Riemannian surfaces as one dimensional complex manifolds

TrickyDicky

Welcome back Lavinia.
Ok, this is understood. Let's change flat by euclidean and let's think about a situation where a euclidean surface could have a extrinsic positive curvature, for instance a negatively curved ambient space, what would be wrong in this picture in your opinion?

Ok, let me try to ask again: Can you give any example of a non orientable conformal surface in R^3? I'm not trying to cheat by adding this but I think the WP page is referring to euclidean ambient space because it defines orientability as:" a property of surfaces in Euclidean space". Not trying to defend wikipedia here, just pointing out that in this case there seems to be a contextual ambiguity.



Ok,I think this is related with what I write above on the context of conformal surfaces in Euclidean space and the klein bottle impossibility to be embedded in euclidean space (R^3). It seems in R^3 all surfaces conformal structures are orientable and therefore determine complex structures.







See above.



Yes, I'm using this problematic way of dealing with topology in reference to Gauss-Bonnet, rather than to the "Riemannian metric" use of curvature. I see you caught my drift here.



Never mind this, bad choice of words again. What I meant was the topology is of course not changed by embedding in a hyperbolic manifold, but the metric induced on that topology by the Negatively curved ambient Riemannian metric could change and admit a euclidean metric on the topology.








I see this, that's why I keep talking about the "one-point compactification of the complex plane" topology, but of course my expressions lack mathematical rigor.

I know the globally flat part doesn't sound right, I'm not sure about it either but all the time I've had the feeling that I'm not being able to get across what I mean, I don't mind being wrong but it bothers me not to be able to express what i mean in a sound mathematical way, so I guess I have to study this harder.

lavinia

The Mobius band is a non-orientable surface in 3 space. It naturally has a flat metric. There are no closed surfaces without boundary in 3 space that are unorientable. For instance the Klein bottle can not be embedded in 3 space.

Ben Niehoff

I will try to help if you're willing to work with me.

I haven't the faintest idea how to answer this question, because those words do not mean anything when put together in that order.

There are many extrinsic curvatures a surface can have, depending on the dimension of the ambient manifold. For example, I can pinch the north and south poles of a sphere, pull on them, and twist them by any arbitrary amount. All of these are measured by various extrinsic curvatures.

There is one special extrinsic curvature, the Gauss curvature, which Gauss proved is actually intrinsic; that is, independent of the ambient manifold into which the surface is embedded. From the extrinsic point of view, the Gauss curvature is the product of the principal curvatures (which are curvatures of curves on the surface, measured with respect to the ambient space). From the intrinsic point of view, the Gauss curvature is one-half the Ricci scalar intrinsic to the surface. But the two are always equal to each other. This is why it is quite strange to me (and likely everyone else) that you keep harping on embeddings in H^3 being somehow different from embeddings in R^3. They are not.

I think the simple explanation here is that Wikipedia is wrong (or in this case, incomplete). Orientability is a property that can be defined without reference to any ambient space. An n-dimensional manifold is orientable if and only if it admits a nowhere-vanishing n-form.

In two dimensions, it turns out that a nowhere-vanishing 2-form can also be interpreted as a complex structure. Hence all orientable 2-surfaces are complex manifolds (in fact, they are Kaehler).

I think you've missed Homeomorphic's point. A sphere can have different metrics, yes; but the total curvature of any metric on the sphere must be positive! You can imagine pulling on the ends of a sphere and pinching it in the middle to give it regions of negative curvature, but the regions of positive curvature at the ends must win out when you integrate over the whole sphere. This is a basic fact of topology, a direct consequence of the Gauss-Bonnet theorem; it does not depend on the embedding and hence you cannot negate this fact by embedding the sphere in H^3 or anywhere else.

The total curvature of any topological sphere must be positive.

This is where you've gone wrong. These things called "horospheres" are not, in fact, embeddings of the sphere into H^3. Look closely at the definition of "embedding". The embedded manifold must map entirely into the ambient space, and there must be a tubular neighborhood around it in the ambient space.

Using the Poincare ball model, H^3 is the open ball on the interior. It does not include the boundary. In fact, H^3 is homeomorphic to R^3; it has no boundary! (More on that later).

Looking at a horosphere in the Poincare ball model, the horosphere is represented by what looks like a round sphere touching the boundary. However, one point of this round sphere lies on the boundary of the ball model, and therefore does not properly lie within H^3. Therefore, a horosphere is not actually an embedding of the sphere into H^3, because there is one point of the sphere which is not inside H^3.

More properly speaking, the horosphere is the embedding in H^3 of a "sphere with one point removed". Or in other words, it is a 1-point de-compactification of the sphere; i.e., a horosphere is actually an embedding of the infinite plane! As I have pointed out before, this is why it is possible for the horosphere to be globally flat.



OK, so then what is the boundary of the Poincare ball model? This is easiest to explain after we see what the Poincare ball model is.

Start with a more natural model of hyperbolic space: the hyperboloid model. The hyperboloid model is easiest to visualize from by embedding it isometrically in Minkowski space. Starting with the metric

[tex]ds_4^2 = dx^2 + dy^2 + dz^2 - dw^2[/tex]
consider one sheet of the hyperboloid

[tex]w^2 - x^2 - y^2 - z^2 = 1[/tex]
This is a 3-hyperboloid that lies entirely within the future light cone (and asymptotically approaches it at infinity). If we compute the induced metric on this hyperboloid, we get

[tex]ds_3^2 = d\rho^2 + \sinh^2 \rho \, d\theta^2 + \sinh^2 \rho \, \sin^2 \theta \, d\phi^2[/tex]
The reason I call this a more natural model of H^3 is because now it is clear that H^3 is homogenous, isotropic, and infinite in extent in all directions. Also, it embeds isometrically within the future lightcone, so you can imagine what it looks like without distorting distances (whereas, the Poincare ball model must distort distances in order to fit all of H^3 within a ball).

To map between this model and the Poincare ball model, one actually uses stereographic projection! Unfortunately I can't draw a picture here. But imagine lines emanating from the origin of Minkowski space and intersecting the H^3 hyperboloid. Then map each point of H^3 to the unique point where each of these lines intersects the hyperplane [itex]w = 1[/itex]. It is easy to show that the result is the Poincare ball model, with the usual metric.

We can also easily see that the "boundary sphere" of the Poincare ball model is the image of the lightcone itself under this stereographic projection. Since stereographic projection is a conformal transformation, we can call this boundary a "conformal boundary". Under stereographic projection, H^3 itself maps to an open ball in E^3, whereas the lightcone maps to the boundary of this ball. To obtain a closed ball, we must add a whole 2-sphere's worth of points to H^3.

It is important to realize that H^3 itself is not a closed ball. To make it into one, we must conformally compactify H^3 and add a sphere's worth of points. The result turns H^3 into a closed ball in Euclidean space. Therefore if we want to consider spheres in the Poincare ball model that actually touch the boundary, then we can't use the hyperbolic metric to discuss them! After the conformal compactification required to give us the boundary, we will have a Euclidean metric inside the ball, and the spheres touching the boundary are simply ordinary, round spheres.

Alternatively, we can de-compactify, which turns the open interior of the ball into an honest H^3, and turns the horospheres into honest R^2's.

mathwonk

this is becoming more and more fun! thanks for the question, tricky. this is the way real math conversations go, when everyone is feeling her/his way, (with all respect).

lavinia

I thought about the metric on the sphere obtained from the usual Euclidean distance that you mentioned. I think that it can not be a metric that is derived from a Riemannian matric on the tangent bundle. The reason I think is that the distance between near by points along a geodesic must be additive. The distance from a to c must equal the distance from a to b plus the distance from b to c. But it seems with this metric and any curve the sum of the distances is always greater. What do you think?

lavinia

Sometimes an embedding does determine a Riemannain metric that is different than the one that the manifold inherits by restricting the ambient inner product to its tangent planes. Take the case of a surface embedded in Euclidean 3-space that has positive Gauss curvature. Then the eigen values of the second fundamental form are strictly positive so they determine a new Riemannian metric. I do not know if the manifold with this new Riemannian metric can be embedded in 3 space. I guess if the change is small it will still have positive curvature.

homeomorphic

The one-point compactification gives you the same topology as usual. Makes no difference topologically whether it's the one point compactification of ℂ or the unit sphere as a subspace of ℝ^3. The two are homeomorphic, via stereographic projection.


I think you mean in the presence of a suitable Riemannian metric. The 2-form by itself is just a symplectic structure. With a compatible metric, you can then define an almost complex structure, which will turn out to be integrable, so you get a complex structure (actually, a Kahler structure). An orientable surface only has 2 possible orientations, but it has lots and lots of complex structures--actually, a whole moduli space of them.

Ben Niehoff

Yes, I agree.

TrickyDicky

Sure I'm willing, I'm clarifying lots of stuff here, I admit that includes being frustrating at times, learning is hard and more so in a forum, but as mathwonk says it can be fun too.

I really agree with this. I'll try asking specific questions, so I can get closer to what I mean, but certainly I am aware that the topology is not affected by the embedding, my doubt is about metrics, as to whether a euclidean 2-dimensional metric can be thought of as having positive curvature wrt the ambient space or not. Can this be directly addressed by anyone?



I can see now that mathwonk was completely right to complain about this.


Fully agree, see above
Ok, I see this. It is also true I've made an effort to differentiate between the H^3 space and the H^3 manifold (quotient space) that includes the boundary. Can then be said according to you that a horosphere is not embedded in the H^3 manifold (not just the H^3 space)? And I would like for you or someone to comment on the fact that the Riemann sphere is the conformal boundary 2-manifold of the hyperbolic 3-manifold. And correct me if I'm wrong here but the R^3 representation of this boundary is a sphere (see Poincare ball model of H^3). Are horospheres not homeomorphic to this conformal boundary?

TrickyDicky

Nicely explained, thanks, it agrees with my previous knowledge about hyperbolic geometry.



You are describing precisely the back and forth between a round metric for the stereographic projection of the extended complex plane in E^3 and a flat metric in H^3 that I've been talking about all the time.
Just one point, as I said in a hyperbolic 3-manifold I would say we'd have a compact conformal boundary that will not require the space in the ball to be euclidean, as it is evident the ambient space in a hyperbolic manifold has a hyperbolic metric.

Maybe it is worth recalling here that there are some differences about some notions in topology and in manifold theory, for instance the concept of boundary in topology differs slightly from that in manifolds, or the concepts of closed and open in certain contexts.

Ben Niehoff

No, the curvature of a Riemannian metric (assuming Levi-Civita connection) is an intrinsic property; it doesn't care about the ambient space. If you choose some other connection, it is still an intrinsic property, because the connection is an intrinsic object as well. If the surface is isometrically embedded, then its extrinsic Gauss curvature will agree with its intrinsic Ricci scalar.

In this context, "space" and "manifold" mean the same thing, so I don't know what you're trying to say. Also, there's no quotient space anywhere in sight...to form a quotient space, you would quotient out by some family of submanifolds, which isn't being discussed here. (However, for example, the double torus and higher-genus Riemann surfaces can be thought of as quotients of H^2 by a discrete symmetry group).

H^3 as a manifold does not include the conformal boundary. It can't include the conformal boundary, because you can't just add points that are infinitely far away (any interval of infinite length must be open). You can add these points to the conformal compactification of H^3; but the conformal compactification of H^3 is honestly an open ball in E^3. This is analogous to the fact that the conformal compactification of R^2 is honestly an open region of a round sphere. The conformal transformation that compactifies the space actually changes the curvature. It makes H^3 flat, and it makes R^2 curved.

In fact, the coordinates in the Poincare ball model can be thought of as exactly the map from H^3 to E^3 that accomplishes this conformal compactification.

It's easy to see that conformal compactifications change curvature: just stick a conformal factor in front of the metric and recompute the curvature, you'll find it changes in a simple way:

http://en.wikipedia.org/wiki/Ricci_c...rmal_rescaling

TrickyDicky

Maybe you'll understand better my question if I remind you that I was not referring in this case to a Riemanninan metric but to the conformal geometry class of metrics since I'm considering the extended complex plane as a conformal manifold rather than a Riemannian manifold. So these facts apply according to WP:"Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can. Another feature is that there is no Levi-Civita connection because if g and λ2g are two representatives of the conformal structure, then the Christoffel symbols of g and λ2g would not agree. Those associated with λ2g would involve derivatives of the function λ whereas those associated with g would not."


Right.

In the context that I'm presenting it is of great importance to distinguish a space like H^3 from an hyperbolic 3-manifold. My question from the beginning is about manifolds, complex manifolds as submanifolds of hyperbolic manifolds. What do you mean there is no quotient space in sight?: WP:
"A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously." This subgroup happens to be isomorphic to the the automorphism group of the Riemann sphere.

Do you mean that a hyperbolic 3-manifold can't have a 2-manifold compact boundary? This boundary manifold being what you call the conformal compactification of H^3.

I looked at the link and it says:"For two dimensional manifolds, the above formula shows that if f is a harmonic function, then the conformal scaling g ↦ e2ƒg does not change the Ricci curvature."

Ben Niehoff

My previous response assumed you were talking about H^3. In your above post, you have changed all the definitions, quite literally changing the rules of the game.

You must be clear what you're talking about in the first place.

Ben Niehoff

I do know a bit about conformal geometry of surfaces, since this is important to string theory. I can try to respond to your new questions later.

For now, it might help for you to go back and read some of Mathwonk's and Homeomorphic's earlier comments concerning local vs. global conformal transformations. I think the discussion on Wikipedia is incomplete again. Wikipedia seems to focus on conformal classes of local metrics, but these kinds of conformal transformations do not preserve global topology.

For now, I can respond to this:

You need to make up your mind. The Poincare ball model is a model of H^3, which is the quotient of H^3 by the trivial group. That's why I say there are no quotient manifolds in play.

If you do quotient H^3 by a discrete group, you get a closed manifold without boundary (and with constant negative curvature). This is analogous to the quotient of R^2 by a discrete group (giving you, e.g., the torus), or the quotient of H^2 by a discrete group (giving you the double torus and higher-genus Riemann surfaces).

You can imagine a quotient of H^3 by a discrete group as a tesselation of H^3 by regular polyhedra, where the quotient manifold is one such polyhedron with various faces identified.

lavinia

For a surface embedded in a 3 dimensional Riemannian manifold calculation shows that the Riemann curvature tensors of the 3 dimensional manifold and the surface are related by a formula.

R[itex]_{ambient}[/itex](x,y,x,y) = R[itex]_{surface}[/itex](x,y,x,y) + A(x,y)[itex]^{2}[/itex]-A(x,x)A(y,y)

where A(x,y) is the normal component of the covariant derivative of y with respect to x.

The Gauss curvature of the surface thus satisfies the equation,

G = R[itex]_{ambient}[/itex](x,y,x,y)/|x^y|[itex]^{2}[/itex] - (A(x,y)[itex]^{2}[/itex]-A(x,x)A(y,y))/|x^y|[itex]^{2}[/itex]

If the sectional curvature of the ambient manifold is constant,K, as in Eudlidean space or some other flat 3 manifold or in hyperbolic 3 space then this equation becomes

G = K - (A(x,y)[itex]^{2}[/itex]-A(x,x)A(y,y))/|x^y|[itex]^{2}[/itex]

If k = 0 this is the usual equation for the Gauss curvature with respect to the determinant of the second fundamental form.

For the horosphere, G = 0 so the normalized determinant of the second fundamental form is equal to 1.

It is important to realize though that this equation in no way says that if the surface is a sphere that G can be identically 0. G is still the intrinsic curvature no matter how it is embedded in another manifold.

TrickyDicky

Thanks for the formula, it is actually a tensorial form to express the formula I brought from the Cartan book in post #10.

lavinia

Another example of this is the flat torus embedded in the 3 sphere. Here G = 0 so the normalized determinant of the second fundamental form is constantly -1.

I wonder if this means that an observer in the three sphere looking at a flat torus would see it as negatively curved - assuming light followed geodesics in the 3 sphere.