|
By a cone is meant a warped
product I ×gF, where I is an interval and the warping function g : I → ℝ≥0
lies in ℱK, i.e., satisfies g′′ + Kg = 0. Cones include metric products and
linear cones (K = 0), hyperbolic, parabolic, and elliptical cones (K < 0),
and spherical suspensions (K > 0). A cone over a geodesic metric space
supports a natural K-affine function, that is, a function whose restriction to
every unit-speed geodesic is in ℱK. Conversely, the main theorems of this
paper show that on an Alexandrov space X of curvature bounded below
or above, the existence of a nonconstant K-affine function f forces X to
split as a cone (subject to a boundary condition or geodesic completeness,
respectively).
For K = 0 and curvature bounded below, X splits as a metric product with a line;
this case is due to Mashiko (2002). Some special cases for complete Riemannian
manifolds were discovered much earlier: by Obata (1962), for K > 0, with the strong
conclusion that X is a standard sphere; and by Innami (1982), for K = 0.
For K < 0, with the additional assumption that f has a critical point, our
theorem now gives the dual to Obata’s theorem, namely, X is hyperbolic
space.
|
