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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.
紀要論文
収録刊行物
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- Ryukyu mathematical journal
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Ryukyu mathematical journal 24 1-17, 2011-12
Department of Mathematical Science, Faculty of Science, University of the Ryukyus
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詳細情報 詳細情報について
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- CRID
- 1050855676756480000
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- NII論文ID
- 120003851590
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- NII書誌ID
- AA10779580
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- ISSN
- 1344008X
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- HANDLE
- 20.500.12000/23589
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- 本文言語コード
- en
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- 資料種別
- departmental bulletin paper
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- データソース種別
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- IRDB
- CiNii Articles