Isometric Semiparallel Immersions of Two-Dimensional Riemannian Manifolds into Pseudo-Euclidean Spaces

Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

“…Here R(X, Y ) is the curvature operator of the van der Waerden-Bortolotti connection. The Riemannian submanifolds M m in n satisfying this condition are called semisym-metric (extrinsically) [8]- [11], or more often semiparallel [4], [14]. Intrinsically every semiparallel submanifold is a semisymmetric Riemannian manifold; this follows again from the Gauss equation and the expressions for the curvature tensors of ∇ ⊥ (see [4], [15]).…”

“…Up to now only the two-dimensional case has been investigated in [14]. The result can be summarized in the following way.…”

“…It is shown in [9] that if n = 5 then the only semisymmetric surfaces of the class (iii) in 5 are the parallel ones, namely the single Veronese surfaces, every one of which has constant positive Gaussian curvature K. In [14] there is added that into pseudoEuclidean space 3 5 (with 3 minus signs in the canonical form of ds 2 ) also M 2 of negative constant K can be immersed isometrically as a semiparallel surface of the class (iii), again as a parallel one.…”

“…The main result in [14] is that into s 6 with s ∈ {0, 3, 4} also a Riemannian M 2 of non-constant Gaussian curvature K can be immersed isometrically as a non-parallel semiparallel surface of the class (iii), but in s 7 with s ∈ {0, 3, 4, 5} every Riemannian M 2 can be isometrically realized in the class (iii).…”

“…It can be shown (see [13]) that the pole of every of these logarithmic spirals coincides with the common centre of the concentric spheres above. In the limit case when a = 0 these logarithmic spirals reduce to circles and then M 3 is a parallel orbit, namely the Segre orbit (see [13], [15] Recall that partial differential prolongation of this system leads to the equations (4.9) and (4.10), but by the further prolongation the following consequences can be obtained (see [11]): by differential prolongation (see [9], [11], [14]) dA = Bω …”