PlanetPhysics/Test OCR2

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 $$ (22)= = \quad c_{f^{\prime\prime}k^{\prime\prime}}^{i^{\prime\prime}}=0. $$ \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 $$ (23)= = \quad d\omega^{i}=\Sigma_{\rho,k}a_{\rho k}^{i}\pi^{\rho}\wedge\omega^{k}. $$ By taking the exterior derivative of (23) and using (18), we get $$ (24)= = \quad \Sigma_{\rho,k}a_{\rho k}^{i}\Pi\rho_{\mathrm{A}\omega^{k}=0_{;}} $$ where we put $$ (25)= = \quad \Pi\rho=d\pi^{\rho}+\#\Sigma_{\sigma.\tau}\gamma_{\sigma\tau}^{\rho}\pi^{\sigma}\mathrm{A}\pi^{\tau}. $$ For a fixed value of $$k$$ we multiply the above equation by $$ \omega^{1}= = A.=. . = A= = \omega^{k-1}= = A= = \omega^{k+1_{\Lambda}}\ldots= = A= = \omega^{n}, $$ getting $$ \sum_{\rho}a_{\rho k^{\prod\rho}}^{i}= = A= = \omega^{1}= = A.=. . = A= = \omega^{n}=0, $$ 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 $$ \Pi\rho\equiv 0,= = \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{j}. $$ It followo that II $$\rho$$ is of the form $$ IA= = \rho_{=\Sigma_{j}\phi_{J^{\mathrm{A}\omega^{f}}}^{\rho}} $$ 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 $$ \Sigma_{\rho}(a_{\rho k}^{i}\phi_{f}^{\rho}-a_{\rho j}^{1}\phi_{k}^{\rho})\equiv 0,= = \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{\prime}. $$ Since $$G$$ has the property $$(C)$$, the above equations imply that $$ \phi_{f}^{\rho}\equiv 0,= = \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{k}. $$

OCR based on this tiff scan