Busemann function

In Riemannian geometry , a branch of mathematics , the Busemann function is a function that measures the "distance to infinitely distant points". It is named after Herbert Busemann .

definition

Let be a Riemannian manifold and a geodesic parameterized according to arc length . The Busemann function is defined by ${\displaystyle M}$ ${\displaystyle \gamma :\left(0,\infty \right)\rightarrow M}$ ${\displaystyle b_{\gamma }:M\rightarrow \mathbb {R} }$

${\displaystyle b_{\gamma }(x):=\lim _{t\rightarrow \infty }(t-d(x,\gamma (t))),\quad x\in M}$.

The limit exists because it increases monotonically and is bounded by above. ${\displaystyle t-d(x,\gamma (t))}$ ${\displaystyle d(x,\gamma (0))}$

In a way, it measures the distance of a point from the infinitely distant point . ${\displaystyle b_{\gamma }}$ ${\displaystyle \gamma (\infty )}$

Horospheres

The Poincaré model of the hyperbolic plane , various geodesics ending in the same point (in red) and a corresponding horosphere (in blue); the horosphere does not depend on the geodesic, but only on the end point.

The levels of the Busemann function are called horospheres . In the case of surfaces, the (then one-dimensional) horospheres are also referred to as horocycles.

The sub-level sets for are called horoballs . So a horosphere is the edge of a horoball. ${\displaystyle b_{\gamma }^{-1}(\left[a,\infty \right])}$ ${\displaystyle a\in \mathbb {R} }$

The end point at infinity of the geodesics defining the Busemann function is called the midpoint or center of the horospheres and horoballs thus defined. ${\displaystyle \gamma (\infty )}$ ${\displaystyle b_{\gamma }}$

properties

${\displaystyle b_{\gamma }}$is a Lipschitz function with Lipschitz constant . ${\displaystyle 1}$

If is a Hadamard manifold , then is twice continuously differentiable and concave (for every geodesic ). ${\displaystyle M}$ ${\displaystyle b_{\gamma }}$ ${\displaystyle \gamma }$

Conversely, it is convex if has nonnegative section curvature . If has nonnegative Ricci curvature , then it is subharmonic , and if a Kähler manifold with nonnegative holomorphic bicutral curvature is plurisubharmonic . ${\displaystyle b_{\gamma }}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle b_{\gamma }}$ ${\displaystyle M}$ ${\displaystyle b_{\gamma }}$

literature

• Herbert Busemann : The geometry of geodesics. Academic Press Inc., New York, NY, 1955.

Web links

• Busemann function (Encyclopedia of Mathematics)