scholarly journals Lie applicable surfaces and curved flats

2021 ◽  
Author(s):  
Francis Burstall ◽  
Mason Pember
Keyword(s):  
Darboux Transformation ◽  
Lie Sphere Geometry ◽  
One To One ◽  
Sphere Geometry ◽  
Lie Sphere

AbstractWe investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation.

scholarly journals Discrete channel surfaces

2019 ◽  
Vol 294 (1-2) ◽  
pp. 747-767
Author(s):  
Udo Hertrich-Jeromin ◽  
Wayne Rossman ◽  
Gudrun Szewieczek
Keyword(s):  
Discrete Version ◽  
Smooth Channel ◽  
Lie Sphere Geometry ◽  
Discrete Channel ◽  
Channel Surface ◽  
Sphere Geometry ◽  
Definition Of ◽  
Lie Sphere

Abstract We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete channel surface that may be used to reconstruct the underlying particular discrete Legendre map. As an application we investigate isothermic discrete channel surfaces and prove a discrete version of Vessiot’s Theorem.

scholarly journals Classification of surfaces in three-sphere in Lie sphere geometry

1996 ◽  
Vol 143 ◽  
pp. 59-92
Author(s):  
Takayoshi Yamazaki ◽  
Atsuko Yamada Yoshikawa
Keyword(s):  
Plane Curves ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Three Sphere ◽  
Lie Sphere Geometry ◽  
Sphere Geometry ◽  
Necessary And Sufficient ◽  
Lie Sphere

We studied plane curves in Lie sphere geometry in [YY]. Especially we constructed Lie frames of curves in S2 and classified them by the Lie equivalence. In this paper we are concerned with surfaces in S3. We construct Lie frames and classify them. We moreover obtain the necessary and sufficient condition that two surfaces are Lie equivalent.

Dupin Submanifolds in Lie Sphere Geometry

1989 ◽  
pp. 269-330
Author(s):  
Thomas E. Cecil ◽  
Shiing-Shen Chern
Keyword(s):  
Lie Sphere Geometry ◽  
Sphere Geometry ◽  
Lie Sphere
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