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

Abstract

We consider rotational hypersurface in the four dimensional Euclidean space. We calculate the mean curvature and the Gaussian curvature, and some relations of the rotational hypersurface. Moreover, we define the third Laplace-Beltrami operator and apply it to 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.…”

Classification of Conformal Surfaces of Revolution in Hyperbolic 3-Space

“…Further, the notion of finite type can be extended to any smooth function on a submanifold of a Euclidean space or a pseudo-Euclidean space. The theory of submanifolds of finite type has been studied by many geometers [7,11].…”