I liked this quote to give a feeling why the euclidian metric for the gradient might be wrong. https://hips.seas.harvard.edu/blog/2013/01/25/the-natural-gradie

I liked this quote to give a feeling why the euclidian metric for the gradient might be wrong. https://hips.seas.harvard.edu/blog/2013/01/25/the-natural-gradient/ https://hips.seas.harvard.edu/blog/2013/01/25/the-natural-gr... "As a concrete example, imagine two people standing on two different mountain tops. If one of the people is Superman (or anybody else who can fly) then the distance they would fly directly to the other person is the Euclidean distance (in R^3). If both people were normal and needed to walk on the surface of the Earth the Riemannian metric tensor tells us what this distance is from the Euclidean distance. Don’t take this illustration too seriously since technically all of this needs to take place in a differential patch (a small rectangle whose side lengths go to 0)."