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The aim of this paper is to present a complete description of all rotational linear Weingarten surface into the Euclidean sphere S 3 . These surfaces are characterized by a linear relation aH +bK = c, where H and K stand for their mean and Gaussian curvatures, respectively, whereas a, b and c are real constants.

Abstract. The aim of this work is to show that a compact smooth starshaped hypersurface Σ n in the Euclidean sphere S n+1 whose second function of curvature S 2 is a positive constant must be a geodesic sphere S n (ρ). This generalizes a result obtained by Jellett in 1853 for surfaces Σ 2 with constant mean curvature in the Euclidean space R 3 as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere S n+1 with constant mean curvature. In order to prove our theorem we shall present a formula for the operator L r (g) = div (P r ∇g) associated with a new support function g defined over a hypersurface M n in a Riemannian space form M n+1 c .

CaWO 4 nanocrystals were synthesized by the coprecipitation method of with different solvent ratio (water/ethylene glycol)

In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.2010 Mathematics Subject Classification. Primary 53C42; Secondary 53A07, 53B20.

The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand nonnegative Ricci curvature. Under a suitable choice, we have$area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.

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