課程名稱 |
微分幾何(一), (二) |
授課對象 |
研究生與高年級大學部同學 |
預備知識 |
高等微積分, 線性代數, 微分方程, 幾何學 |
其他條件 |
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微分幾何是高維度空間的研究中最重要的理論,也是廣義相對論以及部分理論物理學的數學基礎。本課程旨在提供足夠的理論,使同學修完一年的課程之後,有能力學習更專業的幾何研究課程如辛幾何,複幾何,低維度微分拓樸或將微分幾何應用在其他學科上。基本課程包含
- Differentiable Manifolds: Partition of unity, tangent spaces, vector fields and flows, Lie derivatives, Frobenius integrability theorem, Sard’s theorem, Whitney imbedding theorem.
- Tensor Calculus: Lie derivatives of tensors, Cartan’s theory on differential forms, Stokes theorem, de Rham cohomology, de Rham theorem. Introduction to the Hodge theory of harmonic forms.
- Riemannian Geometry: Levi-Civita connection, geodesics, exponential map, Riemann curvature tensor, 1st and 2nd variations of arc length, Hopf-Rinow theorem, Bonnet-Synge theorem, Jacobi fields, Cartan-Hardamard theorem, Cantan-Ambrose-Hicks theorem.
下學期若尚有剩餘的時間,則可能著重於某些專題,如以下之一:
- Vector Bundle Theory: Connections and curvature, Chern-Weil theory on Chern classes, Spinors and Dirac operators, the Atiyah-Singer index theorem.
- Geometric Analysis: Comparison geometry, Bochner principle, eigenvalue estimates, analysis on complete manifolds, introduction to Ricci flows.
- Minimal Submanifolds: 1st and 2nd variations, Weierstrass representations, Platean problem, introduction to calibrated and Kaehler geometry.
以下為常用之參考文獻:(前三本為常用教科書)
- Warner: Foundations of Differential Manifolds and Lie Groups.
- Do Carmo: Riemannian Geometry.
- Jost: Riemannian Geometry and Geometric Analysis.
- Berline et. al.: Heat Kernels and Dirac operators.
- Li: Lecture Notes on Geometric Analysis.
- Lawson: Lectures on Minimal Submanifolds.
- Schoen and Yau: Lectures on Differential Geometry.
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大學部課程介紹 |
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研究所課程介紹 |
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