A Restatement of the Algebraic Classification of Area Metrics on 4-Manifolds

Abstract

An area metric is a`0 4´-tensor with certain symmetries on a 4-manifold that represents a non-dissipative linear electromagnetic medium. A recent result by Schuller, Witte and Wohlfarth gives a pointwise algebraic classification for such area metrics. This result is similar to the Jordan normal form theorem for`1 1´-tensors, and the result shows that pointwise area metrics divide into 23 metaclasses and each metaclass requires two coordinate representations. For the first 7 metaclasses, we show that only one coordinate representation is needed. For the remaining 16 metaclasses we find an additional third coordinate representation.

Determination of an electromagnetic medium from the Fresnel surface

“…Equation (46) states that components G i jkl 0 define a twisted 4 0 -tensor density G 0 on N of weight 1. The Tamm-Rubilar tensor density [1,2] is the symmetric part of G 0 and we denote this twisted tensor density by G .…”

“…A main result of [13] is that such a 6 × 6 transformation matrix (which a priori has 36 degrees of freedom) can for skewon-free constitutive tensors essentially be realized using a coordinate transformation on N (which has only 16 degrees of freedom). See equation ( 14) and for a further discussion see [13,46]. Let us also note that lemma 4.6 and [15, theorem 2.1] are pointwise results, but theorem 4.3 is a global result on a possibly non-orientable manifold.…”