On the rectilinear congruences establishing a mapping between its focal surfaces which preserves the Gauss curvature

1982 ◽  
Vol 131 (1) ◽  
pp. 255-264 ◽  
Author(s):  
Gr. Tsagas ◽  
B. Papantoniou
Keyword(s):  
Gauss Curvature ◽  
Rectilinear Congruences

Regularity of almost extremal solutions of Monge–Ampère equations

1991 ◽  
Vol 117 (1-2) ◽  
pp. 21-29 ◽  
Author(s):  
John I. E. Urbas
Keyword(s):  
Large Class ◽  
Convex Domain ◽  
Gauss Curvature ◽  
Extremal Solutions ◽  
Classical Solutions ◽  
Uniformly Convex ◽  
Suitable Sense ◽  
Monge Ampere

SynopsisWe show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.

Geometry of 1-lightlike submanifolds in anti-de Sitter n-space

2013 ◽  
Vol 143 (5) ◽  
pp. 1089-1113 ◽  
Author(s):  
Zhigang Wang ◽  
Donghe Pei ◽  
Liang Chen
Keyword(s):  
Differential Geometry ◽  
Geometric Property ◽  
Gauss Curvature ◽  
De Sitter ◽  
Lightlike Submanifolds

In this paper we investigate the differential geometry of 1-lightlike submanifolds in anti-de Sitter n-space as an application of the theory of Legendrian singularities. Based on some theory of lightlike submanifolds, we also introduce the notion of 1-lightlike horospherical Gauss curvature, which is important for us to study the singularities of 1-lightlike horospherical hypersurfaces. Moreover, we discuss the related geometric property of 1-lightlike horospherical hypersurfaces in anti-de Sitter n-space.

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