Covariant derivatives on a sphere

Question

I have a question to taking covariant derivatives on a sphere. On http://mathworld.wolfram.com/SphericalCoordinates.html the following formula is given:

$A_{j;k}=\frac{1}{g_{kk}}\frac{\partial A_j}{\partial x_k}-\Gamma^i{_{jk}}A_i$.

My question is, why does the factor $1/g_{kk}$ appear here? It doesn't appear in the usual definition of the covariant derivative...

Answer 1

The expression $A_{j;k} = A_{j,k} - \Gamma^i_{jk}A_i$ is valid for tensors in any manifold, with any coordinate system. There is nothing particular about the sphere, and I think that $1/g_{kk}$ is probably a mistake or a typo.

Also note that the presence of that $g_{kk}$ breaks the index balance of Einstein's summation convention (which is in force, in view of the $\Gamma^i_{ij}A_i$ term).