**FRANCESCO MERCURI**

*Parallel and semi-parallel immersions into space forms*

**Pages** 91-108

**Received:** 25 June 1992

**Mathematics Subject Classification:** 53C42

**Abstract**
The geometry of a submanifold of a space form is described by the second fundamental form. Therefore it is natural to study
isometric immersions whose second fundamental form are simple in some sense. In this paper we will describe some (by now classical) results on
parallel immersions, i.e. immersions whose second fundamental form is covariantly constant, and some more recent results on semi-parallel immersions,
i.e. immersions whose second fundamental form verifies the corresponding integrability condition.
It is probably worthwhile to observe that the above condition are the analog, in submanfolds theory, of what symmetric and semi-symmetric spaces are in intrinsic
riemannian geometry.