Investigation: b_1=0

We compute the geometric tensors:

eqs = ShowGeo[{b_1→0}] ;

We consider the scalar product ℬ = ( {{0, b_2, b_3, b_4}, {b_2, b_5, b_6, b_7}, {b_3, b_6, b_8, b_9}, {b_4, b_7, b_9, b_10}} )

with determinant<br /> |ℬ| = b_4^2 (b_6^2 - b_5 b_8) + (b_3 b_7 - b_2 b_9)^2 - 2 b_4 (b_3 (b_6 b_7 - b_5 b_9) + b_2 (-b_7 b_8 + b_6 b_9)) - (b_3^2 b_5 - 2 b_2 b_3 b_6 + b_2^2 b_8) b_10

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

Reduce[eqs]

(b_3 == -b_5 || b_3 == b_5) &&b_2 == 0&&b_4^2 b_6^2 - 2 b_3 b_4 b_6 b_7 + b_5^2 b_7^2 - b_4^2 b_5 b_8 + 2 b_3 b_4 b_5 b_9 - b_5^3 b_10≠0

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