Subbundle 2007-03-18 17:11:54 2007-03-18 17:11:54 Definition subbundle subbundle vector bundle sub-bundle sub-vector bundle % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a vector bundle $\pi \!:\! \mathcal E \rightarrow M$, a subbundle $\mathcal E^\prime$ is a subset of the total space, $\mathcal E^\prime \subset \mathcal E$, so that \[ \pi\vert_{\mathcal E^\prime}\!:\! \mathcal E^\prime \rightarrow M\] is a vector bundle, and for each point $p\in M$, the fibre at $p$ \[{\pi\vert_{\mathcal E^{\prime}}}^{-1}(p) = \mathcal E^\prime_p\] is a vector subspace of $\mathcal E_p = \pi^{-1}(p)$