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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.

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    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?2011-11-11
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    Like @Hans said, there are at least three different uses of the $*$ in Chow's book. First is the usual notation for [pushforwards](http://en.wikipedia.org/wiki/Pushforward_%28differential%29) and [pullbacks](http://en.wikipedia.org/wiki/Pullback_%28differential_geometry%29) 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](http://en.wikipedia.org/wiki/Hodge_star) on the exterior algebra. Which page are you looking at?2011-11-11
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    thanks,please see the book "the hamilton's ricci flow" of bennet chow and penglu, section 2.7, namely, 口Rm=Rm*Rm+Rc*Rm,this is the evolution of rimemann curvature tensor under ricci flow ,so can you tell me for two tensor A and B,what does A*B represent like Rc*Rm?Thanks!2011-11-11
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    $\frac{\partial Rm}{\partial t}=\Delta Rm+ Rm \star Rm + Rc \star Rm $2011-11-11
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    so what does $ Rm \star Rm \ and \ Rc \star Rm \ $ mean?2011-11-11
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    @ 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?2011-11-12
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    @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 $A \otimes B$."2011-11-12
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    @Yuri Vyatkin,thans very much!2011-11-17

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