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
- .
The limit exists because it increases monotonically and is bounded by above.
In a way, it measures the distance of a point from the infinitely distant point .
Horospheres
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.
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.
properties
is a Lipschitz function with Lipschitz constant .
If is a Hadamard manifold , then is twice continuously differentiable and concave (for every geodesic ).
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 .
literature
- Herbert Busemann : The geometry of geodesics. Academic Press Inc., New York, NY, 1955.
Web links
- Busemann function (Encyclopedia of Mathematics)