In other words, it is the identity transformation. That is, it is a symmetric tensor . Specific examples of these involve the Kronecker delta, and are in the homework. The extra term on the right violates the transformation rules. We will see more interesting examples later.
Y j determine a scalar field.
If X i and Y j are any contravariant fields on M, then their dot product X . Y is a scalar, and X . Exercise Set 4 1. Compute the transformation rules for each of the following, and hence decide whether or not they are tensors.