Tip (differential geometry)

Manifold with a tip

In the mathematical field of differential geometry , certain "very rapidly narrowing ends" are called peaks .

definition

A peak is one end of a Riemannian manifold , the surrounding area of which is a distorted product ${\displaystyle E}$ ${\displaystyle (M,g_{M})}$

${\displaystyle E\cong N\times \left[0,\infty \right)}$

can be parameterized with

${\displaystyle g_{M}=dt^{2}+e^{-2t}g_{N}}$.

Here is a Riemannian manifold of dimension and is the parameter of the second factor in . ${\displaystyle (N,g_{N})}$ ${\displaystyle \operatorname {dim} (N)=\operatorname {dim} (M)-1}$ ${\displaystyle t\in \left[0,\infty \right)}$ ${\displaystyle N\times \left[0,\infty \right)}$

The manifold is called the cusp cross section. ${\displaystyle N}$

Examples

• Peaks of -dimensional hyperbolic manifolds have a -dimensional flat manifold as cross section . ${\displaystyle n}$ ${\displaystyle (n-1)}$
• Peaks of -dimensional complex-hyperbolic manifolds have a manifold as a cross-section, the universal superposition of which is isometric to the -dimensional Heisenberg group . ${\displaystyle 2n}$ ${\displaystyle (2n-1)}$
• Peaks of quaternionic-hyperbolic manifolds have a manifold as cross section, the universal superposition of which is isometric to the quaternionic Heisenberg group .
• In a locally symmetrical space of finite volume and rank , all ends are points; their cross-section can be determined from the Langlands decomposition . ${\displaystyle \Gamma \backslash G/K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle rk_{\mathbb {Q} }(\Gamma )=1}$

Vertices of hyperbolic manifolds

The tips of a hyperbolic manifold are isometric to a submanifold of shape ${\displaystyle M=\Gamma \backslash H^{n}}$

${\displaystyle \Gamma _{c}\backslash H_{c}}$,

where is a horoball around a point at infinity and a discrete group of parabolic isometries with a fixed point . ${\displaystyle H_{c}}$ ${\displaystyle c\in \partial _{\infty }H^{n}}$ ${\displaystyle \Gamma _{c}\subset \Gamma }$ ${\displaystyle c}$

So that it is a point in the sense of the above definition, must

${\displaystyle \operatorname {rank} (\Gamma _{c})=\operatorname {dim} (H_{c})=\operatorname {dim} (M)-1}$ be.

In hyperbolic geometry (and more generally in the theory of locally symmetrical spaces ), an edge point at infinity is often referred to as a point even if there is a non-trivial, discrete group of parabolic isometries with a common fixed point. So you don't ask that is. ${\displaystyle X}$ ${\displaystyle c\in \partial _{\infty }X}$ ${\displaystyle \Gamma _{c}}$ ${\displaystyle c}$ ${\displaystyle \operatorname {rank} (\Gamma _{c})=\operatorname {dim} (M)-1}$

For a peak defined in this sense , one can also consider the quotient , which is one end of the manifold . If is, then that end has infinite volume and is not of the shape defined above. One then speaks of a top rank . ${\displaystyle \Gamma _{c}\backslash H_{c}}$ ${\displaystyle \Gamma \backslash X}$ ${\displaystyle r:=\operatorname {rank} (\Gamma _{c})<\operatorname {dim} (X)-1}$ ${\displaystyle r}$

From Margulis' lemma it follows that the thin part of an orientable hyperbolic 3-manifold is either a tip of rank or or a tube environment of a closed geodesic . The peaks of rank 1 are homeomorphic to with , the peaks of rank 2 are homeomorphic to for the torus . ${\displaystyle M_{<\epsilon }}$ ${\displaystyle M}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle D^{*}\times R_{+}}$ ${\displaystyle D^{*}=\left\{z\in \mathbb {C} \colon 0<z<1\right\}}$ ${\displaystyle T^{2}\times \mathbb {R} _{+}}$ ${\displaystyle T^{2}}$

literature

Michail Kapovich : Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition. Modern Birkhäuser Classics. Birkhauser Boston, Inc., Boston, MA, 2009. ISBN 978-0-8176-4912-8

See also

• Point (hyperbolic geometry)