Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 $B$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.