Investigation: b_1≠0,b_5≠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, b_7→0} ;

Τα[B/.subst, 0, EXP[λ M[[4]]]/.λ→λ]//MF

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

We compute the geometric tensors:

eqs = ShowGeo[{b_4→0, b_7→0}] ;

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

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

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

Reduce[eqs]

False

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