  We consider isometric immersions of complete connected Riemannian manifolds
into space forms of nonzero constant curvature. We prove that if such an
immersion is compact and has semi-definite second fundamental form, then it is
an embedding with codimension one, its image bounds a convex set, and it is
rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as
well as by M. do Carmo and F. Warner. It also settles affirmatively a
conjecture by do Carmo and Warner. We establish a similar result for complete
isometric immersions satisfying a stronger condition on the second fundamental
form. We extend to the context of isometric immersions in space forms a
classical theorem for Euclidean hypersurfaces due to Hadamard. In this same
context, we prove an existence theorem of hypersurfaces with prescribed
boundary and vanishing Gauss-Kronecker curvature. Finally, we show that
isometric immersions into space forms which are regular outside the set of
totally geodesic points admit a reduction of codimension to one.

Differential Geometry