the formula $$ trace\nabla du = \sum_i \nabla du (e_i,e_i) = (g^{ij}\nabla_i\nabla_ju^k) \dfrac{\partial}{\partial x^k} $$
what do $\nabla_i,\nabla_j$ mean? where can I find the detailed description of this symbol? Is there anything like $\nabla^i,\nabla^j$?