Picture below is from the 113 page of Jost's RIEMANNIAN GEOMETRY AND GEOMETRIC ANALYSIS. In 3.3.49 , why there is not term with $\Gamma_{kl}^j\Gamma_{mn}^r$ ?
2026-04-07 09:06:13.1775552773
Calculation of inner product of star operator
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The last term in (3.3.49) has uncontracted indices $k$ and $l$, and one extra index $m$, so there is obviously a typo in it. The penultimate term has a similar problem.