2
votes
1answer
168 views

Is this sphere bundle over SL3/SO3 trivial?

The space $SL_3(\mathbb{R})/SO_3(\mathbb{R})$ can be though of as some kind of 5-(real)dimensional generalized upper half-space. Fix any copy of $SO_2(\mathbb{R})$ sitting inside $SO_3(\mathbb{R})$, ...
3
votes
0answers
110 views

Orthonormal frame bundle orthogonal to a curve

This is a duplicate of this question on math.stackexchange, since I got there not a single answer. Let $M$ be a $n$-dimensional smooth riemannian manifold and ...
0
votes
1answer
289 views

Cotetrad, spin connection and Dirac operator

Let us consider an orientable smooth 4-manifold $M$. Pick a vector bundle $T$ that's isomorphic to the tangent bundle $TM$. We then equip $T$ with a cotetrad (or coframe field) $e$ and (spin) ...
1
vote
2answers
1k views

Vector bundles vs principal $G$-bundles

It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres $$F=\pi^{-1}(x), \ \ \ x\in B $$ over any $x\in B$, are ...
1
vote
1answer
219 views

Intersection of subvector bundles

Suppose we have a smooth vector bundle $\pi: E \rightarrow B$ and two sub vector bundles $\pi_1: E_1 \rightarrow B_1$ and $\pi_2: E_2 \rightarrow B_2$ such that the bases $B_1$ and $B_2$ are ...
2
votes
2answers
438 views

Elementary transformations of ruled surfaces as maps of vector bundles

This comes as a question in Beauville's Algebraic surfaces book (III.24 (2)). We work over $\mathbb{C}$. All geometrically ruled surfaces (grs) $p:S\longrightarrow C$ over a curve $C$ can be seen as ...
1
vote
3answers
576 views

Topology of maps between fibers of vector bundles

First of all sorry for the (possible) incorrect english. I don't know english very well. I'm with a doubt about topology of maps between fibres of vector bundles. Consider $E$ and $F$ vector bundles ...
0
votes
1answer
456 views

Is a fibre bundle over a vector bundle trivializable on each fibre?

Let $\pi:E\to M$ be a vector bundle over a closed smooth manifold and supose $\Pi:F\to E$ is a fibre bundle over the total space of $\pi$. I'd like to know if, restricted to $E_p$, the second bundle ...
1
vote
2answers
359 views

Relative minimality for conic bundles

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces. The definition is ok: the fiber over ...
4
votes
3answers
2k views

Grassmannian bundle theorem

Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$. ...
14
votes
4answers
1k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...