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Title  :  COMPUTERAIDED VERIFICATION OF THE GAUSSBONNET FORMULA FOR CLOSED SURFACES 
Authors  :  Kamiyama, Yasuhiko 
Authors alternative  :  神山, 靖彦 
Issue Date  :  Dec2011 
Abstract  :  If X, a compact connected closed C^∞surface with EulerPoincaré characteristic _X(X), has a Riemannian metric, and if K : X → R is the Gausscurvature and dV is the absolute value of the exterior 2form which represents the volume, then according to the theorem of GaussBonnet, which holds for orientable as well as nonorientable surfaces, (2π)/1 ∫_xKdV=_X(X). When X is the standard sphere or torus in R^3 , the Gaussian curvature is wellknown and we can compute the lefthand side explicitly. Let X be a compact connected closed C^∞surface of any genus. In this paper, we construct an embedding of X into R^3 or R^4 according as X is orientable or nonorientable. We equip X with the Riemannian metric as a Riemannian submanifold of R^3 or R^4. Then, with the aid of a computer, we compute the lefthand side numerically for the cases that the genus of X is small. The computer data are sufficiently nice and coincide with the righthand side without errors. Such nice data are obtained by converting double integrals to infinite integrals. 
Type Local  :  紀要論文 
ISSN  :  1344008X 
Publisher  :  Department of Mathematical Science, Faculty of Science, University of the Ryukyus 
URI  :  http://hdl.handle.net/20.500.12000/23589 
Citation  :  Ryukyu mathematical journal Vol.24 p.1 17 
Appears in Collections  :  Vol.24 (2011)

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