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_6→0} ;

Τα[B/.subst, 0, EXP[λ M[[3]]]/.λ→λ]//MF

We compute the geometric tensors:

eqs = ShowGeo[{b_4→0, b_6→0}] ;

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

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

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

Reduce[Append[eqs, b_7 == 0]]

False

Reduce[Append[eqs, b_7≠0]]

False

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