In a series of works, one of the authors has developed with J.-
M. Hwang a geometric theory of uniruled projective manifolds,
especially those of Picard number 1, basing on the study of varieties
of minimal rational tangents. A fundamental result in this
theory is a principle of analytic continuation under very mild assumptions,
called Cartan-Fubini extension, of biholomorphisms
between connected open subsets of two Fano manifolds of Picard
number 1 which preserve varieties of minimal rational tangents.
In this article we develop a generalization of Cartan-Fubini extension
for non-equidimensional holomorphic immersions from a
connected open subset of a Fano manifold of Picard number 1
into a uniruled projective manifold, under the assumptions that
the map sends varieties of minimal rational tangents onto linear
sections of varieties of minimal rational tangents and that it satisfies
a mild geometric condition formulated in terms of second
fundamental forms on varieties of minimal rational tangents. Formerly
such a result was known only in the very special case of
irreducible Hermitian symmetric manifolds of rank at least two,
and the proof relied on the existence of flattening coordinates, viz.,
Harish-Chandra coordinates, with respect to which the varieties
of minimal rational tangents form a constant family. The proof of
the main result, which is based on the deformation theory of rational
curves, is differential-geometric in nature and is applicable
to the general situation of uniruled projective manifolds without
any assumption on the existence of special coordinate systems. As
an application, we give a characterization of standard embeddings
for certain pairs of rational homogeneous manifolds in terms of
embeddings of varieties of minimal rational tangents.