It can only be stressed that the curvature tensor
of a general affine connection
allows for two contractions,
corresponding to an (asymmetric) Ricci tensor
and to what is called “homothetic curvature”
.
Schrödinger had included it in a theory; later Tonnelat (cf. Section 10.2) and Sciama [565*] would also use homothetic
curvature.