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Vol.24 (2011) >
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| Title | : | COMPUTER-AIDED VERIFICATION OF THE GAUSS-BONNET FORMULA FOR CLOSED SURFACES |
| Authors | : | Kamiyama, Yasuhiko |
| Authors alternative | : | 神山, 靖彦 |
| Issue Date | : | Dec-2011 |
| Abstract | : | If X, a compact connected closed C^∞-surface with Euler-Poincaré characteristic _X(X), has a Riemannian metric, and if K : X → R is the Gauss-curvature and dV is the absolute value of the exterior 2-form which represents the volume, then according to the theorem of Gauss-Bonnet, which holds for orientable as well as non-orientable surfaces, (2π)/1 ∫_xKdV=_X(X). When X is the standard sphere or torus in R^3 , the Gaussian curvature is well-known and we can compute the left-hand 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 left-hand side numerically for the cases that the genus of X is small. The computer data are sufficiently nice and coincide with the right-hand side without errors. Such nice data are obtained by converting double integrals to infinite integrals. |
| Type Local | : | 紀要論文 |
| ISSN | : | 1344-008X |
| 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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| Vol24p001.pdf | | 946Kb | Adobe PDF | View/Open |
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