We study the so-called inverse problem. Namely, given a prescribed skew-symmetric Ricci tensor we find (locally) a respective linear connection.
@article{bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_37,
author = {Jan Kurek and W\l odzimierz Mikulski},
title = {On the existence of connections with a prescribed skew-symmetric Ricci tensor},
journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
volume = {72},
year = {2018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_37}
}
Jan Kurek; Włodzimierz Mikulski. On the existence of connections with a prescribed skew-symmetric Ricci tensor. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 72 (2018) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_37/
Dusek, Z., Kowalski, O., How many are Ricci flat affine connections with arbitrary torsion?, Publ. Math. Debrecen 88 (3-4) (2016), 511-516.
Gasqui, J., Connexions a courbure de Ricci donnee, Math. Z. 168 (2) (1979), 167-179.
Gasqui, J., Sur la courbure de Ricci d’une connexion lineaire, C. R. Acad. Sci. Paris Ser A–B 281 (11) (1975), 389-391.
Kobayashi, S., Nomizu, K., Foundation of Differential Geometry, Vol. I, J. Wiley-Interscience, New York, 1963.
Opozda, B., Mikulski, W. M., The Cauchy-Kowalevski theorem applied for counting connections with a prescribed Ricci tensor, Turkish J. Math. 42 (2) (2018), 528-536.