Hi, folks! A while ago (circa 2002) while I was trying to understand
the difference(s) between covariant and contravariant tensors, I made
myself a little table concerning the tensor's components; I recently
came across it, and now I don't understand one of my own notations!
So I'm hoping one of you might be able to tell me what I was probably
thinking; here's the table:

Metric "I" not "I"
Basis

Orthogonal co = contra possible?
co = contra?

Non-orthogonal possible: yes co ~= contra
co = contra?

The notation I don't remember is the capital "I" (the ?'s refer to
questions I had about the tensors and their components, not the
notation I don't understand). Given the context, what did "I"
represent? Isometric? Thanks for your help.

DG

PS: While you're at it, I never did resolve the ?'s, so if you can
provide those answers as well, that'd be nice. Thanks again!

You were considering the metric and whether it took the form of the
Identity operator.
If you choose an ortho-normal basis then the metric components a

1 0 0 ...
I = 0 1 0 ... = diag(1,1,1,...)
0 0 1 ....
...

In said basis, then the covariant and contravariant components of a
vector will
be the same. However you've indicated "I" vs "not I" as independent of
orthogonal vs
non-orthogonal bases. They are related. An orthogonal basis gives
diagionalized
metric, and if you can normalize all vectors to square magnitude of +1
(in a definite metric space) then you have the I case.

In short you can't have a "non-orthogonal case with metric I". You
can have an othogonal (non-ortho-normal) basis and thus have a
non-identity form.
In that case with contravariant metric:
g = diag(g1,g2,g3,...)
then co != contra, but rather contra-variant components are related to
covariant components by:
vectors covariant components X = (x1,x2,....)
vectors contra-variant components X = (g1 x1,g2 x2, ...)

Regards,
James

Regards,
James

Are you sure? It fits awfully well.

No, I'm not sure, that may have been it, I just _think_ I was thinking
of something a little more general and/or geometrical than that. (Yes,
I know, whether or not the metric "takes the form of the identity
operator" has geometric implications - pretty much anything you can say
about a metric tensor is going to have some sort of geometric
implication(s). ). More to the point, I _think_ those column
headings, like the row headings, were meant to describe something about
the basis in which the tensor's components were calculated, not
something about the tensor itself (the latter was the content of the
table itself). In other words, I was trying to get a handle on, given
certain properties of a basis and an arbitrary tensor, what was the
resulting difference between the components of the tensor in the co- v.
contra- representation. Thanks for your patience; if you don't respond
again, I'll understand (I'm a little surprised anyone responded at all
- thanks!)

DG