Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Proof.
We only prove (3.154 Ω i j α β = − Ω i j β α \Omega_{ij\alpha\beta}=-\Omega_{ij\beta\alpha}
Ω i j α β ( 1 2 r d y β α ^ − 1 2 r 3 y μ y β d y μ α ^ ) \displaystyle\Omega_{ij\alpha\beta}(\frac{1}{2r}dy_{\widehat{\beta\alpha}}-\frac{1}{2r^{3}}y_{\mu}y_{\beta}dy_{\widehat{\mu\alpha}})
= \displaystyle= Ω i j α , α + 1 ( − 1 r d y α + 2 − 1 2 r 3 ( y α + 2 y α + 1 d y y α + 1 − y α + 1 2 d y α + 2 ) + 1 2 r 3 ( y α 2 d y α + 2 − y α y α + 2 d y α ) ) \displaystyle\Omega_{ij\alpha,\alpha+1}\Big(-\frac{1}{r}dy_{\alpha+2}-\frac{1}{2r^{3}}(y_{\alpha+2}y_{\alpha+1}dy_{y_{\alpha+1}}-y_{\alpha+1}^{2}dy_{\alpha+2})+\frac{1}{2r^{3}}(y_{\alpha}^{2}dy_{\alpha+2}-y_{\alpha}y_{\alpha+2}dy_{\alpha})\Big)
= \displaystyle= Ω i j α , α + 1 ( − 1 2 r d y α + 2 − 1 2 r 2 y α + 2 d r ) \displaystyle\Omega_{ij\alpha,\alpha+1}\Big(-\frac{1}{2r}dy_{\alpha+2}-\frac{1}{2r^{2}}y_{\alpha+2}dr\Big)
(3.156) = \displaystyle= − 1 2 Ω i j α , β ( 1 2 r d y α β ^ + 1 2 r 2 y α β ^ d r ) . \displaystyle-\frac{1}{2}\Omega_{ij\alpha,\beta}\Big(\frac{1}{2r}dy_{\widehat{\alpha\beta}}+\frac{1}{2r^{2}}y_{\widehat{\alpha\beta}}dr\Big).
∎