Investigation: b_1≠0

We reduce the scalar product: By applying a Lie-algebra automorphism, we yield the implication b_1≠0⇒b_3=0∧b_4=0.

subst = {b_3→0, b_4→0} ;

Τα[B/.subst, 0, EXP[λ M[[1]]]/.λ→λ]//MF

( {{b_1, b_2, λ b_1, 0}, {b_2, b_5, λ b_2 + b_6, b_7}, {λ b_1, λ b_2 + b_6, λ^2 b_1 + b_8, b_9}, {0, b_7, b_9, b_10}} )

We compute the geometric tensors:

eqs = ShowGeo[{b_3→0, b_4→0}] ;

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

with determinant<br /> |ℬ| = 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]

False

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