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

Abstract

In metric spaces of curvature bounded above in the sense of Alexan-

drov, 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) ≤ lim inf κ(γm), where γm is a sequence of curves that converges m→∞

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 curva- ture bounded above.