Investigation: b_1=0

We compute the geometric tensors:

eqs = ShowGeo[{b_6→0, b_1→0}] ;

We consider the scalar product ℬ = ( {{0, b_2, b_3, b_4}, {b_2, b_5, 0, b_7}, {b_3, 0, b_8, b_9}, {b_4, b_7, b_9, b_10}} )

with determinant<br /> |ℬ| = -b_4^2 b_5 b_8 + 2 b_2 b_4 b_7 b_8 + 2 b_3 (b_4 b_5 - b_2 b_7) b_9 + b_3^2 (b_7^2 - b_5 b_10) + b_2^2 (b_9^2 - b_8 b_10)

The conditions det[B]≠0, and Ric=0 imply

Reduce[eqs]

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