1
vote
0answers
30 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
2
votes
0answers
25 views

How to put a structure of Fréchet space on $\Gamma(E)$?

Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms ...
2
votes
1answer
53 views

Characteristic classes not defined on vector bundles

If you read the definition on Wikipedia, you'll see that they allow characteristic classes to be defined on general principal $G$-bundles (vector bundles being subsumed in this general case by looking ...
2
votes
0answers
58 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
0
votes
0answers
38 views

Sections of the dual bundle of a smooth vector bundle

Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$. In this question, it is proven that the canonical map ...
0
votes
0answers
60 views

Two books defined two Chern(Euler) classes yet differed by a negative sign, what's wrong?

In the book 'Principles of Algebraic Geometry' P141 and the book 'Differential Forms in Algebraic Topology' P72-73, they defined the Chern(Euler) classes of line bundles using the patching data ...
0
votes
1answer
19 views

generation of sub bundle

Let $M$ differentiable manifold with $\dim M=n$. If $(TM,\pi,M)$ be the fiber bundle tangent. Consider the family $E=\lbrace E_x\rbrace _{x\in M}$ such that $E_x \subset T_xM$ and $\dim E_x=k$ for ...
2
votes
1answer
55 views

vector bundles and their cross-sections

Let $B$ be a compact Hausdorff space and $C^0(B)$ be the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$, let $\Gamma(\xi)$ denote the $C^0(B)$-module consisting ...
2
votes
1answer
67 views

Section of a vector bundle as a submanifold

I am currently working on one part of a problem surrounding sections and submanifolds. Given a real vector bundle $\pi: E\rightarrow M$ of rank k, with a smooth global section $s:M\rightarrow E$, can ...
1
vote
0answers
41 views

Restriction of a line bundle to a a two-cycle

I am reading a paper on Chiral Differential Operators http://arxiv.org/pdf/hep-th/0604179v3.pdf and it says on page 23 that a line bundle over a manifold $C$ can be characterized completely by its ...
3
votes
2answers
181 views

Hairy ball theorem : a counter example ?

Recall the hairy ball theorem : any continuous vector field on a sphere $S^n$ of even dimension must vanish at least once. Now consider the Riemann sphere $\mathbb{C} \cup \infty \simeq S^2$. Let ...
1
vote
0answers
65 views

$\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle

This was asked several times on math.se but none of them was answered. I'm trying to construct an explicit isomorphism from $E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}$ to $T = [0, 1] × R/ ∼$ where ...
1
vote
1answer
68 views

Real line bundle is smoothly isomorphic to Möbius bundle

I'm stuck on this question and tried to follow the partial answer of Neal. Erno's answer is fine too but it seems like I need to find the local trivializations of the Mobius bundle, which requires a ...
1
vote
0answers
81 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 ...
1
vote
1answer
44 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 ...
1
vote
0answers
62 views

Relative Euler class

In this topic http://mathoverflow.net/questions/93438/euler-class-in-the-non-compact-case you can read about relative Euler class. Can you show me some example of calculation of this class? Do you ...
0
votes
0answers
54 views

Constructing vector bundles from local covers and transitions functions

Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the ...
8
votes
1answer
141 views

Can the diagonal of a manifold be expressed as the zero set of a section of a vector bundle?

Let $\mathbb{C} \mathbb{P}^2$ be the two dimensional complex projective space and $$M:= \mathbb{C} \mathbb{P}^2 \times \mathbb{C} \mathbb{P}^2.$$ Let $ \gamma \rightarrow \mathbb{C} \mathbb{P}^2 $ be ...
0
votes
1answer
167 views

Understanding the definition and meaning of cotangent space

I would like to understand definition and also meaning of cotangent space. Let's first of all define tangent space: generally we know that equation of tangent line of function $f(x)$ at point $x_0$ ...
3
votes
0answers
84 views

Almost complex structures on a manifold and its exotic copy

Consider a compact simply-connected smooth $4$-manifold $M$ and its exotic copy $M'$. Identify the underlying topological manifolds. Dold-Whitney theorem guaranties ($\mathbb R$-linear) isomorphism of ...
8
votes
2answers
212 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
7
votes
1answer
143 views

Tangent bundles of exotic manifolds

Consider a pair of homeomorphic but not diffeomorphic smooth manifolds $M_1$ and $M_2.$ Fix a homeomorphism $\phi\colon M_1\rightarrow M_2.$ If I understand correctly, two bundles $TM_1$ and ...
1
vote
0answers
73 views

Isomorphism of vector bundles (exercise 6.2 of Bott, Tu)

I'm self-studying the book by Bott & Tu "Differential forms in algebraic topology" and I'm having problems with exercise 6.2. It says "Show that two vector bundles on $M$ are isomorphic iff their ...
2
votes
1answer
64 views

Every diffeomorphism induces a bundle isomorphism?

I'm starting to learn some vector bundle theory and I have the next question. If I have a diffeomorphism $f:M \rightarrow M$ and $E$ is a vector bundle with base $M$, is it true that there exists a ...
3
votes
2answers
65 views

Showing that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is a smooth vector bundle over $S^1$

I want to show that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is the total space of a smooth vector bundle over $S^1$. Here $\sim$ is defined as follows: fix a linear isomorphism $L$ of ...
7
votes
0answers
169 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
4
votes
2answers
232 views

Describe tangent and normal bundle to a manifold

Consider the set $X:=\{(x,y,z)\in\mathbb{R}^3 | x^2+3y^2=1+z^2\}$ I have to show that $X$ is a submanifold of $\mathbb{R}^3$ (and this is trivial); then, using on $T\mathbb{R}^3$ the standard ...
3
votes
1answer
171 views

Orthogonal complement of a vector bundle

Let $E \rightarrow X$ be a vector bundle with an inner product. If $F$ is a sub-bundle, we can define an orthogonal complement bundle $F^\perp$ (see http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf ...
4
votes
1answer
148 views

Image of smooth vector bundle morphism

Let $\pi_1:V_1 \rightarrow B_1$ and $\pi_2:V_2 \rightarrow B_2$ be smooth vector bundles and write $\phi_V: V_1 \rightarrow V_2$ as well as $\phi_B:B_1 \rightarrow B_2$ respectively for the total and ...
3
votes
2answers
449 views

Understanding differential form

Let $M$ be a smooth manifold. A differential form of degree $k$ is a smooth section of the $k$th exterior power of the cotangent bundle of $M$. Does it mean that a differential form of degree ...
4
votes
3answers
504 views

Trying to draw the tautological line bundle ($\subseteq \mathbb{CP}^1\times \mathbb{C}^2$)

In order to learn about vector bundles, I would like to draw the tautological vector bundle over the complex projective line $$ E = \{(x,v) \in \mathbb{CP}^1 \times \mathbb{C}^2 : v \in x \} .$$ ...
6
votes
2answers
635 views

How to draw a complex line bundle?

The most basic example of a topologically non-trivial real line bundle is the well-known Möbius strip. Everyone who learns about vector bundles will be confronted by it, if only because it has the ...