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Abstract:
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In the present work we classify left invariant metrics of arbitrary
signature on four-dimensional nilpotent Lie groups. Their geometry is extensively
studied with special emphasis on holonomy groups and decomposability of metrics.
Also, isometry groups are completely described and we give examples of metrics
where strict inequalities Isplit < Iaut < I hold. It is interesting that Walker metrics
appear as the underlying structure of neutral signature metrics on the nilpotent Lie
groups with degenerate center. We nd necessary and su cient condition for them
to locally admit nilpotent group of isometries.
Finally, we solve the problem of projectively equivalent metric on four-dimensional
nilpotent Lie groups by showing that left invariant metric is either geometrically rigid
or have projectively equivalent metrics that are also a nely equivalent. All
a nely equivalent metrics are left invariant, while their signature may change. |