- WARPED PRODUCTS AND 2-DIMENSIONAL SOLITONS 13
where r is the standard coordinate on an interval I c ~' g is a given metric
on an n-dimensional manifold N, and w(r) > 0 is the warping function,
which scales distances along the N-factors in the product.
Several well-known metrics may be expressed as warped products, at
least on an open subset. When N is the unit circle S^1 , with g = d()^2 forthe standard coordinate(), then w(r) = r, w(r) = sinr, w(r) = sinhr give,
respectively, the Euclidean plane, the round sphere (with Gauss curvature+1), and the hyperbolic plane (with Gauss curvature -1) with the origin
omitted. When N = sn with the standard metric g = 9can of constant
curvature +1, then w(r) = r gives the standard (flat) metric on ~n+l with
the origin omitted. In each of these cases, the metric extends smoothly
across the origin. When N = sn, Lemma 2.10 on p. 29 in Volume One
(i.e., Lemma A.2 of this volume) gives sufficient conditions for smoothly
closing off the Riemannian manifold M, by a point at one (or both) ends
of the interval I. Namely, assuming we wish to close off as r "\, 0, we need
limr-+O w(r) = 0 and limr-+O w'(r) = 1 (the prime denotes the derivative
with respect to r).
For constructing Ricci solitons, it is convenient that g be a sufficiently
'nice' metric on N; for the rest of this section, we will assume that g is an
Einstein metric, with Re(§) = pg. An easy calculation in moving frames
gives the following (see Proposition 9.106 on p. 266 of Besse [27]).
LEMMA 1.21 (Ricci tensor and Hessian of warped product). If g is an
Einstein metric on a manifold Nn, with Einstein constant p, then the Ricci
tensor of the warped product metric (1.37) is given by^5
(1.38)
w"
Rc(g) = -n-dr^2 + (p - ww" - (n - l)(w')^2 ) g.
wFurthermore, if f is any function of the radial coordirpate r, then the Hessian
off with respect to g is given by
(1.39) \7\lf = J"(r)dr^2 +ww'f'g.
For example, if f(r) = J; w(t)dt, then \7\lf = w'(r)g. Conversely, by
an argument of Cheeger and Colding, warped products may essentially be
characterized by the existence of a function whose Hessian is some function
times the metric; see §1 in [71] for details.
3.3. Constructing the cigar and other 2-dimensional solitons.
For the sake of starting with something familiar, before we study the Bryant
soliton, we will briefly repeat the construction of the cigar metric, which is
discussed at greater length in Chapter 2 of Volume One.
Suppose we wish to construct a complete, steady, rotationally symmetric
gradient soliton metric on ~^2. Such a metric will be a warped product (1.37),
(^5) rn the case where (N, g) is the unit n-sphere, this follows from (l.58).