Ideal simplex

The ideal simplex is a term from geometry and describes a simplex with "corners at infinity".

Ideal simplices in hyperbolic geometry

definition

Let it denote the -dimensional hyperbolic space and its geodetic boundary . ${\ displaystyle H ^ {n}}$ ${\ displaystyle n}$ ${\ displaystyle \ partial _ {\ infty} H ^ {n}}$

An ideal simplex is a geodetic simplex in whose corners are in. ${\ displaystyle {\ overline {H ^ {n}}}: = H ^ {n} \ cup \ partial _ {\ infty} H ^ {n}}$ ${\ displaystyle \ partial _ {\ infty} H ^ {n}}$

One can show that for every tuple there is an ideal -Simlex with corners . ${\ displaystyle (v_ {0}, \ ldots, v_ {k}) \ in (\ partial _ {\ infty} H ^ {n}) ^ {k + 1}}$ ${\ displaystyle k}$ ${\ displaystyle v_ {0}, \ ldots, v_ {k}}$

Dimension 2

All ideal triangles in the hyperbolic plane (or in a higher-dimensional hyperbolic space) are isometric. This is a direct consequence of the fact that the group of orientation- preserving isometries of the hyperbolic plane has a transitive effect on it. ${\ displaystyle PSL (2, \ mathbb {R})}$ ${\ displaystyle 3}$ ${\ displaystyle \ partial _ {\ infty} H ^ {2} = \ mathbb {R} P ^ {1}}$

Dimension 3

Non-degenerate ideal tetrahedra in 3-dimensional hyperbolic space are classified, except for isometry, by the double ratio of their corners. This, too, is a direct consequence of the fact that the group of orientation- preserving isometries of hyperbolic space has a transitive effect on. ${\ displaystyle 4}$ ${\ displaystyle PSL (2, \ mathbb {C})}$ ${\ displaystyle 3}$ ${\ displaystyle \ partial _ {\ infty} H ^ {3} = \ mathbb {C} P ^ {1}}$

Regular simplices

An ideal simplex with vertices is regular , if for every permutation of the corners is an isometric view with there. ${\ displaystyle (v_ {0}, \ ldots, v_ {k}) \ in (\ partial _ {\ infty} H ^ {n}) ^ {k + 1}}$ ${\ displaystyle \ pi}$ ${\ displaystyle g \ colon H ^ {n} \ to H ^ {n}}$ ${\ displaystyle g (v_ {i}) = \ pi (v_ {i})}$

volume

The ideal simplices of maximum volume are exactly the regular ideal simplices. In particular, there is an upper bound for the volume of ideal simplices in hyperbolic space. ${\ displaystyle n}$

Generalizations

More generally, ideal simplices in simply connected spaces of non-positive section curvature can also be defined as geodetic simplices with corners in the edge at infinity .

In simply connected spaces of negative section curvature there is an ideal simplex with these corners for every tuple of points in the boundary at infinity. In general, this does not have to be the case with only non-positive cutting curvature.

literature

Benedetti, Riccardo; Petronio, Carlo: Lectures on hyperbolic geometry. University text. Berlin etc .: Springer-Verlag. (1992)
Ratcliffe, John G .: Foundations of hyperbolic manifolds. 2nd ed. Graduate Texts in Mathematics 149. New York, NY: Springer ( ISBN 0-387-33197-2 ) (2006).
Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor: Manifolds of nonpositive curvature. Progress in Mathematics, 61. Boston-Basel-Stuttgart: Birkhäuser. (1985)

Individual evidence

↑ Haagerup, Uffe; Munkholm, Hans J .: Simplices of maximal volume in hyperbolic n-space. (English) Acta Math. 147, 1-11 (1981).
↑ Peyerimhoff, Norbert: Simplices of maximal volume or minimal total edge length in hyperbolic space. J. Lond. Math. Soc., II. Ser. 66, no. 3, 753-768 (2002).

[1] Haagerup, Uffe; Munkholm, Hans J .: Simplices of maximal volume in hyperbolic n-space. (English) Acta Math. 147, 1-11 (1981).

[2] Peyerimhoff, Norbert: Simplices of maximal volume or minimal total edge length in hyperbolic space. J. Lond. Math. Soc., II. Ser. 66, no. 3, 753-768 (2002).