in this paper, by modifying cheng-yau$'$s technique to complete hypersurfaces in $s^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [h. li, hypersurfaces with constant scalar curvature in space forms, {em math. ann.} {305} (1996), 665--672].

In [12], we treated the Christoffel-Minkowski problem as a convexity problem of a spherical hessian equation on S via Gauss map. In this paper, we study the curvature equations of radial graphs over Sn. Our main concern is the existence of hypersurface with prescribed Weingarten curvature on radial directions. For a compact hypersurface M in Rn+1, the kth Weingarten curvature at x ∈ M is define...

We derive the equations of Gauss and Weingarten for a non-degenerate hypersurface of a semi-Riemannian manifold admitting a semi-symmetric metric connection, and give some corollaries of these equations. In addition, we obtain the equations of Gauss curvature and Codazzi-Mainardi for this non-degenerate hypersurface and give a relation between the Ricci and the scalar curvatures of a semi-Riema...

For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover through the inverse Gauss map when some information is prescribed on S ([27]). This question has attracted much attention for more than a hundred years. The most notable example is probably the Minkowski...

In the present study, we deal with canal hypersurfaces according to extended Darboux frame field of second kind in Euclidean 4-space (E4) and this context, firstly obtain Gaussian, mean principal curvatures hypersurface give some results for flatness minimality these E4. Also, Weingarten E4 finally, construct an example.

We study null hypersurfaces of indefinite Kähler manifolds and by taking the advantages almost complex structure $J$, we select a suitable rigging $\zeta$, which call $J-$rigging, on hypersurface. This enables us to build an associated Hermitian metric $\breve{g}$ ambient space is restricted into non-degenerated $\widetilde{g}$ normalized derive Gauss-Weingarten type formulae for hypersurface $...

In this paper we review some author’s results about Weingarten surfaces in Euclidean space R 3 and hyperbolic space H 3 . We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property. First, we consider Weingarten surfaces in R 3 that are foliated by circles, proving that the surface is rotational, a Riemann example or a generalized cone. Next...

A linear Weingarten surface in Euclidean space R 3 is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ R. Such a surface is said to be hyperbolic when a + 4bc < 0. In this paper we classify all rotational linear Weingarten surfaces of hyperbolic type. As a consequence, we obtain a family of complete hyperbolic linear Weingarte...