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Bu çalışmada, ilk olarak 4-boyutlu Öklidyen uzayında bir Monge hiperyüzeyinin ortalama ve Gaussian eğriliklerini verdik. Ardından, farklı yoğunluklara sahip uzayında Monge hiperyüzeylerini çalıştık. Bu bağlamda, ve hepsi aynı anda sıfır olmayan sabitler olmak üzere, (lineer yoğunluk) ve yoğunluklu uzayında ağırlıklı minimal ve ağırlıklı flat Monge hiperyüzeylerini ve sabitlerinin farklı seçimleri yardımıyla elde ettik. Anahtar Kelimeler:Yoğunluklu manifold, ağırlıklı ortalama eğrilik, ağırlıklı gaussian eğriliği, monge yüzeyleri.

The object of the present paper is to classify N(κ)-contact metric manifolds admitting the semi-symmetric non-metric connection with certain curvature conditions the projectively curvature tensor. We studied projective flat, ξ −projectively flat, φ −projectively flat N(κ)contact metric manifolds admitting the semi-symmetric non-metric connection. Also, we examine such manifolds under some local symmetry conditions related to projective curvature tensor.

 Rotational hypersurfaces in Euclidean 4-space with density have considered.  Weighted minimal and weighted flat rotational hypersurfaces in Euclidean 4-space with density have obtained.  Some examples for these hypersurfaces have constructed.

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In this paper, we characterize N(k)-contact metric manifolds with generalized Tanaka-Webster connection. We obtain some curvature properties. It is proven that if an N(k)-contact metric manifold with generalized Tanaka-Webster connection is K-contact then it is an example of generalized Sasakian space form. Also, we examine some flatness and symmetric conditions of concircular curvature tensor on an N(k)-contact metric manifolds with generalized Tanaka-Webster connection.

In this study, we study rotational hypersurfaces in 4-dimensional Lorentz-Minkowski space. We find the rotational hypersurfaces about spacelike axis according to Gaussian and mean curvatures in E 4 1 and give some results with the aid of the Gaussian and mean curvatures. After that, we deal with the Gauss map of rotational hypersurface about spacelike axis by obtaining the Gaussian and mean curvatures. We obtain the second and third Laplace-Beltrami operators on rotational hypersurface about spacelike axis in E 4 1 . Also, we give these characterizations for rotational hypersurfaces about timelike and lightlike axes, too.

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