  The main purpose of the paper is twofold: First, to extend a well known
theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly,
to apply this result to extend Hoffman-Osserman-Schoen Theorem (HOS Theorem) to
3-dimensional symmetric spaces. Precisely, it is defined a Gauss map of a
hypersurface M^{n-1} immersed in a symmetric space N^n taking values in the
unit pseudo sphere S^m of the Lie algebra g of the isometry group of N,
dim(g)=m+1, and it is proved that M has CMC if and only if its Gauss map is
harmonic. As an application, it is proved that if dim(N)=3 and the image of the
Gauss map of a CMC surface S immersed in N is contained in a hemisphere of S^m
determined by a vector X, then S is invariant by the one parameter subgroup of
isometries of N of the Killing field determined by X. In particular, it is
obtained an extension of HOS Theorem to the 3-dimensional hyperbolic space.
  In the paper it is also shown that the holomorphic quadratic form induced by
the Gauss map coincides (up to a sign) with the Hopf quadratic form when the
ambient space is a space form and with the Abresch-Rosenberg quadratic form
when the ambient space is H^2 x R and S^2 x R providing, then, an unified way
of relating Hopf's and Abresch-Rosenberg's quadratic form with the quadratic
form induced by a harmonic Gauss map of a CMC surface in these 5 spaces.
