Hyperbolic Manifold With Geodesic Boundary?

In summary, the conversation discusses how to give a surface a hyperbolic metric with geodesic boundary. It mentions that there are genus constraints and that the surface can be obtained by gluing pairs of pants. The concept of a manifold with totally geodesic boundary is explained, and examples of surfaces with constant negative Gauss curvature and geodesic boundary are given. The possibility of slicing a hyperbolic manifold without boundary along closed geodesic circles is also mentioned.
  • #1
WWGD
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Hi All,
I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric. So far I know:

1) There are genus constraints for the surface. Does this have to see with Gauss-Bonnet?

2) Something; not sure exactly what, can be done by gluing pairs-of-pants http://en.wikipedia.org/wiki/Pair_of_pants_(mathematics [Broken])
but not fully sure how this works.

Not much more. Any ideas, refs., please?
 
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  • #2
Not sure what you mean by the genus of a surface with boundary but maybe this will help.

Every compact hyperbolic surface without boundary that has constant Gauss curvature, -1, is the quotient of the Poincare disk by a group of isometries. Remove a small geodesic polygon from the disk and the quotient will be a surface of constant negative curvature with a geodesic boundary.

For surfaces, the Gauss curvature times the volume element is cohomologous to the Euler class. One can see this from the Gauss Bonnet theorem or more simply by observing that the connection 1 form on the tangent circle bundle is a global angular form so its exterior derivative is the pullback (under the bundle projection map) of the Euler class.

It follows that the Euler characteristic of a hyperbolic surface is always negative so this rules out the torus and the sphere. It is a simple exercise show that also must be even.
 
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  • #3
There are classical surfaces in 3 space that have constant negative Gauss curvature. Examples are the tractroid and Dini's surface.

See if you can slice off part of one of these with a geodesic knife to get a surface with geodesic boundary.
 
  • #4
Thanks Lavinia, for the sake of completeness, let me state the definition of manifold M with totally geodesic boundary is:

A manifold M with non-empty boundary that admits an atlas {## \phi_{\alpha}: U_{\alpha} \rightarrow B_{\alpha} ##} to hyperbolic half-spaces bounded by geodesic hyperplanes ##H_{\alpha} \subset \mathbb H^n ## so that {## U_{\alpha} ##} covers M and every chart satisfies ##\phi_{\alpha}(U_{\alpha} \cap \partial M)=\phi_{\alpha}(U_{\alpha} \cap H_{\alpha}) ## , and overlap maps are restrictions of isometries.
 
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  • #5
lavinia said:
Not sure what you mean by the genus of a surface with boundary but maybe this will help.

Every compact hyperbolic surface without boundary that has constant Gauss curvature, -1, is the quotient of the Poincare disk by a group of isometries. Remove a small geodesic polygon from the disk and the quotient will be a surface of constant negative curvature with a geodesic boundary.

.

Do you mean, if g is the geodesic polygon and P is the Poincare disk , that (P-g)/g has geodesic boundary, and so will this have totally geodesic boundary, i.e., will every boundary component (a simple-closed curve here) be a geodesic?
If so, is this I guess the quotient metric?
Thanks.
 
  • #6
WWGD said:
Do you mean, if g is the geodesic polygon and P is the Poincare disk , that (P-g)/g has geodesic boundary, and so will this have totally geodesic boundary, i.e., will every boundary component (a simple-closed curve here) be a geodesic?
If so, is this I guess the quotient metric?
Thanks.
Yes.

- I think also that with a tractroid you can cut off a neighborhood of the singular circle to get a surface in 3 space of constant negative curvature and geodesic boundary.- You could also just slice a hyperbolic manifold without boundary along one or more closed geodesic circles. For a surface of genus 2, three circles suitably chosen would give you your pair of pants.
 
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1. What is a hyperbolic manifold with geodesic boundary?

A hyperbolic manifold with geodesic boundary is a geometric object that has a constant negative curvature and is bounded by geodesic lines, which are the shortest paths between two points on the surface. It is a type of non-Euclidean manifold and is often studied in the field of differential geometry.

2. How is a hyperbolic manifold with geodesic boundary different from a Euclidean manifold?

A Euclidean manifold has a constant zero curvature, while a hyperbolic manifold with geodesic boundary has a constant negative curvature. Additionally, a Euclidean manifold has no boundary, while a hyperbolic manifold with geodesic boundary is bounded by geodesic lines.

3. What are some real-life applications of hyperbolic manifolds with geodesic boundary?

Hyperbolic manifolds with geodesic boundary have many applications in theoretical physics, such as in the study of black holes and cosmology. They are also used in computer graphics and video game design to create visually interesting 3D environments. In mathematics, they are studied for their geometric properties and can be used to model complex networks, such as the internet.

4. How are hyperbolic manifolds with geodesic boundary studied?

Hyperbolic manifolds with geodesic boundary are studied using various mathematical techniques, including differential geometry, algebraic topology, and complex analysis. Computer simulations and visualizations are also used to better understand their properties and behavior.

5. Are there any open questions or unsolved problems related to hyperbolic manifolds with geodesic boundary?

Yes, there are still many open questions and unsolved problems related to hyperbolic manifolds with geodesic boundary. These include finding new examples of such manifolds, understanding their global structure and properties, and characterizing their behavior under different transformations and deformations.

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