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

✍ Scribed by Chaiwat Maneesawarng; Yongwimon Lenbury

• Book ID: 104358216
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 141 KB
• Volume: 19
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(03)00031-7

✦ Synopsis

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.