1.1(2).2

In [Ko01] p.46, the homogeneous pair (g,h) with index 1.1(2).2 is defined as

ShowAd[ad = Table2ad[bas = {e_1, u_1, u_2, u_3, u_4}, ( {{0, u_3, 0, -u_1, 0}, {-u_3, 0, 0, 0, u_1}, {0, 0, 0, 0, p u_2}, {u_1, 0, 0, 0, u_3}, {0, -u_1, -p u_2, -u_3, 0}} )], 4]

e_1 u_1 u_2 u_3 u_4
e_1 0 u_3 0 -u_1 0
u_1 -u_3 0 0 0 u_1
u_2 0 0 0 0 p u_2
u_3 u_1 0 0 0 u_3
u_4 0 -u_1 -p u_2 -u_3 0

for p∈R. Any ρ-invariant scalarproduct B is of the form

Clear[B] ;

B = LSolve[ΗadInv[ad, B = Τℊ★[g, 4], 0], B] ;

MF[B = B/.Vars[B] → {b, c, a, d, e, f}]

( {{a, 0, 0, 0}, {0, b, 0, c}, {0, 0, a, 0}, {0, c, 0, d}} )

The tensor ν:g×g→m is determined by

Showν[ΗΡν[ad, B], bas]

e_1 u_1 u_2 u_3 u_4
e_1 0 u_3/2 0 -u_1/2 0
u_1 u_3/2 (a c u_2)/(-c^2 + b d) + (a b u_4)/(c^2 - b d) 0 0 u_1/2
u_2 0 0 (b c p u_2)/(-c^2 + b d) + (b^2 p u_4)/(c^2 - b d) 0 ((c^2 + b d) p u_2)/(-2 c^2 + 2 b d) + (b c p u_4)/(c^2 - b d)
u_3 -u_1/2 0 0 (a c u_2)/(-c^2 + b d) + (a b u_4)/(c^2 - b d) u_3/2
u_4 0 u_1/2 ((c^2 + b d) p u_2)/(-2 c^2 + 2 b d) + (b c p u_4)/(c^2 - b d) u_3/2 (c d p u_2)/(-c^2 + b d) + (c^2 p u_4)/(c^2 - b d)

The Levi-Civita connection Λ:g→gl(m) is determined by

ShowΛ[ΗΡΛ[ad, B]]

Λ_ (j, 1)^i ( {{0, 0, -1, 0}, {0, 0, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, 0}} )
Λ_ (j, 2)^i ( {{0, 0, 0, 1}, {(a c)/(-c^2 + b d), 0, 0, 0}, {0, 0, 0, 0}, {(a b)/(c^2 - b d), 0, 0, 0}} )
Λ_ (j, 3)^i ( {{0, 0, 0, 0}, {0, (b c p)/(-c^2 + b d), 0, (b d p)/(-c^2 + b d)}, {0, 0, 0, 0}, {0, (b^2 p)/(c^2 - b d), 0, (b c p)/(c^2 - b d)}} )
Λ_ (j, 4)^i ( {{0, 0, 0, 0}, {0, 0, (a c)/(-c^2 + b d), 0}, {0, 0, 0, 1}, {0, 0, (a b)/(c^2 - b d), 0}} )
Λ_ (j, 5)^i ( {{0, 0, 0, 0}, {0, (c^2 p)/(-c^2 + b d), 0, (c d p)/(-c^2 + b d)}, {0, 0, 0, 0}, {0, (b c p)/(c^2 - b d), 0, (c^2 p)/(c^2 - b d)}} )

The non-zero evaluations of the Riemannian-curvature tensor R:m×m→gl(m) are determined by

Showℛ[ℛ = ΗΡℛ[ad, B]]

ℛ_ (1, 2, j)^i ( {{0, (b^2 p)/(c^2 - b d), 0, (b c p)/(c^2 - b d)}, {(a b p)/(-c^2 + b d), 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}} )
ℛ_ (1, 3, j)^i ( {{0, 0, (a b)/(c^2 - b d), 0}, {0, 0, 0, 0}, {(a b)/(-c^2 + b d), 0, 0, 0}, {0, 0, 0, 0}} )
ℛ_ (1, 4, j)^i ( {{0, (b c p)/(c^2 - b d), 0, -1 + (c^2 p)/(c^2 - b d)}, {(a c (-1 + p))/(-c^2 + b d), 0, 0, 0}, {0, 0, 0, 0}, {(a b)/(-c^2 + b d), 0, 0, 0}} )
ℛ_ (2, 3, j)^i ( {{0, 0, 0, 0}, {0, 0, (a b p)/(c^2 - b d), 0}, {0, (b^2 p)/(-c^2 + b d), 0, (b c p)/(-c^2 + b d)}, {0, 0, 0, 0}} )
ℛ_ (2, 4, j)^i ( {{0, 0, 0, 0}, {0, (b c p^2)/(c^2 - b d), 0, (b d p^2)/(c^2 - b d)}, {0, 0, 0, 0}, {0, (b^2 p^2)/(-c^2 + b d), 0, (b c p^2)/(-c^2 + b d)}} )
ℛ_ (3, 4, j)^i ( {{0, 0, 0, 0}, {0, 0, (a c (-1 + p))/(-c^2 + b d), 0}, {0, (b c p)/(c^2 - b d), 0, -1 + (c^2 p)/(c^2 - b d)}, {0, 0, (a b)/(-c^2 + b d), 0}} )

The Ricci-curvature Ric:m×m→R is determined by

MF[Ric = ΜRic[ℛ]]

We demand Ric=0. Thus, b=0 and p=1.

sol = Union[Solve[Ric≡0]]

Solve :: svars : Equations may not give solutions for all \"solve\" variables. Mehr…

{{b→0, p→1}}

But then, the Riemannian curvature tensor vanishes also.

Showℛ[ℛ/.sol[[1]]]

ℛ-Flat

Note, this result is independent of the signature of B.

Created by Mathematica  (August 15, 2006) Valid XHTML 1.1!