  After establishing the uniqueness of the continuation of local Cauchy data
for harmonic maps between two Riemannian manifolds M and N, we prove (i) a
reflection principle for a smooth minimal submanifold Y of a Riemannian
manifold M that contains a reflective submanifold of M as a hypersurface and
(ii) the reflection principle of for a class of smooth harmonic maps between
Riemannian manifolds that admit reflective submanifolds. A reflection principle
for harmonic functions on M with respect to a reflective hypersurface of M
follows easily from the above reflection principle. To consider reflection
principles for harmonic and holomorphic maps from general Hermitian symmetric
spaces M, the notion of recursive real forms is introduced. Hermitian symmetric
spaces admitting such recursive real forms include Hermitian manifolds of
constant holomorphic curvatures, the complex hyper-quadrics and their
respective non-compact duals. Let M be an irreducible Hermitian symmetric space
of complex dimension n with a recursive real form B1 and corresponding
involutive isometry s1, X be a smooth Riemannian manifold with B2 as a
reflective submanifold and corresponding involutive isometry s2 . Also let h be
a harmonic map from M to X such that (i) h(B1) is in B2 and (ii) h maps tangent
vectors normal to B1 to vectors normal to B2 . We will prove that h(s1(p)) =
s2(h(p)) all p in M. A special case of the above assertion is that X is a
Hermitian symmetric space, B_2 is a real form of X and h is a holomorphic map.
These two assertions generalize the classical Schwartz reflection principle for
harmonic functions and holomorphic functions on the complex plane to harmonic
maps and holomorphic maps from a class of Hermitian symmetric spaces admitting
recursive real forms. A reflection principle for meromorphic functions defined
on the complex plane and taking on real values on the real line is also proved.
