FLUCTUATION OF SECTIONAL
CURVATURE FOR
CLOSED HYPERSURFACES

MARIUS OVERHOLT

Abstract:

Liebmann proved in 1899 that the only closed surfaces in Euclidean three-space that have constant Gauss curvature are round spheres. Thus, if a closed surface in three-space is not a topological sphere, its Gauss curvature must fluctuate. We consider quantitative formulations of this fact, also in higher dimensions.