We can train SOM network with different distance metrics: euclidian, taxicab, angle between vectors, mahalanobis. But the formula for weights update is: w(t+1) = w(t) + A*(x-w(t)) Will it work with non-euclidian like distance metrics (angle,mahalanobis)?

Why wouldn't it?

cause there is difference in euclidian and angle metrics, one compares distances between points other angles between them!
I puzzled that with SOM training formula we end up with weights vectors ajusted closer to input vectors that is equal to reducing euclidian distance but with angle metric we need training formula to reduce angle between weight and input vectors. It works anyway and the vectors far apart but with closer angles becomes closer in euclidian distance, I wonder if it needs closer angles training formula?