I will introduce some calculus for working with tensors in complex spaces. The syntax for such spaces is the same as standard tensor notation but with one important and major distinction. We will use the over-bar to indicate elements that are in the complex conjugate space.
To begin, we start with a metric as we do in real spaces:
Since we are working in a complex space, we will also make Cauchy-Riemann requirements of any tensor in the space. Namely, that there are no mixing between regular complex indices and their complex conjugates. Explicitly:
Conjugation is itself a “coordinate transform” of sorts, but in keeping things distinct must be treated separately from regular coordinate transformations. In that way, we can write rules for coordinate changes in our tensor space.
The pattern here should be familiar and obvious. The metric also has the same properties that the metric of real spaces. It can raise and lower indices. It should also be obvious that if a vector is constant in one coordinate system it will NOT be constant in another, prompting us to define a more general form for the derivative. This gives rise to a covariant and contravarient derivative via a set of connections that, as in the real space, are derived from the metric.
Applying this to our equation for the square length gives the following for the metric:
This has an easy solution
If we assume the metric is either Hermitian or anti-hermitian, then equation 7 can be derived from equation 6. We can also add equation 6 and 7 to give us an equation for the metric that only contains a derivative of the real variable x:
If we drop all imaginary components of our connection and metric this equation reduces to what we had in the real space.
It should also be noted here that our affine connection is NOT torsion free.