Total curvature and length estimate for curves in CAT( K) spaces

We introduce the notion of total curvature of curves (which agrees with the usual one in the piecewise smooth case) in spaces of Alexandrov curvature bounded above. Basic properties of total curvature, including rectifiability of curves of finite total curvature and additivity of total curvature, ar...

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Bibliographic Details
Published in:Differential geometry and its applications 2003-09, Vol.19 (2), p.211-222
Main Authors: Maneesawarng, Chaiwat, Lenbury, Yongwimon
Format: Article
Language:English
Subjects:
CAT( K)
Curvature bounded above
Length estimate
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Summary:We introduce the notion of total curvature of curves (which agrees with the usual one in the piecewise smooth case) in spaces of Alexandrov curvature bounded above. Basic properties of total curvature, including rectifiability of curves of finite total curvature and additivity of total curvature, are then obtained. A sharp upper estimate of a type due to Schmidt on the length of a curve in a CAT( K) space is also given in terms of its total curvature and the distance between its endpoints.
ISSN:0926-2245
1872-6984
DOI:10.1016/S0926-2245(03)00031-7