I was learning about tensors and vectors and it was mentioned in class that they were "invariant" objects that take on specific representations in different bases. I get this for vectors -- I can compute the magnitude of a vector and it shouldn't change under basis change. And for a 2 index tensor, the trace or determinant are invariant.
But what about tensors with more indices -- is there something that remains invariant under isometrics?
And are the invariants for mixed tensors (e.g. rank(1,1) -- linear transformation) the same as totally co/contravariant tensors (e.g. rank(0,2) -- metric tensor) of same form (e.g. determinants and traces)?
thanks!