E^5_2 Yarı-Öklid Uzayındaki Biharmonik Hiperyüzeyler

Bu çalışmada, $\mathbb E^5_2$ yarı-Öklid uzayının indisi 2 olan biharmonik hiperyüzeyleri, $\nabla H$ gradyenti ışıksal olan $H$ ortalama eğriliğine sahip olmaları, yani $\left\langle \nabla H,\nabla H\right\rangle = 0$ ve $\nabla H\neq0$ koşullarının sağlanması varsayımı altında incelenmiştir.

Anahtar Kelimeler:

Biharmonik hiperyüzeyler, Yarı-Öklid uzay, Yarı-Riemannsal alt manifoldlar, Şekil operatörü

Biharmonic Hypersurfaces in the Pseudo-Euclidean Space E^5_2

In this work, biharmonic hypersurfaces of index 2 in pseudo-Euclidean space E52 are studied under the assumption of having mean curvature H whose gradient ÑH is light-like, i.e. hÑH;ÑHi = 0 and ÑH 6= 0. In the first two sections, the problem is introduced and some basic definitions and formulas that we will use in other part of the paper are recalled. Moreover, all possible canonical forms of the shape operator of a hypersurface of index 2 are obtained. In the third section of this work, for each of these cases, some of geometrical properties of hypersurfaces is investigated. In particular, there are 2 possible canonical forms of the shape operator for a biharmonic hypersurface such that whose gradient ÑH is light-like are obtained. After that, the non-existance of biharmonic hypersurface of index 2 in pseudo-Euclidean space E52 with the light-like ÑH is proved. In the last section, the results from this work is summarized and the discussion part is given.

Keywords:

Biharmonic hypersurfaces, Pseudo-Euclidean space, Semi-Riemannian submanifolds, Shape operator,

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