Tagged Questions

3
votes
1answer
39 views

normal form of an n-form

It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. ...
7
votes
1answer
57 views

Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$

Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?
1
vote
0answers
69 views

An example of a differentiable manifold class $C^k$ but not class $C^{k +1} $

I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where ...
2
votes
0answers
65 views

Relations between elliptic curves and topological quantum field theory

I heard that there are relations between elliptic curves and topological quantum field theory (TQFT). I googled and found that something called "elliptic genus" might be the key word to relate these ...
2
votes
0answers
42 views

smoothness structure on a set

I'm reading Milnor's 'characteristic classes', and in the first chapter he defines smoothness structure on a set M, which is confusing for me, as what follows:(these are not Milnor's words so please ...
8
votes
1answer
101 views

Classification of all 28 Exotic 7-Spheres

In http://en.wikipedia.org/wiki/Exotic_sphere#Explicit_examples_of_exotic_spheres Wikipedia says "As shown by Egbert Brieskorn (1966, 1966b) (see also (Hirzebruch & Mayer 1968)) the intersection ...
1
vote
1answer
158 views

where can I find solutions to A comprehensive introduction to differential geometry by Spivak?

I have tried google and I fail to find solutions to the exercises in the book A comprehensive Introduction to differenial geometry volume I by Spivak. Does anyone know about a site with solutions to ...
2
votes
1answer
206 views

control of the $C^{1}$ norm of a diffeomorphism

Let $\mathcal{E}$ be the set of smooth manifolds with boundary $E\subset \mathbb{R}^{3}$ which are perturbations of the unit ball whose volume $V$, diameter $d$ and area of the boundary $A$ satisfy: ...
1
vote
0answers
68 views

Reference Request: Poincare-Hopf Index Theorem

When I read Griffiths' Algebraic Curves page 23, it states Poincare-Hopf formula for differential forms that is smooth except finite isolated singularities. I tried to find reference for it, but ...
2
votes
1answer
83 views

Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
14
votes
1answer
210 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
5
votes
1answer
120 views

Proving that $O(n,m)$ is simply connected.

My question is the following: Under which conditions on given integers $n\le m$ is $$O(n,m) = \{A \in \mathbb R^{m\times n} : A^TA = \mathbf 1\}$$ simply connected? Does anyone know a reference for ...
0
votes
1answer
98 views

Co-homology Groups of the Torus

I wanted to explain to me, or give me a reference of how to calculate the cohomology groups of the complex and real, torus $\mathbb{T}^2$. I want to use this as an example in a seminar that I will ...
2
votes
1answer
84 views

Reference for topology and fiber bundle

I am looking for an introductory book that explains the relations of topology and bundles. I know a basic topology and algebraic topology. But I don't know much about bundles. I want a book that ...
3
votes
1answer
100 views

On the converse of Sard's theorem

Let $f: M \rightarrow N$ be a smooth map between two submanifolds of $\mathbb{R}^{m}$, $\mathbb{R}^{n}$ respectively. Sard's famous theorem asserts that the set of critical values $C$ of $f$ has ...
3
votes
3answers
244 views

Prerequisites for studying smooth manifold theory?

I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to ...
4
votes
1answer
118 views

Any manifold admits a morse function with one minimum and one maximum

I have heard the claim: "Any closed manifold admits a Morse function which has one local minimum and one local maximum" often used in talks without a reference. This does not seem to be very easy to ...
1
vote
0answers
133 views

Hairy ball theorem: references to applications

I'm looking for references to applications of the Hairy ball theorem. I already visited wikipedia and cited references, but I need a little more explanation in both meteorology and applications in ...
2
votes
0answers
124 views

Poincaré-Hopf theorem using Stokes

The wiki entry on the Poincaré-Hopf theorem claims that it "relies heavily on integral, and, in particular, Stokes' theorem". However, in the sketch of proof given there which is more or less the one ...
2
votes
1answer
96 views

Showing $[f^{-1}(y)]$ is Poincare dual to $f^*(\operatorname{vol})$.

Let $f: N^n \to M^m$ be a smooth map between closed oriented manifolds. Then I'm trying to show that for almost all $y \in M$, the homology class $[f^{-1}(y)] \in H_{n-m}(N)$ is Poincare dual to $f^* ...
2
votes
0answers
182 views

Higher-order derivatives in manifolds

If $E, F$ are real finite dimensional vector spaces and $\mu\colon E \to F$, we can speak of a (total) derivative of $\mu$ in Fréchet sense: $D\mu$, if it exists, is the unique mapping from $E$ to ...
4
votes
1answer
87 views

Tubular neighbourhood style theorem reference request

Let $X$ be a smooth manifold and $Y$ be a closed submanifold. Then there exists a neighbourhood $U$ of $Y$ in $X$ such that $Y$ is a deformation retract of $U$ right? I can only find (stronger forms ...
1
vote
1answer
103 views

reference for poincare-hopf theorem

I am an graduate student interested in fluid dynamics and have almost zero background in differential and algebraic topology. I must say that I do know some analysis (Lebesgue integration plus basics ...
18
votes
10answers
2k views

“Immediate” Applications of Differential Geometry

My professor asked us to find and make a list of things/facts from real life which have a differential geometry interpretation or justification. One example is this older question of mine. Another ...
5
votes
3answers
430 views

Texts on Principal Bundles, Characteristic Classes, Intro to 4-manifolds / Gauge Theory

I am looking for a textbook that might serve as an introduction to principal bundles, curvature forms and characteristic classes, and perhaps towards 4-manifolds and gauge theory. Currently, the only ...
11
votes
2answers
350 views

Books on topology and geometry of Grassmannians

Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective ...
9
votes
5answers
454 views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
2
votes
0answers
118 views

Topology of pseudo projective space

I don't know if "pseudo projective space" is a general accepted term, but I once read a book on general topology where the term was used for $\mathbb{S}^n / (\mathbb{Z}/m\mathbb{Z})$ (where you get ...
3
votes
0answers
111 views

non-orientable 4-manifolds

Most of the books and texts I read about classfication problems surrounding 4-manifolds which are closed and orientable (with a occasional side-track to open orientable 4-manifolds). This is ...
2
votes
2answers
107 views

Looking books about the topology of n-manifold ($n > 4$)

There are a lot of books dealing with the strangeness of the topology of 4-dimensional topology. I wonder if there are books or overview references on the topology of n-manifolds (where n > 4) ? ...
4
votes
1answer
189 views

Fiber of jet bundle of a fiber bundle

Given a fiber bundle $p:E\to B$ with fiber $V$ and structure group $G$, one can define the corresponding $k$-jet bundle $E^k\subset J^k(B,E)$ of jets of local sections of $E$. On Wikipedia there is a ...
16
votes
3answers
809 views

Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
2
votes
0answers
89 views

Question on the transversality between sections

Let $M^n$ be a differentiable manifold and $\pi\colon E\to M$ is $n$-dimensional vector bundle over $M$. We have a zero section $s\colon M\to E$ of $\pi$. How can I make a section $s'$ which is ...
4
votes
1answer
296 views

Classification of lens space

Let $L(p,q)$ be the lens space, that is $L(p,q)=S^3/\mathbb{Z}_p$. Here, $\mathbb{Z}_p$ acts on $S^3$ by $(z_1,z_2)\mapsto (\rho z_1,\rho^q z_2)$, $ \rho=e^{\frac{2\pi i}{p}}$. It is well known ...
4
votes
2answers
375 views

The Strong Whitney Embedding Theorem-Any Recommended Sources?

Just about all of the standard textbooks on manifold theory give proofs of weak versions of the Whitney Embedding theorem. But other then Whitney's original 1944 paper,are there any standard sources ...
31
votes
8answers
6k views

Teaching myself differential topology and differential geometry

I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology. I have decided to fix this lacuna once for ...