You should raise the indices of the contravariant components a_j, b_j, otherwise this does not make sense in terms of Einstein summation.
In Euclidean space they’re equal since the metric is kronecker
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It's not only about non-euclidean metric; lower/upper indices are about co/contra-variance.
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Okay fair but they transform identically when the metric is identity
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Also here the equation makes zero sense if there’s no summation implied, though I should clarify
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