I am trying to understand the following estimate from a book of John Roe. The setting is as follows. Suppose $\nabla$ is a connection on a vector bundle $S$ over a closed manifold $M$. We want to make a local estimate of the $L^2$-norm
$$\|\nabla s\|_0^2$$
of $\nabla s\in C^\infty(T^*M\otimes S)$. Here the notation $\|\cdot\|_k$ means the $k$-th Sobolev norm, defined locally as
$$\|f\|_k=\sum_{i=0}^k\|\frac{\partial^\alpha f}{\partial x^\alpha}\|_0,$$
for a section $f$ of $S$ supported over the local coordinate chart. The $0$-th Sobolev space is identified with $L^2(S)$.
The author writes:
Now I understand the equality, but not the next inequality. In particular, why does there exist a positive constant $C_2$ for which this is true? It seems to me that when $i\neq j$, the term $\displaystyle\int g^{ij}(\frac{\partial s}{\partial x^i},\frac{\partial s}{\partial x^j})$ could be negative and large. How can we estimate the size of this term?