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課程名稱 |
微分幾何二
Differential Geometry (Ⅱ) |
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開課學期 |
104-2 |
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授課對象 |
理學院 數學系 |
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授課教師 |
蔡宜洵 |
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課號 |
MATH7302 |
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課程識別碼 |
221 U2940 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期三9(16:30~17:20)星期五3,4(10:20~12:10) |
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上課地點 |
天數102天數102 |
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備註 |
研究所數學組基礎課。
總人數上限：30人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1042MATH7302_ |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
http://www.math.ntu.edu.tw/courses/super_pages.php?ID=allcourses |
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課程目標 |
1.Jacobi fields, 2nd variation, Jacobi equation, conjugate points,minimizing property of geodesics, Index Lemma, Jacobi’s theorem, two proofs
2. Myers-Bonnet theorem, Cartan-Hadamard theorem, Rauch comparison theorem with applications to injectivity radius
3. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields
4. Cartan-Ambrose-Hicks Theorem
5, 6, 7. Miscellaneous
a) flows and transformations
b) Killing vector fields
c) volume element and divergence
d) Ricci curvature and volume growth
e) 2nd Bianchi identity applied to Einstein manifolds
f) Cut locus, injectivity radius, Klingenberg’s lemma
8. vector bundles, bundle maps, pull-back bundles, complex vector bundles
9. connection, curvature form, Bianchi identity
10. Chern classes, invariant polynomials, Chern character, unitary connection
11. Examples and application of Chern classes, immersions and embeddings in complex projective spaces
12. Pontrjagin classes, Euler class, relation with Chern classes,Todd class, A-hat genus
13. star operator, Hodge decomposition theorem, Poincare duality,
14. Kunneth formula, Bochner-Weitzenbock formula, proof
15. divergence, application of B-W formula to topology of manifolds, index of de Rham complex, remark on Index theorem
16. Gauge theory, Erlanger Program, historical remarks, principle bundles
17. Examples of Lie groups, SU(2)-bundles, Yang-Mills equation, self-duality equation
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課程要求 |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
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參考書目 |
[dC] do Carmo, Riemannian geometry.
[GHL] Gallot-Hulin-Lafontaines, Riemannian geometry.
[H] Helgason, Differential geometry, Lie groups and symmetric spaces.
[N] Noel J.Hicks, Notes on Differential geometry.
[CE] Cheeger and Ebin, Comparison theorems in Riemannian geometry.
[W] Warner, Foundations of differentiable manifolds and Lie groups.
[BT] Bott and Tu, Differential forms in algebraic topology.
[T]Taubes,C., “Differential geometry”:discusses bundle theory in great
detail
[KN]Kobayashi-Nomizu, “Foundations of Differential geometry”:bundle,principle bundles and related topics.
[BGV]Berline, N., Getzler, E., Vergne, M.,“Heat kernels and Dirac
operators”:a more specialized book on bundle theory and used it for various
index theorems
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評量方式
(僅供參考) |
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