Investigation: b_1≠0

We reduce the scalar product: By applying a Lie-algebra automorphism, we yield the implication b_1≠0⇒b_4=0.

subst = {b_4→0} ;

Τα[B/.subst, 0, EXP[λ M[[2]]]/.λ→λ]//MF

( {{b_1, b_2, b_3, λ b_1}, {b_2, b_5, b_6, λ b_2 + b_7}, {b_3, b_6, b_8, λ b_3 + b_9}, {λ b_1, λ b_2 + b_7, λ b_3 + b_9, λ^2 b_1 + b_10}} )

We compute the geometric tensors:

eqs = ShowGeo[{b_4→0}] ;

We consider the scalar product ℬ = ( {{b_1, b_2, b_3, 0}, {b_2, b_5, b_6, b_7}, {b_3, b_6, b_8, b_9}, {0, b_7, b_9, b_10}} )

with determinant<br /> |ℬ| = b_3^2 (b_7^2 - b_5 b_10) + 2 b_2 b_3 (-b_7 b_9 + b_6 b_10) + b_2^2 (b_9^2 - b_8 b_10) - b_1 (b_7^2 b_8 - 2 b_6 b_7 b_9 + b_5 b_9^2 + (b_6^2 - b_5 b_8) b_10)

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

Reduce[eqs]

b_6 == 0&&b_5 == 0&&b_8≠0&&b_2 == 0&&b_1 == b_3^2/(2 b_8) &&b_3 b_7≠0

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