1. Riemann, G. F. B., (1953). Über die Hypothesen, welche der Geometrie zu Grunde liegen, Abhand. K. Ges. Wiss. Göttingen, 13, 133, 1868; English translation by Clifford, W. K. Nature 8, 14, 1873; reprinted and edited by Weyl, H., Springer, Berlin, 1920. Included in its Gesammelte Mathematische Werke, wissenschaftlicher Nachla_ und Nachträge, eds. Weber, H., Dedekind, R., Teubner, B. G., Leipzig, 1892; 2d ed. Dover Publ., New York.
2. Eisenhart, L. P., (1960). A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, Inc. New York, p 93.
3. In spite of its interest, the known results on diagonalization of three-dimensional metrics do not belong to this type: in all of them, in addition to the f D 3 scalars, an orthogonal tetrad is also involved with the (more o less implicit) flat metric. See Cartan, E., Les syst`emes differentials ext´erieurs et leurs applications g´eom´etriques, Hermann, Paris, 1945 and 1971 for an analytic proof on the existence of orthogonal coordinates; Deturck, M., Duke Math. Jour., 51, p 243–60, (1984) for a C1 proof;Walberer, P., Abhandl. Math. Sem. Univ. Hamburg, 10, p 169–79, 1934 for the decomposition in a given orthogonal frame; Bel, L., Gen. Rel. Grav., 28, p 1139–50, (1996) for decompositions in principal frames; Tanno, S., J. Differential Geometry, 11, p 467–74, (1976) for other particular decompositions.
4. Coll, B., (1999). A Universal Law of Gravitational Deformation for General Relativity, in Proceedings of the Spanish Relativity Meeting in honour of the 65th Birthday of Lluis Bel “Gravitation and Relativity in General” ed. by Martin, J., Ruiz, E., Atrio, F., Molina, A., World Scientific.
5. See for example O'Neill, B., Elementary Differential Geometry, Academic Press, 1966.