Recognitions:

## Riemannian surfaces as one dimensional complex manifolds

 Quote by TrickyDicky But they are. "A horosphere has a critical amount of (isotropic) curvature." according to wikipedia. And as I wrote in post #90 and has not been refuted yet: The theorem of Micallef-Moore says that a compact simply connected manifold of positive isotropic curvature is homeomorphic to the sphere.
The horosphere is not compact so the theorem does not apply.

 Quote by homeomorphic Horospheres are not topological spheres. They are spheres missing a point because that point is not in H^3. No, they are spheres minus a point, by convention, since we want them to live in hyperbolic space. Didn't we just say they were Euclidean (i.e. flat)? They are not compact. If you insist on including that extra point that makes them compact, you get the same problem we have been talking about. The metric won't extend to that point. It's Euclidean. There's no such thing as a Euclidean sphere, by Gauss-Bonnet. Therefore, by your own assertions, we must leave out that point at infinity and define the horosphere to be a sphere minus that point, which is then, non-compact and homeomorphic to R^2.
I corrected my previous assertions severl posts above, the horosphere is completely embedded in H^3, in contrast to what I thought, it is not missing any point, it doesn't have any point at the boundary, how could it have the same problem in both situations, looks as if you weren't acknowledging the change.

 Quote by homeomorphic There's no such thing as a Euclidean sphere
From "Three-dimensional geometry and topology, Volumen 1" page 61 by Thurston:

"A horizontal Euclidean plane is not a plane in hyperbolic geometry, it lies entirely on one side of a true hyperbolic plane tangent to it, wich is a Euclidean sphere... These horizontal surfaces are examples of horospheres"

 From "Three-dimensional geometry and topology, Volumen 1" page 61 by Thurston: "A horizontal Euclidean plane is not a plane in hyperbolic geometry, it lies entirely on one side of a true hyperbolic plane tangent to it, wich is a Euclidean sphere... These horizontal surfaces are examples of horospheres"
Yes, he means sphere minus a point. You know Gauss-Bonnet, so I don't see what your objection is.

 I corrected my previous assertions severl posts above, the horosphere is completely embedded in H^3, in contrast to what I thought, it is not missing any point, it doesn't have any point at the boundary, how could it have the same problem in both situations, looks as if you weren't acknowledging the change.
Yes, it IS missing a point. It doesn't have a point at the boundary, but that is because it is approaching that point at the boundary.

Recognitions:
 Quote by TrickyDicky From "Three-dimensional geometry and topology, Volumen 1" page 61 by Thurston: "A horizontal Euclidean plane is not a plane in hyperbolic geometry, it lies entirely on one side of a true hyperbolic plane tangent to it, wich is a Euclidean sphere... These horizontal surfaces are examples of horospheres"
ok. so can you say in your own words what Thurston means by a Euclidean sphere? Tell us.

Recognitions:
 Quote by TrickyDicky In post #23 I said this: I think it is easy to see now that horospheres are embedded in H^3 and are basically spheres of infinite radius, so they have euclidean geometry on them and at the same time I would say they are homeomorphic to the topological sphere (and therefore to the Riemann sphere). Would you guys agree with this?
horospheres are not homeomorphic to the topological sphere. But... you have insisted that they are despite repeated explanations that they are not. So there must be a homeomorphism that you have in mind. Could you tell us what you are thinking of? What is the homeomorphism?

 Quote by lavinia ok. so can you say in your own words what Thurston means by a Euclidean sphere? Tell us.
I certainly don't think he meant a sphere with a point missing or else he would have stated it so.
He seems to refer to a sphere the way we are used to in euclidean embedding. I think he means horospheres are homeomorphic to the topological sphere, that they have positive isotropic curvature, thus his insistence that, in hyperbolic space, the euclidean plane is different from euclidean space.

Recognitions:
 Quote by TrickyDicky I certainly don't think he meant a sphere with a point missing or else he would have stated it so. He seems to refer to a sphere the way we are used to in euclidean embedding. I think he means horospheres are homeomorphic to the topological sphere, that they have positive isotropic curvature, thus his insistence that, in hyperbolic space, the euclidean plane is different from euclidean space.
so you are not sure

 Quote by lavinia horospheres are not homeomorphic to the topological sphere. But... you have insisted that they are despite repeated explanations that they are not. So there must be a homeomorphism that you have in mind. Could you tell us what you are thinking of? What is the homeomorphism?
Ever heard about a Bryant surface, the horosphere is one very important example of a Bryant surface.
http://en.wikipedia.org/wiki/Bryant_surface
It says in WP:"In Riemannian geometry, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional hyperbolic space with constant mean curvature equal to 1."
Horospheres are the only Bryant surfaces in wich all of the surface points are umbilical points. And it is a well known fact of differential geometry that only for umbilical points the gaussian curvature equals the square of the mean curvature at that point, as a conclusion horospheres have positive gaussian curvature and therefore gauss-bonnet theorem is fine with them. Ben should correct his calculation of the Euler characteristic in a previous post.

Some References:
R. Aiyama, K. Akutagawa, Kenmotsu-Bryant type representation formula for
constant mean curvature spacelike surfaces in H3, Differential Geom.
Appl. 9 (1998), 251–272.
L. Bianchi, Lezioni di Geometria Differenziale, Ast´erisque, 154-155 (1987),
321–347.
R.L. Bryant, Surfaces of mean curvature one in hyperbolic space, terza Edizione,
Bologna (1987).
P. Collin, L. Hauswirth, H. Rosenberg, The geometry of finite topology Bryant
surfaces, Ann. of Math. 153 (2001), 623–659.
J.A. G´alvez, A. Mart´ınez, F. Mil´an, Flat surfaces in the hiperbolic 3-space,
Math. Ann., 316 (2000), 419–435.

 Besides what is said above: WP:"In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric." A horosphere is the boundary of a horoball, wich is the limit of a sequence of increasing balls in hyperbolic space, a ball in hyperbolic space is homeomorphic to a ball in Euclidean space (since hyperbolic space is homeomorphic to Euclidean space), so this should prove a horosphere is homeomorphic to a sphere in Hyperbolic space (therefore also to the Riemann sphere logically) even if it has Euclidean metric. I know is counterintuitive, and hard to swallow but I don't know how else explain it.

Recognitions:
 Quote by TrickyDicky Besides what is said above: WP:"In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric." A horosphere is the boundary of a horoball, wich is the limit of a sequence of increasing balls in hyperbolic space, a ball in hyperbolic space is homeomorphic to a ball in Euclidean space (since hyperbolic space is homeomorphic to Euclidean space), so this should prove a horosphere is homeomorphic to a sphere in Hyperbolic space (therefore also to the Riemann sphere logically) even if it has Euclidean metric. I know is counterintuitive, and hard to swallow but I don't know how else explain it.
I am tired of your answering with quotes from Wikipedia. what's the matter? Can't you explain things in your own words? Tell me - without some language that you find in a book or on line - what is meant by a topological sphere? How is a horosphere homeomorphic to a topological sphere? What is the homeomorphism? write it down.

 Quote by lavinia I am tired of your answering with quotes from Wikipedia. what's the matter? Can't you explain things in your own words? Tell me - without some language that you find in a book or on line - what is meant by a topological sphere? How is a horosphere homeomorphic to a topological sphere. What is the homeomorphism? write it down.
I do so to avoid errors, I'm not a mathgematician nor have any formal training in math, so I can only use my intuition and try to search for places where it comes explained in a better way than I can do it. Did you read the previous post?
 Since Ben works with strings he might find interesting this quote from an article on strings on page 10: "Triangulated Surfaces in Twistor Space:A Kinematical Set up for Open/Closed String Duality" http://arxiv.org/abs/hep-th/0607146 "Let γ(∞) denote the endpoint of γ on the sphere at infinity ∂H3 ≃ S2, a closed horosphere centered at γ(∞) is a closed surface Ʃ ⊂ H3 which is orthogonal to all geodesic lines in H3 with endpoint γ(∞)." So since the horosphere is closed it has no boundary term (it is compact without boundary) and your calculation was incorrect on that too: We only need the integral of the gaussian curvature to obtain the Euler characteristic, and since it has positive gaussian curvature due to its mean curvature +1 and being totally umbilical: \begin{align} 2 \pi \, \chi(M) &= \int_Ʃ K \, dV = 4 \pi \\ \chi(M) &= 2 \end{align}