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[parent] ocr 2 proofreading test (Definition)

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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Cross-references: formulas, tensor, Lie algebra, type

This is version 1 of ocr 2 proofreading test, born on 2009-02-14.
Object id is 523, canonical name is Ocr2ProofreadingTest.
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