I attempted to make a computer graphical rendering C1 isometric
embedding of the hyperbolic plane.
I am not sure if the result really is a C1 embedding, but the pictures
are cool:

http://www.xs4all.nl/~westy31/Geomet...try.html#Embed

Gerard

Gerard Westendorp wrote:
I attempted to make a computer graphical rendering C1 isometric
embedding of the hyperbolic plane.
I am not sure if the result really is a C1 embedding, but the pictures
are cool:

http://www.xs4all.nl/~westy31/Geomet...try.html#Embed

A general question:
Does a surface exist in the form z = f(x,y) that is an isometric C1
embedding of the hyperbolic plane?

Gerard

No. The hyperbolic plane cannot be isometrically embedded in R^3. This
is a theorem by Hilbert. See the book Differential Geometry of Curves
and Surfaces by M.P. do Carmo.

On Jul 16, 9:49*am, Gerard Westendorp wrote:
Gerard Westendorp wrote:
I attempted to make a computer graphical rendering C1 isometric
embedding of the hyperbolic plane.
I am not sure if the result really is a C1 embedding, but the pictures
are cool:

http://www.xs4all.nl/~westy31/Geomet...try.html#Embed

A general question:
Does a surface exist in the form z = f(x,y) that is an isometric C1
embedding of the hyperbolic plane?

Gerard

Hello geometers,

Contrary to what Jairo Bochi wrote here (Thu, 17 Jul 2008), I
believe that D. Hilbert merely proved that the real hyperbolic
plane admits no C^k isometric embedding in R^3 for k large (k \geq
2 ?). See, for example, M.D. Spivak's "Comprehensive introduction
to differential geometry".

Using John Nash's 1954 embedding ideas

MR0065993 (16,515e) Nash, John $C\sp 1$ isometric imbeddings. Ann. of Math.
(2) 60, (1954). 383--396. (Reviewer: S. Chern) 53.0X

Nicolas Kuiper proved that the real hyperbolic plane does admit a
C1 isometric embedding in R^3; see

Kuiper, Nicolaas H. On $C^1$-isometric imbeddings. I, II.
(English) Nederl. Akad. Wet., Proc., Ser. A 58, 545-556, 683-689
(1955).

Gromov gives another proof in his differerential inequalities
book of the 1970s.

These two proofs are hard and incredibly abstract. I have
never seen a genuine explicit C1 isometric embedding constructed.
Nor have I heard anyone claim to have one until Gerard
Westendorp's impressive

http://www.xs4all.nl/~westy31/Geomet...try.html#Embed

now tempered by the admission (8 Jul 2008):

I am not sure if the result really is a C1 embedding,
but the pictures are cool...

Such claims are of considerable interest and deserve to be
established by precise description --- and by careful proof that
the whole plane is isometrically embedded and C1 smooth.

Larry Siebenmann