PlanetPhysics/Test OCR2

necessary to consider the second bundle. The curvature form of our connection is a tensorial quadratic differential form in $M$, of [[../Bijective/|type]] $ad(G^{\prime })$ and with values in the [[../TopologicalOrder2/|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/|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 $(22)==c_{f^{\prime \prime }{k}^{\prime \prime }}^{i^{\prime \prime }}=0.$ \quad We proceed to derive the analytical [[../Formula/|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 $(23)==d{\omega }^i={\mathrm{\Sigma }}_{\rho ,k}{a}_{\rho k}^i{\pi }^{\rho }\wedge {\omega }^k.$ By taking the exterior derivative of (23) and using (18), we get $(24)=={\mathrm{\Sigma }}_{\rho ,k}{a}_{\rho k}^i\mathrm{\Pi }{\rho }_{\mathrm{A}{\omega }^k=0_{;}}$ where we put $(25)==\mathrm{\Pi }\rho =d{\pi }^{\rho }+\mathrm{\#}{\mathrm{\Sigma }}_{\sigma .\tau }{\gamma }_{\sigma \tau }^{\rho }{\pi }^{\sigma }\mathrm{A}{\pi }^{\tau }.$ For a fixed value of $k$ we multiply the above equation by ${\omega }^1==A.=..=A=={\omega }^{k-1}==A=={\omega }^{k+1_{\mathrm{\Lambda }}}\dots ==A=={\omega }^n,$ getting $\sum _{\rho }{a}_{\rho k^{\prod \rho }}^i==A=={\omega }^1==A.=..=A=={\omega }^n=0,$ or ${\mathrm{\Sigma }}_{\rho }{a}_{\rho k^{\mathrm{\Pi }\rho }}^l\equiv 0,\mathrm{m}\mathrm{o}\mathrm{d}{\omega }^{;}$.

\noindent Since the infinitesimal transformations $X_{\rho }$ are linearly independent, this implies that $\mathrm{\Pi }\rho \equiv 0,==\mathrm{m}\mathrm{o}\mathrm{d}{\omega }^j.$ It followo that II $\rho$ is of the form $IA=={\rho }_{={\mathrm{\Sigma }}_j{\varphi }_{J^{\mathrm{A}{\omega }^f}}^{\rho }}$ where ${\varphi }_j^{\rho }$ are Pfaffian forms. Substituting these expressions into (24), we get ${\mathrm{\Sigma }}_{\rho ,j,k(a_{\rho k}^i{\varphi }_j^{\rho }-a_{\rho j}^i{\varphi }_k^{\rho })\mathrm{A}{\omega }^j\mathrm{A}{\omega }^k=0}.$ It follows that ${\mathrm{\Sigma }}_{\rho }(a_{\rho k}^i{\varphi }_f^{\rho }-a_{\rho j}^1{\varphi }_k^{\rho })\equiv 0,==\mathrm{m}\mathrm{o}\mathrm{d}{\omega }^{\prime }.$ Since $G$ has the property $(C)$, the above equations imply that ${\varphi }_f^{\rho }\equiv 0,==\mathrm{m}\mathrm{o}\mathrm{d}{\omega }^k.$

OCR based on this tiff scan