For questions on vector bundles.

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7
votes
3answers
700 views

Vector bundle transitions and Čech cohomology

I have read that transition maps $g_{\alpha\beta}:U_\alpha\cap U_\beta\to GL(n)$ of a vector bundle of rank $n$ are related to the Čech cohomology group $H^1\left(M,GL(n,\mathcal{C}^\infty_M)\right)$ ...
11
votes
2answers
472 views

Conditions such that taking global sections of line bundles commutes with tensor product?

Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles $L, ...
33
votes
2answers
924 views

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have ...
4
votes
1answer
250 views

Global sections of a tensor product of vector bundles on a smooth manifold

This question is similar to Conditions such that taking global sections of line bundles commutes with tensor product? and Tensor product of invertible sheaves except that I am concerned with ...
4
votes
1answer
122 views

Tangent bundle : is a manifold

I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector ...
3
votes
3answers
1k views

Understanding the trivialisation of a normal bundle

I've been looking for a definition of "trivialisation of normal bundle". I think I understand what a vector bundle, fibre bundle and a local trivialisation of either is. I also know what a tangent ...
3
votes
1answer
84 views

Learning Fibre Bundle from “Topology and Geometry” by Bredon

Bredon defines bundle projection in the following way: $\bf13.1.$ Definition. Let $X,B$ and $F$ be Hausdorff spaces and $p:X\to B$ a map. Then $p$ is called a bundle projection with fiber $F$, if ...
8
votes
2answers
224 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 ...
11
votes
1answer
359 views

Second Stiefel-Whitney Class of a 3 Manifold

This is exercise 12.4 in Characteristic Classes by Milnor and Stasheff. The essential content of the exercise is to show that $w_2(TM)=0$, where $M$ is a closed, oriented 3-manifold, $TM$ its tangent ...
7
votes
1answer
496 views

Understanding the canonical line bundle $H$, and the fact that $(H \otimes H)\oplus 1 \simeq H \oplus H$

I am trying to understand Example 1.13 of Hatcher's book on vector bundles and K-Theory (page 24). The cannonical line bundle $H \to \mathbb{C}P^1$ satisfies the relation $(H \otimes H)\oplus 1 ...
3
votes
2answers
190 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 ...
-4
votes
1answer
291 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
6
votes
1answer
229 views

Möbius strip and $\mathscr O(-1)$. Or $\mathscr O(1)$?

On the real $\textbf P^1$ we have these algebraic line bundles: $\mathscr O(1)$ and $\mathscr O(-1)$. Which one corresponds to the Möbius strip? (Both are $1$-twists of $\textbf P^1\times\textbf ...
6
votes
1answer
389 views

Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle

Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the disjoint union of all these $k$-dimensional subspaces and ...
5
votes
0answers
338 views

Tangent bundle of a quotient by a proper action

Given a compact group $G$ acting freely on a manifold $X$, is there a "nice" way to describe the tangent bundle of the quotient $X/G$ (when it is a manifold)? In the case the group $G$ is finite, or ...
3
votes
2answers
112 views

Line bundle over $S^2$

I'm trying to study line bundle over $S^2$. In this post was outlined the method based on clutching functions. But now I'm interesting in another approach. For the sphere there is two maps : upper ...
3
votes
2answers
151 views

Tangent space and tangent vectors

As I have heard, tangent vector to a smooth manifold $M$ in $p \in M$ is the operator $D_{\xi}$:$f \to D_{\xi}f$, where $f$ is a smooth function $f: M \to R$, with the following properties: ...
3
votes
2answers
599 views

Real line bundle smoothly isomorphic to Möbius bundle

I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that ...
2
votes
1answer
124 views

How a principal bundle and the associated vector bundle determine each other

It seems to me that given a vector bundle, the associated principal bundle is univocally determined. In fact one has to construct a principal bundle given the base, the fibre (the group $G$ in which ...
6
votes
1answer
94 views

Is T($S^2 \times S^1$) trivial?

How would I find out if T($S^2 \times S^1$) is trivial or not? Using the hairy ball theorem I can show that T($S^2$) is not trivial, and it is straight forward to show that T($S^1$) is trivial. ...
5
votes
1answer
193 views

Reduction of a structure group.

Let $X$ be a smooth manifold and $\pi:E\rightarrow X$ a vector bundle of rank $k$ on X. If one manages to redefine $E$ by using a cocycle $\{g_{\alpha,\beta}\}$ whose values are all contained in a ...
3
votes
1answer
68 views

When does the difference between a vector bundle and the associated frame bundle matter?

In the comments to this question How a principal bundle and the associated vector bundle determine each other, it was remarked that while there is a bijective correspondence between rank $n$ vector ...
1
vote
1answer
49 views

Some troubles about topology and definition of a Vector Bundle

Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my ...
0
votes
0answers
73 views

a question about pre-symplectic manifold

Let $(M,\omega)$, is pre-symplectic. Then can we say, ker$ \omega$ is subbundle of tangent bundle $TM$?
8
votes
1answer
162 views

Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)

It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
8
votes
0answers
173 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 ...
5
votes
1answer
134 views

Visualizing topology of a Vector Bundle

I've started reading Milnor, Stasheff - Characteristic Classes and at page $18$ they proved that $\mathbb{R}^n$-bundle $\xi$ is trivial if and only if $\xi$ admits $n$ cross sections $s_1, \dots , ...
4
votes
3answers
192 views

What is a tangent bundle? (Aubin)

Here's what I read in A Course in Differential Geometry by Thierry Aubin. 2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$ And then 2.6. Definition. Let $\Phi$ be a ...
2
votes
1answer
121 views

The Affine Property of Connections on Vector Bundles

Given any two connections $\nabla_1, \nabla_2: \Omega^0 (V) \to \Omega^1 (V)$ on a vector bundle $V \to M$, their difference $\nabla_1 - \nabla_2$ is a $C^\infty (M)$-linear map $\Omega^0 (V) \to ...
1
vote
1answer
79 views

Topology of $GL_n(K)$

I need to show any of the following results: Consider $K=\mathbb{R}$ or $\mathbb{C}$, then, 1) The compact-open topology and the usual topology of $GL_n(K)$ are the same. 2) Taking inverses and ...
1
vote
0answers
303 views

Characterization of Chern classes and Whitney product formula

Let E be a Complex vector bundle of rank r on M. Given $s_1, \cdots , s_r$ generic global sections, i can characterize the i-th Chern class as follows: $ C_i(E)= \eta_{V_i}$, and $V_i$ is the locus ...
0
votes
1answer
59 views

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

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 $(0, t) ∼ (1, −t)$. I verified that $\Bbb{R}P^1$ is homeomorphic to ...