- NOTES AND COMMENTARY 53
Section 4. Einstein metrics are special cases of Ricci solitons. In gen-
eral, if a warped product metric g is Einstein, then the fiber (P, g) is Ein-
stein. If the base is 1-dimensional, then provided w (r) is not constant, the
metric
g=dr^2 +w(r)^2 g
satisfies Re (g) = Ag if and only if
w' (r)^2 + ~w (r)^2 = P,
n n-1
where Re (g) = pg and n = dim P. Without loss of generality (e.g., up to
scaling and diffeomorphism), we have the following cases:
I P = ~~{ \ ,\ = R~) \ w ( r) \ g
- (n - 1) -n coshr including hyperbolic with two ends
0 -n er including hyperbolic cusp
n-1 -n sinhr including hyperbolic space
n-1 0 r Ricci fiat cone
n-1 n smr including sphere
0 0 1 Ricci fiat productAs a consequence, we have the following (see Theorem 9.110 on p. 268 of
[27]).
THEOREM 1.94 (Einstein warped products over 1-dimensional base). If
a warped product (M, g) over a 1-dimensional base, with the dimension of
the fiber at least 2, is a complete Einstein manifold, then either
(1) g is a Ricci fiat product,
(2) M is topologically a cone on P and the fiber is Einstein with positive
scalar curvature, or
(3) the base is JR., the fiber is Einstein with nonpositive scalar curvature,
and g has negative scalar curvature.