This sign is wrong and one index (probably a typo). In Pavel Grinfeld's Introduction to Tensor Analysis and the Calculus of Moving Surfaces, the covariant derivative of a covariant tensor is:
$$\nabla_k A_i=\frac{∂A_i}{∂x^k} - \Gamma^l_{ik}A_l$$
The covariant derivative of a contravariant...
Coincidentally I have a similar doubt but probably it comes from a different perspective. The boundary conditions proposed for electromagnetic fields consider only a spatial normal term. But I want to know what happens with the jump in an arbitrary spacetime surface. Mathematically, we know what...