scholarly journals The relative nullity of complex submanifolds and the Gauss map

2015 ◽  
Vol 144 (4) ◽  
pp. 1689-1695
Author(s):  
Antonio J. Di Scala ◽  
Carlos Olmos
Keyword(s):  
Gauss Map ◽  
Relative Nullity ◽  
Complex Submanifolds

scholarly journals Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

2021 ◽  
Vol 9 ◽  
Author(s):  
Joseph Malkoun ◽  
Peter J. Olver
Keyword(s):  
Convex Hull ◽  
Convex Geometry ◽  
Gauss Map ◽  
Parameter Family ◽  
Convex Hulls ◽  
Dimensional Sphere ◽  
Continuous Maps ◽  
Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

scholarly journals A note on surfaces with prescribed oriented Euclidean Gauss map

2005 ◽  
Vol 2005 (4) ◽  
pp. 537-543
Author(s):  
Ricardo Sa Earp ◽  
Eric Toubiana
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Gauss Map ◽  
Compatibility Relation ◽  
Hyperbolic Gauss Map ◽  
Conformal Immersion

We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss mapGand formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determine a conformal immersion, proved in a previous paper.

scholarly journals Modified defect relations for the Gauss map of minimal surfaces

1989 ◽  
Vol 29 (2) ◽  
pp. 245-262 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Minimal Surfaces ◽  
Gauss Map

On the geometry of the Gauss map of conformal foliations by lines

2004 ◽  
Vol 136 (1) ◽  
pp. 247-255
Author(s):  
JEAN-MARIE BUREL ◽  
SIGMUNDUR GUDMUNDSSON
Keyword(s):  
Gauss Map

Ruled submanifolds with finite type Gauss map

1994 ◽  
Vol 49 (1-2) ◽  
pp. 42-45 ◽  
Author(s):  
Christos Baikoussis
Keyword(s):  
Finite Type ◽  
Gauss Map

The Gauss Map for Surfaces: Part 2. The Euclidean Case

1986 ◽  
Vol 293 (2) ◽  
pp. 447
Author(s):  
Joel L. Weiner
Keyword(s):  
Gauss Map ◽  
Euclidean Case

scholarly journals The tension field of the Gauss map

1970 ◽  
Vol 149 (2) ◽  
pp. 569-569 ◽  
Author(s):  
Ernst A. Ruh ◽  
Jaak Vilms
Keyword(s):  
Gauss Map

scholarly journals Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature

Topology ◽  
2002 ◽  
Vol 41 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Mohammad Ghomi
Keyword(s):  
Gauss Map

A Gauss map and hybrid degree formula for compact hypersurfaces in Minkowski space

1989 ◽  
Vol 32 (1) ◽  
Author(s):  
Marek Kossowski
Keyword(s):  
Minkowski Space ◽  
Gauss Map
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