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Doubly ruled surfaces in hyperbolic 3-space

A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of ...
Brendan Guilfoyle's user avatar
4 votes
1 answer
321 views

A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$

I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
Zaragosa's user avatar
  • 143
10 votes
2 answers
1k views

Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
Willemien's user avatar
  • 305
17 votes
2 answers
2k views

Geodesics on the twisted pseudosphere (Dini's surface)

I wonder how difficult it is to compute geodesics on Dini's Surface, a twisted pseudosphere? Here is one parametrization, from Alfred Gray's Modern Differential Geometry of Curves and Surfaces, p....
Joseph O'Rourke's user avatar