1.1(2).8

In [Ko01] p.48, the homogeneous pair (g,h) with index 1.1(2).8 is defined as

0 | 0 | 0 | |||

0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 | ||

0 | 0 | 0 | 0 |

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 | |||

0 | 0 | 0 | 0 | ||

0 | 0 | 0 |

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

zero…4×4 | |

zero…4×4 | |

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.

But then, the Riemannian curvature tensor vanishes also.

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

Created by Mathematica (August 15, 2006) |