1.4(1).2

In [Ko01] p.65, the homogeneous pair (g,h) with index 1.4(1).2 is defined as

0 | 0 | ||||

0 | 0 | 0 | 0 | ||

0 | 0 | 0 | |||

0 | 0 | 0 | |||

0 |

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

The tensor ν:g×g→m is determined by

0 | 0 | 0 | |||

0 | 0 | 0 | |||

0 | 0 | ||||

0 | |||||

0 |

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

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

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

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

But then, the Riemannian curvature tensor vanishes also.

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

Created by Mathematica (August 15, 2006) |