Submanifolds of almost poly-Norden Riemannian manifolds

Selcen Yüksel Perktaş

doi:10.3906/mat-1901-58

Abstract

Our aim in the present paper is to initiate the study of submanifolds in an almost poly-Norden Riemannian manifold, which is a new type of manifold first introduced by Sahin [17]. We give fundamental properties of submanifolds equipped with induced structures provided by almost poly-Norden Riemannian structures and find some conditions for such submanifolds to be totally geodesics. We introduce some subclasses of submanifolds in almost poly-Norden Riemannian manifolds such as invariant and antiinvariant submanifolds. We investigate conditions for a hypersurface of almost poly-Norden Riemannian manifolds to be invariant and totally geodesic, respectively, by using the components of the structure induced by the almost poly-Norden Riemannian structure of the ambient manifold. We also obtain some characterizations for totally umbilical hypersurfaces and give some examples of invariant and noninvariant hypersurfaces.

Highlights

In Riemannian manifolds, different geometric structures such as almost complex structures, almost product structures, almost contact structures, and almost paracontact structures allow significant results to emerge while investigating differential and geometric properties of submanifolds. √As a generalization of the golden mean, the number φ = 1+ 2is known as a solution of the equation x2 − x − 1 = 0, Spinadel introduced a family of metallic proportions in [5]
As a generalization of the golden mean, the number φ is known as a solution of the equation x2 − x − 1 = 0, Spinadel introduced a family of metallic proportions in [5]
Poyraz Önen and Yaşar [16] initiated the study of lightlike geometry in golden semi-Riemannian manifolds by investigating lightlike hypersurfaces of a golden semi-Riemannian manifold

Summary

Introduction

In Riemannian (as well as semi-Riemannian) manifolds, different geometric structures such as almost complex structures, almost product structures, almost contact structures, and almost paracontact structures allow significant results to emerge while investigating differential and geometric properties of submanifolds. Considering the study on a Riemannian manifold with the golden structure [4] and the bronze mean introduced by [15], Şahin in [17] defined a new type of manifold equipped with the bronze structure and named it an almost poly-Norden manifold. He gave some important geometric results and investigated the constancy of certain maps. In [17], by using the bronze mean given in (2.1), the author defined a new type of differentiable manifold equipped with a bronze structure.

Submanifolds of almost poly-Norden Riemannian manifolds

Antiinvariant submanifolds of almost poly-Norden Riemannian manifolds