Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to...

In this paper, we study curves in the lightlike cone. We first obtain the conformally invariant arc length in the (n+1)-dimensional lightlike cone and then characterize some curves in the 2-dimensional lightlike cone and 3-dimensional lightlike cone. MSC 2000: 53A04, 53A30, 53B30

Using screen distributions and lightlike transversal vector bundles we develop a theory of degenerate foliations of semiRiemannian manifolds. We build lightlike foliations of a semiRiemannian manifold by suspension of a group homomorphism φ : π1(B, x0) → Isom(T ). We compute the basic cohomology groups of the flow determined by a lightlike Killing vector field on a complete semi-Riemannian mani...

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which characterized by the condition that second form proportional to metric. It also generalization of notion geodesic submanifold. In this paper, we classify congruent classes full submanifolds non-flat space forms, and consider their moduli spaces. As consequence, show some spaces isometric immersions bet...

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-CartanVranceanu family. This gives a positive answer to a conjecture formulated in [2]. As a partial result, we prove that totally umbilical surfaces only exist if the space is a Riemannian product of a surface of constant Ga...

In this paper, we obtain some classification theorems for totally umbilical semi-invariant submanifolds in locally decomposable metallic Riemannian manifolds. We also prove that there exist no proper a posivitely or negatively curved manifold