  Various curvature conditions are studied on metrics admitting a symmetry
group. We begin by examining a method of diagonalizing cohomogeneity-one
Einstein manifolds and determine when this method can and cannot be used.
Examples, including the well-known Stenzel metrics, are discussed. Next, we
present a simplification of the Einstein condition on a compact four manifold
with $T^{2}$-isometry to a system of second-order elliptic equations in
two-variables with well-defined boundary conditions. We then study the Einstein
and extremal Kahler conditions on Kahler toric manifolds. After constructing
explicitly new extremal Kahler and constant scalar curvature metrics, we
demonstrate how these metrics can be obtained by continuously deforming the
Fubini-Study metric on complex projective space in dimension three. We also
define a generalization of Kahler toric manifolds, which we call fiberwise
Kahler toric manifolds, and construct new explicit extremal Kahler and constant
scalar curvature metrics on both compact and non-compact manifolds in all even
dimensions. We also calculate the Futaki invariant on manifolds of this type.
After describing an Hermitian non-Kahler analogue to fiberwise Kahler toric
geometry, we construct constant scalar curvature Hermitian metrics with
$J$-invariant Riemannian tensor. In dimension four, we write down explicitly
new constant scalar curvature Hermitian metrics with $J$-invariant Ricci
tensor. Finally, we integrate the scalar curvature equation on a large class of
cohomogeneity-one metrics.
