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Recently I am reading the book The Ricci Flow: techniques and applications: Part I: geometric aspects by Bennett Chow et al.. Here I have encountered a result (Proposition 1.13) due to Hamilton which states that "Any expanding or steady Ricci soliton $(g,X)$ on a closed manifold $M^n$ is Einstein. Any shrinking Ricci solution on a closed $n$-manifold has positive scalar curvature." In the proof, the general Ricci soliton equation is given by $$\Delta R + 2|\mathrm{Rc}|^2=\mathcal{L}_X(R)-\epsilon R,$$ which can be written as $$\Delta R + 2|\mathrm{Rc}|^2-\langle\nabla R,X\rangle+\epsilon R=0.$$ The proof says that the last equation can be rewritten as $$\Delta\left(R+\frac {n\epsilon} {2}\right)+2\,\left|\mathrm{Rc}+\frac {\epsilon} {2}g\right|^2-\left\langle\nabla\left(R+\frac {n\epsilon} {2}\right),X\right\rangle-\epsilon\left(R+\frac {n\epsilon} {2}\right)=0.$$ I can not understand the last equation. Please some one help me to understand this.

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  • $\begingroup$ Basically I am concerned with the last two terms in the said last equation $\endgroup$ – debabrata chakraborty Aug 2 '15 at 20:35
  • $\begingroup$ yes...I mean that...and also \pounds reads as lie derivative $\endgroup$ – debabrata chakraborty Aug 2 '15 at 20:54
  • $\begingroup$ Well my edit was rejected I think, not sure why maybe lie derivative use square brackets and not angled ones. $\endgroup$ – marshal craft Aug 2 '15 at 21:26
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It is a direct simple computation We use the followings :

$$\Delta C=\nabla C=0$$ for constant $C$

$$|Rc +Cg|^2=|Rc|^2+ 2C(Rc,g)+C^2 |g|^2=|Rc|^2+2CR +nC^2$$

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