[en] A complete local classification and a geometric description are given for hypersurfaces with semiparallel Ricci tensor in Euclidean spaces

[en] Recently there has been a proposal for modified gravitational f(R) actions which include a direct coupling between the matter action and the Ricci scalar, R. Of particular interest is the specific case where both the action and the coupling are linear in R. It is shown that such an action leads to a theory of gravity which includes higher order derivatives of the matter fields without introducing more dynamics in the gravity sector and, therefore, cannot be a viable theory for gravitation

[en] In this note we study the hypersurfaces of a real space form M-bar(c) whose Ricci operator coincides with the second fundamental form. In particular we obtain characterizations of such hypersurfaces for the case c>0. (author)

[en] Starting with the algebraic classification of the energy-momentum tensor given by Plebansky, it is established that this classification is unstable under versal deformations and a new (stable) classification is given. In order to keep the text to reasonable length, we just write the basic ideas and some results. (Author) (Author)

[en] We study the geometrical properties of null congruences generated by an aligned null direction of the Weyl tensor (WAND) in spacetimes of Weyl and Ricci type N (possibly with a non-vanishing cosmological constant) in an arbitrary dimension. We prove that a type N Ricci tensor and a type III or N Weyl tensor have to be aligned. In such spacetimes, the multiple WAND has to be geodetic. For spacetimes with type N aligned Weyl and Ricci tensors, the canonical form of the optical matrix in the twisting and non-twisting cases is derived and the dependence of the Weyl and the Ricci tensors and Ricci rotation coefficients on the affine parameter of the geodetic null congruence generated by the WAND is obtained. (paper)

[en] This work contributes to the classification of some Einsteinian 4-submanifolds by determining the second fundamental form. This leads to a characterization of some flat submanifolds of the Euclidean space. In particular, we show that an N-codimensional flat submanifold with flat normal connection in E^n+N is cylindrical. Such a result is important since, in 1919, Elie Cartan asserted without proof the existence of flat, non cylindrical submanifolds of dimension n ≥ 4 in E^n+N; at the same time he proved than an n-dimensional submanifold of E^n+N, n ≤ 3, is flat if and only if it is cylindrical. (author). 12 refs

No abstract available

[en] A class of harmonic n-manifold, denoted by HM_n, is, in fact, focussed on a Riemannian manifold with harmonic curvature. A variety of results, with properties, on HM_n is presented in a fair order. Harmonic manifolds are then touched upon manifolds with recurrent Ricci curvature, biRicci-recurrent curvature and recurrent conformal curvature, and, in consequence, a sequence of theorems are deduced. (author). 21 refs

[en] In this paper, we prove that if M is an n-dimensional closed minimal hypersurface in a unit sphere Sⁿ⁺¹(1) with constant scalar curvature, then S=0, or S=n or S≥2n - 6n/n(n+4)+6, where S is the squared norm of the second fundamental form of M. Our theorem indicates that the conjecture proposed by Chern and Peng-Terng would be solved affirmatively. (author). 7 refs

[en] Published in summary form only