PlanetPhysics/OCR2 Proofreading Test

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]] $L(G^{\prime })$ 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 [[../Tensor/|tensor]]. If the forms ${\pi }^{\rho }$ in (13) define a $G$-connection, the vanishing of the torsion form is expressed analytically by the conditions $(22)==c_{j^{\prime \prime }{k}^{\prime \prime }}^{i^{\prime \prime }}=0.$ \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 G-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 $(24)=={\mathrm{\Sigma }}_{\rho ,k}{a}_{\rho k}^i{\mathrm{\Pi }}^{\rho }\wedge {\omega }^k=0,$ where we put $(25)=={\mathrm{\Pi }}^{\rho }=d{\pi }^{\rho }+\frac{1}{2}{\mathrm{\Sigma }}_{\sigma .\tau }{\gamma }_{\sigma \tau }^{\rho }{\pi }^{\sigma }\wedge {\pi }^{\tau }.$ For a fixed value of $k$ we multiply the above equation by ${\omega }^1==\wedge .=..=\wedge =={\omega }^{k-1}==\wedge =={\omega }^{k+1}\dots ==\wedge =={\omega }^n,$ getting $\sum _{\rho }{a}_{\rho k}^i{\mathrm{\Pi }}^{\rho }==\wedge =={\omega }^1==\wedge .=..=\wedge =={\omega }^n=0,$ or ${\mathrm{\Sigma }}_{\rho }{a}_{\rho k}^i{\mathrm{\Pi }}^{\rho }\equiv 0,\mathrm{m}\mathrm{o}\mathrm{d}{\omega }^j.$

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