Riemannian Manifolds of Conullity Two

Classification of 4-dimensional homogeneous D’Atri spaces

“…Let e(1, 1) be the Lie algebra of E(1, 1) with a scalar product , 3 . Then there is an orthonormal basis {f 1 , f 2 , f 3 } of e(1, 1) such that…”

“…A smooth Riemannian manifold M is called curvature homogeneous if for any two points p, q ∈ M there exists a linear isometry F : T p M → T q M such that F * R q = R p . This is also equivalent to saying that, locally, there always exists a smooth field of orthonormal frames with respect to which the components of the curvature tensor R are constant functions (see, for instance, I. M. Singer [14], or the monograph [3]). Hence it is obvious that all principal Ricci curvatures are constant.…”

“…Clearly, any homogeneous manifold is curvature homogeneous. On the other hand, the locally homogeneous Riemannian manifolds in dimensions 3 form a "negligible" subclass of all curvature homogeneous spaces (see a survey in [3]). …”

Semiparallel Isometric Immersions of 3-Dimensional Semisymmetric Riemannian Manifolds

“…The classification theorem by Szabó asserts the following (according to the formulation given in [1], [2]). …”

“…According to [6], [2] a foliated M 3 is said to be planar if it admits infinitely many asymptotic foliations. If it admits just two (or one, or none, respectively) asymptotic foliations, it is said to be hyperbolic (or parabolic, or elliptic, respectively).…”