TestOcr2 2009-02-07 00:57:44 2009-02-07 00:57:44 Definition % 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 necessary to consider the second bundle. The curvature form of our connection is a tensorial quadratic differential form in $M$, of type $ad(G^{\prime})$ and with values in the Lie algebra $L(O^{\prime})$ of $G^{\prime}$. Since the Lie algebra $L(O)$ of $G$ is a subalgebra of $L(G^{\prime})$, there is a natural projection of $L(O^{\prime})$ into the quotient space $L(G^{\prime})/L(G)$. The image of the cur- vature form under this proiection will be called the torsion form or the torsion tensor. If the forms $\pi^{\rho}$ in (13) define a $G$-connection, the vanishing of the torsion form is expressed analytically by the con- ditions $$ (22)\text{ }\quad c_{f^{\prime\prime}k^{\prime\prime}}^{i^{\prime\prime}}=0. $$ \quad We proceed to derive the analytical formulas for the theory of a $G$-connection without torsion in the tangent bundle. In general we will consider such formulas in $B_{G}$. The fact that the O-connection has no torsion simplifies (13) into the form $$ (23)\text{ }\quad d\omega^{i}=\Sigma_{\rho,k}a_{\rho k}^{i}\pi^{\rho}\wedge\omega^{k}. $$ By taking the exterior derivative of (23) and using (18), we get $$ (24)\text{ }\quad \Sigma_{\rho,k}a_{\rho k}^{i}\Pi\rho_{\mathrm{A}\omega^{k}=0_{;}} $$ where we put $$ (25)\text{ }\quad \Pi\rho=d\pi^{\rho}+\#\Sigma_{\sigma.\tau}\gamma_{\sigma\tau}^{\rho}\pi^{\sigma}\mathrm{A}\pi^{\tau}. $$ For a fixed value of $k$ we multiply the above equation by $$ \omega^{1}\text{ }A.\text{ . . }A\text{ }\omega^{k-1}\text{ }A\text{ }\omega^{k+1_{\Lambda}}\ldots\text{ }A\text{ }\omega^{n}, $$ getting $$ \sum_{\rho}a_{\rho k^{\prod\rho}}^{i}\text{ }A\text{ }\omega^{1}\text{ }A.\text{ . . }A\text{ }\omega^{n}=0, $$ or $\Sigma_{\rho}a_{\rho k^{\Pi\rho}}^{l}\equiv 0,\ \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{;}$. \noindent Since the infinitesimal transformations $X_{\rho}$ are linearly independent, this implies that $$ \Pi\rho\equiv 0,\text{ }\mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{j}. $$ It followo that II $\rho$ is of the form $$ IA\text{ }\rho_{=\Sigma_{j}\phi_{J^{\mathrm{A}\omega^{f}}}^{\rho}} $$ where $\phi_{j}^{\rho}$ are Pfaffian forms. Substituting these expressions into (24), we get $$ \Sigma_{\rho,j,k(a_{\rho k}^{i}\phi_{j}^{\rho}-a_{\rho j}^{i}\phi_{k}^{\rho})\mathrm{A}\omega^{j}\mathrm{A}\omega^{k}=0}. $$ It follows that $$ \Sigma_{\rho}(a_{\rho k}^{i}\phi_{f}^{\rho}-a_{\rho j}^{1}\phi_{k}^{\rho})\equiv 0,\text{ }\mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{\prime}. $$ Since $G$ has the property $(C)$, the above equations imply that $$ \phi_{f}^{\rho}\equiv 0,\text{ }\mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{k}. $$