The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

Abstract

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

Rotational Hypersurfaces in Euclidean 4-Space with Density

“…Dursun [12]. Also, Güler and et al have studied Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4space [13], second Laplace-Beltrami operator of the rotational hypersurface in 4-space [32] and Cheng-Yau operator and Gauss map of the rotational hypersurface in 4-space [33]. Yüce has studied Weingarten Map of the Hypersurface in Euclidean 4-Space [34].…”

“…Then, the rotational hypersurface in 4 is given by : ( , , ) = ( , , , ( )) (2.10)where : ⊂ − {0} → is a ∞ function for all ∈ and 0 ≤ , ≤ 2 . The Gaussian curvature G and the mean curvature H of rotational hypersurface are obtained as follows[13,32,33].…”

“…Then M can be parameterized by (c 3 − 2 x 3 + 2 ) 2 dx + c 4 ), where 3 , 4 ∈ and 4 > ( 3 − 2 3 + 2 )Corollary Let be a minimal rotational hypersurfaces in4 . Then M can be parameterized by and 4 ∈[13,32,33]. If the mean curvature of rotational hypersurfaces(3.11) in the Euclidean 4-space is ( cos ( )))),(3.12) where 3 = sin(1) ,4 = 0 and 2 > > − show the projections of the rotational hypersurface with ( ) = 2 sin( )+ ( ) −yzt, xzt, and xyt-spaces in (a), (b) and (c), respectively.…”

Dini-Type Helicoidal Hypersurfaces with Timelike Axis in Minkowski 4-Space E14

“…Güler, Magid and Yaylı [28] defined helicoidal hypersurface with the Laplace-Beltrami operator in E 4 . Furthermore, Güler, Hacısalihoglu and Kim [29] worked rotational hypersurface with the III Laplace-Beltrami operator and the Gauss map in E 4 .…”

Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space

“…It is a bit too general since the curve is not located in any subspace before rotation. Güler and Kişi [30] studied the Weierstrass representation, the degree and the classes of surfaces in E 4 , see [31][32][33][34][35][36][37][38] for some previous work.…”