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4 questions
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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 ...
4
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1
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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\...
10
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2
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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 ...
17
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2
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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....