1. GENERAL SOLITONS AND THEIR CANONICAL FORMS 5 (1) (f(t) : A4 —» M is the 1-parameter family of diffeomorphisms gen- erated by X (t) = ^hr grad^o/o that is, —tp (t) (x) = y (grad90/0) (v (*) (x)), (2) ^ (i) is the pull-back by p (t) of go up to the scale factor r (t), (1.11) 9(t) = T(t)p(tygo, (3) / (t) is the pull-back by if (t) of fa: (1-12) f(t) = fo°p(t) = p(t)*f0. Moreover, (1.13) Re (5 (*)) + V » W V ^ / (*) + ^ 5 (t) = 0, where V9^ denotes the covariant derivative with respect to g (t), and (1.14) fHgrad 9 ( t ) /«)£,„• EXERCISE 1.8. Prove Proposition 1.7. HINT: Verify the equations in the order in which they are presented. Because of the proposition above, we shall at some times consider gra- dient Ricci soliton structures and at other times consider gradient Ricci solitons in canonical form. Now we give a couple of examples of Ricci solitons in canonical form. We are already acquainted with solutions which evolve purely by homothety, e.g., these solutions correspond to Einstein metrics. If go is an Einstein metric with Einstein constant A (i.e., Rc(#o) ^#o) then g(t) = (1 - 2\t)g0 satisfies the Ricci flow - g = -2Rc(g), since Rc(#) = Rc(#o) = ^9o- An Einstein manifold with A 0 is necessarily compact (thanks to Myers's theorem) and, as the metric approaches zero, the manifold shrinks to a point in finite time. Einstein metrics are stationary points for the normalized Ricci flow on a closed manifold. Following up on our previous discussion of static Euclidean space, we have the following. EXAMPLE 1.9 (Gaussian soliton). Regard Euclidean space as a Ricci soliton in canonical form, so that (1.4) holds with j(t) = 1 + et for some £ G R . For s ^ O , the Euclidean solution is called the Gaussian soliton. By differentiating (1.4) and multiplying the result by r(t), we obtain (1.15) 0 = - 2 Rc(p c a r i ) = £ V /#ca n + ^Scan, where (1.16) /(*) = _£]£!!
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