What's my mistake in the calculation?

Question

Summation convention holds. If $\frac{\partial}{\partial t}g_{ij}=\frac{2}{n}rg_{ij}-2R_{ij}$, then ,I compute: $$ \frac{1}{2}g^{ij}\frac{\partial}{\partial t}g_{ij}=\frac{1}{2}g^{ij}(\frac{2}{n}rg_{ij}-2R_{ij})=\frac{1}{n}r(\sum\limits_i\sum\limits_jg^{ij}g_{ij})-g^{ij}R_{ij}=nr-R $$

But on the Hamilton's THREE-MANIFOLDS WITH POSITIVE RICCI CURVATURE,the result is : $$ \frac{1}{2}g^{ij}\frac{\partial}{\partial t}g_{ij}=r-R $$

I don't know where I make my mistake,and who can tell me ? Very thanks.

Answer 1

Note:

$$\sum_{i,j=1}^n g^{ij} g_{ij} = \text{tr} (g^{-1}g) = \text{tr} (I_n) = n.$$

Answer 2

$$ \sum_j g^{ij} g_{kj} = \delta^i_k $$ Now, $\delta^i_i = 1$, with no summation, so $$ \sum_i \sum_j g^{ij} g_{ij} = \sum_i \delta_i^i = \sum_i 1 = n. $$ You may be confusing whether or not you are using summation convention.