Harmonic-curvature warped products over surfaces

Abstract: For warped products with harmonic curvature, nonconstant warping functions

p h i

, and compact two-dimensional bases

(M, h)

, we establish a dichotomy: either the Gaussian curvature

K

of the metric

g = p h i^{- 2} h

is constant and negative, or

p h i

equals a specific elementary function of

K

, also depending on the dimension

p

and Einstein constant

v a r e p s i l o n

of the fibre. In both cases the fibre must be an Einstein manifold with

p > 1

and

v a r e p s i l o n > 0

, while the function

f = p h i^{p / 2}

satisfies a Yamabe-type second-order differential equation on

(M, g)

. We prove that both possibilities are realized on every closed orientable surface of genus greater than

1

, and in the latter case -- which also occurs on the

2

-sphere and real projective plane -- the metrics in question constitute uncountably many distinct homothety types.

This page was built for publication: Harmonic-curvature warped products over surfaces