Publication: Total Curvature of Curves in Riemannian Manifolds
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Publication Date
2010
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Springer-Verlag
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Abstract
The total curvature of C2 curves embedded in an arbitrary Riemannian manifold is shown to be the limit of the curvatures of inscribed geodesic polygons. The formula for the total curvature of a curve as the least upper bound of curvatures of inscribed geodesic polygons holds for a manifold of non-positive sectional curvature only.
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A.D. Alexandrov, Yu.G. Reshetnyak, General Theory of Irregular Curves, translated from Russian by L.Ya. Yuzina, Mathematics and Its Applications (Soviet Series), vol. 29, Kluwer Academic Publishers Group, Dordrecht, 1989.
R. Benedetti, M. Dedò, A geometric inequality for the total curvature of plane curves, Geom. Dedicata 22 (1) (1987) 105–115.
R. Benedetti, M. Dedò, Global inequalities for curves and surfaces in three-space, Ann. Mat. Pura Appl. (4) 155 (1989) 213–241.
G.R. Buck, J.K. Simon, Total curvature and packing of knots, Topology Appl. 154 (1) (2007) 192–204.
E. Denne, J.M. Sullivan, Convergence and isotopy type for graphs of finite total curvatures, in: Discrete Differential Geometry, in: Oberwolfach Seminars, vol. 38, Birkhäuser Verlag, Basel/Switzerland, 2008, pp. 163–174.
Y. Diao, C. Ernst, Total curvature, ropelength and crossing number of thick knots, Math. Proc. Cambridge Philos. Soc. 143 (1) (2007) 41–55.
M.P. Do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992.
T. Ekholm, O. Weistrand, Total curvatures of holonomic links, J. Knot Theory Ramifications 9 (7) (2000) 893–906.
M.A. Grayson, The shape of a figure-eight under the curve shortening flow, Invent. Math. 96 (1) (1989) 177–180.
R. Gulliver, Total curvature of graphs in space, Pure Appl. Math. Q. 3 (3) (2007) 773–783, Part 2.
S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, John Wiley & Sons, Inc. (Interscience Division), New York, 1963.
J.W. Milnor, On the total curvature of knots, Ann. of Math. 52 (2) (1950) 248–257.
Yu.G. Reshetnyak, On the theory of irregular curves in n-dimensional Euclidean space, Siberian Adv. Math. 13 (2) (2003) 95–107. Translated from: Proceedings of the Conference “Geometry and Applications” dedicated to the seventieth birthday of V.A. Toponogov, Novosibirsk, 2000, Ross. Akad. Nauk Sib. Otd., Inst. Mat., Novosibirsk, 2001, pp. 158–170 (in Russian).
J.M. Sullivan, Curves of finite total curvature, in: Discrete Differential Geometry, in: Oberwolfach Seminars, vol. 38, Birkhäuser Verlag, Basel/Switzerland, 2008, pp. 137–161.