We can similarly define a hyperbolic structure on a surface, by defining charts as maps from open sets to the hyperbolic plane and requiring that the transition functions be hyperbolic isometries .
Given a real projective or hyperbolic structure on a surface , through analytic continuation, we can define the developing map , a map from the universal cover into or respectively. acts in the obvious way on and in addition we get a representation of the fundamental group into or so that for all , .
There is a model of the hyperbolic plane known as the Klein Model in which points in the hyperbolic plane are thought of as points on the interior of a conic in , and lines as lines on the interior of . The isometry group of the hyperbolic plane here is thought of as the subgroup of preserving this conic.
This shows that a hyperbolic structure on a surface determines a projective structure. Now it appears useful for us to invoke some hyperbolic geometry. If is a closed surface with hyperbolic structure we can cut it along geodessics, breaking it into pieces which are topologically 3-punctured spheres (also know as pairs of pants).
|A pair of pants decomposition of a genus two surface|
Every hyperbolic 3-punctured sphere can further be broken along geodessics into two identical right hexagons. The right hexagons are determined by the lengths of every other side, and so the hyperbolic structure on a 3-punctured sphere is determined by the lengths of the geodessics around the punctures.
|Constructing a pair of pants from two right hexagons|
Thus we can construct any closed hyperbolic surface by gluing together 3-punctured spheres along their boundaries. But, we have freedom in how we glue. The obvious gluing is to line up the seams where the two right hexagons meet inside 3-punctured spheres, but there is no reason why we must do it this way. After constructing our surface we can change its hyperbolic structure later by cutting along one of our geodessics, and reattaching the (now) two circles in a different way (intuitively we will rotate one of the circles dragging part of the surface along with it and regluing). This process is called Dehn surgery.
|Dehn Surgery on a surface|
Now lets return to projective geometry. We start with hyperbolic structures on 3-punctured spheres. These hyperbolic structures give us projective structures via the Klein model. Now we would like to glue 3-punctured spheres together. At this point it is not obvious what gluing freedoms we have available. For instance, can we glue two such projective 3-punctured spheres together in such a way as to get a new projective structure on a surface which is not induced by a hyperbolic structure? To answer this question, we must return to our discussion of the developing map.
Suppose we cut a surface with a hyperbolic structure along an embeded geodessic loop in the interior of our surface dividing the surface into two pieces, and . In this loop lifts to several geodessics, pick one and call it . We have the induced projective structure on this surface as well. We know from our description of the Klein model that is a line in dividing the interior of the conic into two disjoint pieces.
Now consider cutting along and then regluing. We can take a closed collar nieghborhood of and look at its boundary components, call them and and assume that they are homotopically equivalent (in , ) and lie in and resepectively. Choose the right lifts and to and they will lie in seperate pieces of \. Now instead of regluing exactly the way the surface was before consider deforming the second piece by a transformation then regluing, that is a point in this piece are first sent to and then we reglue the image. We obviously would like to preserve the line segment but we require a little more.
Because and are homotopically equivalent, in terms of the representation , . So to glue in the way we described above, we need that . One way to ensure that this holds, is to ensure that commutes with .
Notice that if is an hyperbolic action on the Klein model, it will fix a line inside the conic containing the hyperbolic plane. In particular, it will also fix the two points and at which this line intersects the conic boundary. Also because it fixes the conic, it fixes the two lines and tangent to the conic at these point, and therefore also fixes the point at which they meet, . Recall that two projective transformations commute if and only if they have the same three fixed points.
|Fixed points of a hyperbolic translation in the Klein Model|
So we only consider use of which fix the three fixed points described above. The space of projective transformations which fix these is a -dimensional set. After choosing such a and performing our gluing operation, we must also must reglue the rest of the images under the developing map of preimages of in in a similar way. After regluing in this way, there will be a new convex set invariant under often not the interior of a conic.
The images of our conic under maps can be any conic tangent to and at the points and respectively. Our two dimensional space of deformations we can make in this manner, can be thought of as decomposing into two directions, one with hyperbolic deformations (Dehn surgeries along ) and the other projective deformations which change the shape of . It is not necessary that the we preform the surgeries along geodessic loops which divide the surface into two hyperbolic pieces; we can do these surgeries whenever a surface can locally split into two hyperbolic pieces (say into two pairs of pants).
|Possible images of the conic under the map|
Hopefully, more to come
The applet below allows a user to explore this space of projective deformations of a hyperbolic structure on a genus two surface by viewing the developing map. Note that the boundary of the developing map should always be a convex shape whose boundary curve is differentiable but not twice differentiable unless that boundary is a conic.
|The gluing of a hyperbolic surface used. Deformations happen where the two pairs of pants meet.|
The applet begins with a hyperbolic tiling by equilateral right hexagons. Any four adjacent differently colored right hexagons form a fundamental domain for the group action of on the interior of the conic.
You can zoom in and out in the upper left hand box by holding down the mouse button and dragging up or down on the mouse in the box.
You may move the vertices of the white quadrilateral around the box on the left. Possible placements for these vertices represent maps from the interior of the quarilateral to the box on the upper right side of the screen; vertices being mapped to the corners. It is recomended that you attempt to keep the quadrilateral convex and ensure that it contains the tiling.
In the lower left, you may apply the deformations along curves decomposing the surface into a pair of pants- six deformations in all.
An important button in the lower right allows you to load the image on the right into the image on the left. This allows you to explore the infinite nature of the tilings. By choosing a view then loading the image and repeating, you can close in on any hexagon in the tiling.
The animations displayed when deforming do not represent actual projective structures on the surfaces- it is just there as a flashy effect, and is just linearly interpolating between the two images. Drawings after the animation completes however do represent projective structures on the genus two surface.