THE LOWER SEMI-CONTINUITY OF TOTAL CURVATURE IN SPACES OF CURVATURE BOUNDED ABOVE

Abstract

In metric spaces of curvature bounded above in the sense of Alexandrov, the notions of (pointwise) curvature and total curvature can be defined. Certain properties of them were verified true, whereas many are still left unchecked. These include the lower semi-continuity of total curvature. In other words, it has not been known whether the relation κ(γ) ≤ liminf m→∞ κ(γm), where γm is a sequence of curves that converges to a curve γ, holds in a more general setting than that of the Euclidean space. We present here the validity of this statement in spaces of curvature bounded above.