# evolution of curvature under ricci flow , What does the tensor A*B means?

in the book of bennet chow,the definition of the "*" notation is simple ,I can't understand . can someone gives me some examples of this definition? or tell me in which book I can find this?thanks.

-
You'll need to give more context. Looking at Google Books, I see stars used in Section 1.2 for the usual pushforward/pullback. Is that what you mean? – Hans Lundmark Nov 11 '11 at 11:28
Like @Hans said, there are at least three different uses of the $*$ in Chow's book. First is the usual notation for pushforwards and pullbacks of tensor fields under smooth maps. Second is as the duality operator in $T_pM$ or $T^*_pM$: given a vector $V\in T_pM$ we write $V^* := g(V,\cdot) \in T^*_pM$ for its metric dual. And third is as the Hodge star operator on the exterior algebra. Which page are you looking at? – Willie Wong Nov 11 '11 at 11:52
thanks,please see the book "the hamilton's ricci flow" of bennet chow and penglu, section 2.7, namely, 口Rm=RmRm+RcRm,this is the evolution of rimemann curvature tensor under ricci flow ,so can you tell me for two tensor A and B,what does AB represent like RcRm?Thanks! – deng ya Nov 11 '11 at 12:30
@ Willie wong :maybe $\star$ orerator means some linear combination of contractions of tensorS $A\otimes B$ ,it's not a preicise definition, it's just a representation.according to every situation,the representation of $A\star B$ will be not the same. Am I all right? – deng ya Nov 12 '11 at 0:50
@deng ya, you are right, this is an "ad hoc" notation in this book, firstly introduced on p.19 there: "Here, given tensors $A$ and $B$, $A*B$ denotes some linear combination of contractions of $A \otimes B$." – Yuri Vyatkin Nov 12 '11 at 22:03

The notation $A*B$ is used for tensors $A$, $B$ to denote a linear combination of contractions of the tensors $A$ and $B$.