So I've taken two differential topology/geometry classes both from a mathematics department. I see all over this forum a whole lot of talk about indices being up or down and raising/lowering etc. My professors barely ever mentioned these things though I did notice that when they worked in local coordinates they always wrote the indices on certain objects up and other objects down. For example, inner products of vectors always seem to have repeated indices that are up on one object and down on the other like: inner product of x and y = sum_i (x_i*y^i) I've never really understood what we gain from this notation. At least for my example of an inner product no information seems to be gained by writing the indices this way. When talking about more complicated things than inner products I'm at a loss as to how I should arrange the indices and what benefit there is from doing things this way. I initially thought that the notation was up on objects that transform covariantly and down contravariantly (or vice versa) but I'm at a loss as to what this means for objects with mixed indices. It's also not at all clear to me what it means to "raise" or "lower" the indices on some object. If anyone is willing, I'd like clarification both on how the notation actually works as well as on the "philosophy" behind why this notation was chosen.