hi, I have seen that christoffel symbol definition or logic is shown in different ways. For instance, in first attachment ( RED box) you can see a normal vector (n) next to the christoffel symbol, but in second image everything is same except that there is a normal vector. Is there a confusion??? Which is the right representation of christoffel symbol??? Why does these 2 different sources put a normal vector or not put a normal vector???? By the way first image is quoted from "wikipedia", other one is quoted from "Pavel Grinfeld Introduction to Tensor Analysis and the Calculus of Moving Surfaces".