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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 type $ ad(G^{\prime})$ and with values in the 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. If the forms $ \pi^{\rho}$ in (13) define a $ G$ -connection, the vanishing of the torsion form is expressed analytically by the conditions

$\displaystyle (22)$ $\displaystyle \quad c_{j^{\prime\prime}k^{\prime\prime}}^{i^{\prime\prime}}=0.
$

    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 G-connection has no torsion simplifies (13) into the form

$\displaystyle (23)$ $\displaystyle \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

$\displaystyle (24)$ $\displaystyle \quad \Sigma_{\rho,k}a_{\rho k}^{i}\Pi^\rho \wedge \omega^{k}=0,
$

where we put

$\displaystyle (25)$ $\displaystyle \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

$\displaystyle \omega^{1}$ $\displaystyle \wedge.$ . . $\displaystyle \wedge$ $\displaystyle \omega^{k-1}$ $\displaystyle \wedge$ $\displaystyle \omega^{k+1}\ldots$ $\displaystyle \wedge$ $\displaystyle \omega^{n},
$

getting

$\displaystyle \sum_{\rho}a_{\rho k}^{i}{\Pi^\rho}$ $\displaystyle \wedge$ $\displaystyle \omega^{1}$ $\displaystyle \wedge.$ . . $\displaystyle \wedge$ $\displaystyle \omega^{n}=0,
$

or

$\displaystyle \Sigma_{\rho}a_{\rho k}^{i} {\Pi^\rho} \equiv 0,\ \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{j}.$

Since the infinitesimal transformations $ X_{\rho}$ are linearly independent, this implies that

$\displaystyle \Pi^\rho\equiv 0,$ $\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{j}.
$

It follows that $ \Pi^\rho$ is of the form

$\displaystyle \Pi^\rho=\Sigma_{j} \phi_{j}^{\rho} \wedge \omega^{j}
$

where $ \phi_{j}^{\rho}$ are Pfaffian forms. Substituting these expressions into (24), we get

$\displaystyle \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

$\displaystyle \Sigma_{\rho}(a_{\rho k}^{i}\phi_{j}^{\rho}-a_{\rho j}^{i}\phi_{k}^{\rho})\equiv 0,$ $\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{k}.
$

Since $ G$ has the property $ (C)$ , the above equations imply that

$\displaystyle \phi_{j}^{\rho}\equiv 0,$ $\displaystyle \mathrm{m}\mathrm{o}\mathrm{d}\ \omega^{k}.
$



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