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Rotational Hypersurfaces with Constant Gauss-Kronecker Curvature
The authors study rotational hypersurfaces with constant Gauss-Kronecker curvature in ℝ n . They solve the ODE associated with the generating curve of...
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Nomizu–Smyth type results in the hyperbolic and de Sitter spaces via Gauss–Kronecker curvature
In this paper, our purpose is to establish new Nomizu–Smyth type results concerning complete Riemannian immersions in the hyperbolic and de Sitter...
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Isoperimetric Comparison for Sectional Curvature
We consider in this chapter the validity of the Euclidean isoperimetric inequality in a Cartan–Hadamard manifold, a complete, simply connected...
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Double Forms, Curvature Integrals and the Gauss–Bonnet Formula
The Gauss–Bonnet Formula is a significant achievement in nineteenth century differential geometry for the case of surfaces and the twentieth century...
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A Characterization of Separable Hypersurfaces in Euclidean Space
AbstractAs generalizations of classical translation hypersurfaces, in this note we introduce the notion of separable hypersurfaces in an
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Curvature: Differential Geometry
Differential geometry deals with objects that are no longer “straight”, such as curved lines and surfaces. The curvature, which measures the...
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Closed Self-Similar Solutions to Flows by Negative Powers of Curvature
In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers...
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Semi-Riemannian manifolds with linear differential conditions on the curvature
Semi-Riemannian manifolds that satisfy (homogeneous) linear differential conditions of arbitrary order r on the curvature are analyzed. They include,...
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Boundary properties for a Monge-Ampère equation of prescribed affine Gauss curvature
Considering a Monge-Ampère equation with prescribed affine Gauss curvature, we first show the completeness of centroaffine metric on the convex...
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Rigidity of hypersurfaces with constant higher order mean curvature in space forms
We consider complete oriented hypersurfaces M immersed into a Riemannian space form with constant higher order mean curvature and having two...
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Total Curvature and the Isoperimetric Inequality in Cartan–Hadamard Manifolds
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds and develop some applications of this...
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Characterization of manifolds of constant curvature by ruled surfaces
We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve,...
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Annular type surfaces with fixed boundary and with prescribed, almost constant mean curvature
We prove existence and nonexistence results for annular type parametric surfaces with prescribed, almost constant mean curvature, characterized as...
