Talk:PlanetPhysics/OCR2 Proofreading Test

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: ocr 2 proofreading test %%% Primary Category Code: 00. %%% Filename: Ocr2ProofreadingTest.tex %%% Version: 1 %%% Owner: rspuzio %%% Author(s): rspuzio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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necessary to consider the second bundle. The curvature form of our connection is a tensorial quadratic differential form in $M$, of \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} $ad(G^{\prime})$ and with values in the \htmladdnormallink{Lie Algebra}{http://planetphysics.us/encyclopedia/TopologicalOrder2.html} $L(G^{\prime})$ of $G^{\prime}$. Since the Lie algebra $L(G)$ of $G$ is a subalgebra of $L(G^{\prime})$, there is a natural projection of $L(G^{\prime})$ into the quotient space $L(G^{\prime})/L(G)$. The image of the curvature form under this projection will be called the torsion form or the torsion \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html}. If the forms $\pi^{\rho}$ in (13) define a $G$-connection, the vanishing of the torsion form is expressed analytically by the conditions $$ (22)\text{ }\quad c_{j^{\prime\prime}k^{\prime\prime}}^{i^{\prime\prime}}=0. $$ \quad We proceed to derive the analytical \htmladdnormallink{formulas}{http://planetphysics.us/encyclopedia/Formula.html} 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 G-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 $$ (24)\text{ }\quad \Sigma_{\rho,k}a_{\rho k}^{i}\Pi^\rho \wedge \omega^{k}=0, $$ where we put $$ (25)\text{ }\quad \Pi^\rho=d\pi^{\rho}+\frac{1}{2}\Sigma_{\sigma.\tau}\gamma_{\sigma\tau}^{\rho} \pi^{\sigma} \wedge \pi^{\tau}. $$ For a fixed value of $k$ we multiply the above equation by $$ \omega^{1}\text{ }\wedge.\text{. . }\wedge\text{ }\omega^{k-1}\text{ }\wedge\text{ }\omega^{k+1}\ldots\text{ }\wedge\text{ }\omega^{n}, $$ getting $$ \sum_{\rho}a_{\rho k}^{i}{\Pi^\rho}\text{ }\wedge\text{ }\omega^{1}\text{ }\wedge.\text{. . }\wedge\text{ }\omega^{n}=0, $$ or $$\Sigma_{\rho}a_{\rho k}^{i} {\Pi^\rho} \equiv 0,\ \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{j}.$$

\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 follows that $\Pi^\rho$ is of the form $$ \Pi^\rho=\Sigma_{j} \phi_{j}^{\rho} \wedge \omega^{j} $$ where $\phi_{j}^{\rho}$ are Pfaffian forms. Substituting these expressions into (24), we get $$ \Sigma_{\rho,j,k} (a_{\rho k}^{i}\phi_{j}^{\rho}-a_{\rho j}^{i}\phi_{k}^{\rho})\wedge\omega^{j}\wedge\omega^{k}=0. $$ It follows that $$ \Sigma_{\rho}(a_{\rho k}^{i}\phi_{j}^{\rho}-a_{\rho j}^{i}\phi_{k}^{\rho})\equiv 0,\text{ }\mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{k}. $$ Since $G$ has the property $(C)$, the above equations imply that $$ \phi_{j}^{\rho}\equiv 0,\text{ }\mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{k}. $$

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