Abstract |
: |
The author studies for which left invariant diagonal metrics lambda on /b SO/(/b N/), the Euler-Arnold equations /b X/ dot =[/b X/, lambda (/b X/)], /b X/=(/b x//sub ij /) isin /b so/(/b N/), lambda (/b X/)/sub ij/= lambda /sub ij//b x//sub ij /, lambda /sub ij/= lambda /sub ji/ can be linearized on an abelian variety, i.e. are solvable by quadratures. He shows that, merely by requiring that the solutions of the differential equations be single-valued functions of complex time /b t/ isin /b C/, suffices to prove that (under a non-degeneracy assumption on the metric lambda ) the only such metrics are those which satisfy Manakov's conditions lambda /sub ij/=(/b b//sub i/-/b b//sub j/) (/b a //sub i/-/b a//sub j/)/sup -1/. The case of degenerate metrics is also analyzed. For /b N/=4, this provides a new and simpler proof of a result of Adler and van Moerbeke (1982). |