Killing vector fields in three dimensions play important role in the
construction of the related spacetime geometry. In this work we show that when
a three dimensional geometry admits a Killing vector field then the Ricci
tensor of the geometry is determined in terms of the Killing vector field and
its scalars. In this way we can generate all products and covariant derivatives
at any order of the ricci tensor. Using this property we give ways of solving
the field equations of Topologically Massive Gravity (TMG) and New Massive
Gravity (NMG) introduced recently. In particular when the scalars of the
Killing vector field (timelike, spacelike and null cases) are constants then
all three dimensional symmetric tensors of the geometry, the ricci and einstein
tensors, their covariant derivatives at all orders, their products of all
orders are completely determined by the Killing vector field and the metric.
Hence the corresponding three dimensional metrics are strong candidates of
solving all higher derivative gravitational field equations in three
dimensions.Comment: 25 pages, some changes made and some references added, to be
published in Classical and Quantum Gravit
