Let denote the space of vector fields on . An (affine) connection on is a bilinear map such that For every function , The product rule holds. The expression is called the covariant derivative of with respect to .
lEt denote The Space OF veCtoR Fields on . an (affiNe) coNnectIon on iS A biLinEar map such tHat FOr eveRy funCtIon , The PrOduct RuLe Holds. tHe ExPrESsIon iS caLled thE covarIAnt DerIvative of wITh respECt to .