Let \(N^{2n+s}\) be an S-manifold [D. E. Blair, J. Differ. Geom. 4, 155-167 (1970; Zbl 0202.209)] and f an f-structure of rank 2n on \(N^{2n+s}\); let \(M^{m+s}\) be a submanifold immersed in \(N^{2n+s}\). If \(TM^{m+s}\) is the Lie algebra of vector fields on \(M^{m+s}\), we have the following decomposition: \(fX=TX+NX,\) \(X\in TM^{m+s},\) where TX is the tangential component of fX and NX is the normal component. The submanifold \(M^{m+s}\) is called invariant if all structure vector fields are tangent to \(M^{m+s}\) and N is identically zero; \(M^{m+s}\) is called an anti-invariant submanifold if T is identically zero.
The aim of this paper is to prove the theorem: Let \(M^{m+s}\) be a submanifold of an S-manifold \(N^{2n+s}(K)\) of constant invariant f- sectional curvature \(K\neq s\), tangent to the structure vector fields. If the second fundamental form \(\sigma\) of \(M^{m+s}\) is parallel (the covariant derivative of \(\sigma\) is zero), then \(M^{m+s}\) is one of the following submanifolds: 1) an invariant submanifold of constant invariant f-sectional curvature K, immersed in \(N^{2n+s}(K)\) as a totally geodesic submanifold (i.e. \(\sigma\equiv 0)\); 2) an anti-invariant submanifold immersed in \(\bar M^{2m+s}(K)\), where \(\bar M^{2m+s}(K)\) is an invariant and totally geodesic submanifold of \(N^{2n+s}(K)\) of constant invariant f-sectional curvature, \(K\neq s\).