Does any body know how to expand HijHij and write with the summation signs??
Okay, I'm going to change my answer one last time. (If you feel I need negative rep for changing my answer so many times, go ahead.) This time I took some time to look things up so I can be accurate.
I've been thinking about this and I think what has been confusing me so much is that there really isn't a way to apply the summation convention to in general.
When we apply the summation convention we are using the metric tensor to essentially perform an inner product. (Technically an inner product is either positive or negative definite, which most metrics aren't.) The rule for multiplying two vectors together, for example is
Note the position of the indices: The upper index (if I remember right) is a contravariant index and the lower is a covariant index. The only time the summation convention is used is when taking a product of co- and contravariant indices.
So technically we can't do a summation for . However there is one kind of space that we can get away with this: in Euclidean space there is no distinction between covariant and contravariant indices. So in an Euclidean space only we can write