rightarrow N$ is smooth iff for every $p\in M$ there exist charts $(U,\varphi)$ at $p$ and $(V,\psi)$ ...
3
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0answers
47 views

How does a differential act when we identify $T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N$?

It's fairly common to identify the tangent space of a product manifold as $$ T_p(M\times N)\cong T_{p_1}M\times T_{p_2}N $$ where $p=(p_1,p_2)$, and the actual isomorphism is given by $v\in ...
3
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1answer
37 views

If two Curves in $\Bbb R^3$ are transversal then they do not intersect

Proving this is easy: Let $X$ and $Z$ be the two curves in $\Bbb R^3$. Assume $X$ and $Z$ intersect at a point $y$. Then, at most $\dim(T_y(X) + T_y(Z)) = 2$, where $T_x(X)$ and $T_z(Z)$ denotes the ...
2
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1answer
80 views

If $S\subseteq M\times N$ is embedded, and $S$ and $\{p\}\times N$ intersect transversely in one point, then $\pi_M|_S$ is a diffeomorphism?

I'm trying to prove the equivalence of the following statements: Suppose $M^m$ and $N^n$ are smooth manifolds, $S\subseteq M\times N$ immersed, and $\pi_M$ and $\pi_N$ the projection maps. TFAE: ...
2
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0answers
42 views

Transition Functions for Cartesian Coordinate Systems

This is my first time using Mathematics SE (I've only used Physics and Astronomy before), so I apologize if this question is awkwardly phrased or incorrectly presented. I welcome any and all edits and ...
0
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2answers
61 views

Proving that a form is exact

Maybe this question is rather obvious but I didn't manage to solve it myself. Assume $M$ is a closed, oriented manifold. take $$ \Omega^k(M)\ni \omega = \begin{cases} d\beta~,~~~ in~ U\\ 0~,~~~ ...
2
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1answer
34 views

Does Heine-Borel hold for smooth manifolds?

If $M$ is a smooth $n$-manifold, the famous Whitney embedding theorems show that we can view $M$ as an embedded submanifold of some Euclidean space $\mathbb{R}^N$. Does the Heine-Borel theorem still ...
5
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0answers
103 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
1
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1answer
39 views

Why are the basis elements of $T_pM$ a vector field if we let $p$ vary?

At the end of this article on tangent vectors, they say each $\frac{\partial}{\partial x_i}$ is a vector field, if we let the point $p$ vary. However, they are only defined in a particular coordinate ...
1
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1answer
27 views

If $v\in\mathbb{R}^N\setminus\mathbb{R}^{N-1}$, what is the projection $\pi_v$ with kernel $\mathbb{R}v$?

I going over a lemma for the Whitney Embedding theorem which shows that an injective immersion of an $n$-manifold into $\mathbb{R}^N$ can actually be immersed in a lesser dimensional Euclidean space ...
2
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1answer
56 views

If $b$ is a regular value of $f$, $f^{-1}(-\infty,b]$ is a regular domain?

I'm trying to prove the first part of Proposition 5.47 of Lee's Smooth Manifolds, which is left to the reader. It says Suppose $M^m$ is a smooth manifold, and $f\colon M\to\mathbb{R}$ smooth. For ...
0
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1answer
40 views

Show that a set is a smooth curve and find a parameterization for it.

Let $S = \{ (x,y,z) \in \mathbb{R}^3 \mid x - yz + z^3 = 0 \}$. Let $\pi: \mathbb{R}^3 \to \mathbb{R}^2$ be such that $\pi(x,y,z) = (x,y)$. Let $H = \{p \in S \mid \pi_{\mid S}: S \to \mathbb{R}^2 ...
4
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1answer
80 views

Example of a submanifold $S\subseteq M$ that is an immersed submanifold is more than one way?

Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth ...
0
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1answer
37 views

About $C^{0}$ being topological manifold

Is that the reason why $C^{0}$ being topological manifold due to that $C^{0}=\phi$ which contains nothing? Correct me if I am wrong. I am new to differential topology.
1
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2answers
53 views

Is $y^2=x(x-1)^2$ an immersed submanifold?

Is the curve $y^2=x(x-1)^2$ an immersed submanifold in $\mathbb{R}^2$? It's certainly not embedded since it intersects itself at $(1,0)$. I'm aware of techniques to prove a subset is not a immersed ...
2
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2answers
79 views

Why is the irrational winding of the torus not locally path connected?

The irrational winding of the torus given by the map $f\colon\mathbb{R}\to T^2$ where $f(t)=(e^{it},e^{i\alpha t})$ for some irrational $\alpha$. Wikipedia mentions this is not a regular submanifold, ...
5
votes
2answers
95 views

Is $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ diffeomorphic to $S^2$?

I was working on Problem 5-1 of Smooth Manifolds by Professor John Lee, and it lead me to wanting to show that $\{(x,y,z,0)\in\mathbb{R}^4:x^4+y^2+z^2=1\}$ is diffeomorphic to $S^2$, and that is ...
5
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1answer
42 views

Smooth embeddings of the $2$-sphere

I have a past qual question here: given a smooth embedding $f \colon S^2 \to \mathbb{R}^3$, show that there must exist distinct points $p,q \in S^2$ such that the tangent planes to the embedded sphere ...
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1answer
77 views

Video/audio lectures on differential topology?

Do there exist decent online video lectures, or even audio lectures, covering differential topology? I'm aware of Milnor's talk, but it is more like exposition and doesn't go very far.
2
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1answer
47 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
1
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0answers
19 views

Is there a rule for computing the differential of a product of maps?

A lot of partition of unity arguments have some map of the form $f=\sum_i \psi_if_i$. Is there a formula for the differential $df_p$ in terms of its summands? For instance, suppose $f_i:U_i\to V_i$ ...
1
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2answers
40 views

Inward and outward pointing tangent vectors?

If $M^n$ is a smooth manifold with boundary and $p\in\partial M$, then $T_pM$ is the disjoint union of inward and outward point vectors, and $T_p\partial M$. If $(U,(x^i))$ is a smooth boundary chart ...
0
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2answers
55 views

How does $v\Phi^1=\cdots=v\Phi^k=0$ imply $v\in\ker d\Phi_p$?

I'm confused about an immediate corollary in John Lee's Smooth Manifolds. Proposition 5.38 says Suppose $M$ is a smooth manifold and $S\subset M$ is an embedded submanifold. If $\Phi\colon U\to N$ ...
2
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1answer
60 views

Boundary connected sum of manifolds

I have two related questions about the boundary connected sum of manifolds with boundaries. Let $T=S^1 \times S^1$ be a torus and let $X=T \times [0, 1]$ be the cylinder over the torus. Let $X'$ be a ...
3
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0answers
32 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
0
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0answers
14 views

Understanding topological modular forms

I am asking for good references to understanding topological modular forms. Please don't laugh. I am more or less an analyst and differential geometer, and I do know some algebraic topology and very ...
1
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1answer
46 views

Why does $S^n$ satisfy the local $n$-slice condition? From Lee's Smooth Manifolds.

Example 5.9 on page 103 of Lee's Smooth Manifolds says the following: The intersection of $S^n$ with the open subset $\{x:x^i>0\}$ is the graph of the smooth function $$ ...
3
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0answers
62 views

Voisin's proof of Ehresmann's theorem

On p.221 of Voisin's book on Hodge theory, there are two claims: a) Let $B$ be a contractible smooth manifold. There exists a vector field $\chi$ on $B$ whose flow $\Phi_t$ is global and, given any ...