Submanifolds with Parallel Fundamental Form

Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

“…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]). …”

“…planes or spheres or their open parts), (ii) surfaces with zero Gaussian curvature and flat normal connection, (iii) the second order envelopes of Veronese surfaces. (Here the description of the class (iii) is modified using the result of [10]; see also [15]. )…”

“…This classification is made in [16] and [11] (some generalizations for M 3 in Riemannian space forms were made later in [15]). …”

“…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 …”

CURVES AND SURFACES OF AW(k) TYPE

“…Let γ be a Frenet curve of type weak AW (2). If γ is a plane curve then , and the solution of this differential equation is 0 ) ( ) ( ' '…”