  We analyse the relation between two a priori quite different expansions of
the string equations of motion and constraints in a general curved background,
namely one based on the covariant Penrose-Fermi expansion of the metric
G_{\mu\nu} around a Penrose limit plane wave associated to a null geodesic
\gamma, and the other on the Riemann coordinate expansion in the exact metric
G_{\mu\nu} of the string embedding variables around the null geodesic \gamma.
Starting with the observation that there is a formal analogy between the exact
string equations in a plane wave and the first order string equations in a
general background, we show that this analogy becomes exact provided that one
chooses the background string configuration to be the null geodesic \gamma
itself. We then explore the higher-order correspondence between these two
expansions and find that for a general curved background they agree to all
orders provided that one works in Fermi coordinates and in the lightcone gauge.
Requiring moreover the conformal gauge restricts one to the usual class of
(Brinkmann) backgrounds admitting simultaneously the lightcone and the
conformal gauge, without further restrictions.
