微分幾何一

這是一門微分幾何的基礎課程,在這門課我們希望能將近代微分幾何中最基本的知識介紹給同學. 這些基礎知識也是在各種不同的幾何領域(例如黎曼幾何或辛幾何)中所需要的共同基礎.我們期望 這些內容能幫助同學在接下來學習任何相關領域時已具備足夠的幾何基礎背景. This is an introductory course on Differential manifold. We wish to cover some fundamental and commonly used knowledge in varies fields such as Riemannian geometry, Symplectic geometry, etc. so that the students may have enough background for more advanced geometry courses. The material that will be covered in the course includes (but not restricted to) the following: 1. Smooth manifolds, maps between manifolds 2. Tangent spaces and cotangent spaces 3. Vector Bundles 4. Lie groups and Lie algebras 5. Differential forms 6. De Rham cohomology, Differential complex Integration and Stokes's Theorem will be taught in the second semester, together with Hodge theory etc. Course keywords: manifolds, tangent bundles, cotangent bundles, vector fields, differential forms Syllabus will be posted on the course website: http://www.math.nthu.edu.tw/~nankuo/DG2026F.html 暑假建議同學請先讀第1到第4章及附錄A&B，1到4章是在歐氏空間上的情形，只要修過高微與線代 即可了解，附錄A&B是複習高微學過的點集拓樸和反函數定理。 ---------------------------------------------------------------------------- ----------- 一、課程說明(Course Description) 這是一門微分幾何的基礎課程,在這門課我們希望能將近代微分幾何中最基本的知識介紹給同學. 這些基礎知識也是在各種不同的幾何領域(例如黎曼幾何或辛幾何)中所需要的共同基礎.我們期望 這些內容能幫助同學在接下來學習任何相關領域時已具備足夠的幾何基礎背景. This is an introductory course on Differential manifold. We wish to cover some fundamental and commonly used knowledge in varies fields such as Riemannian geometry, Symplectic geometry, etc. so that the students may have enough background for more advanced geometry courses. The material that will be covered in the course includes (but not restricted to) the following: 1. Smooth manifolds, maps between manifolds 2. Tangent spaces and cotangent spaces 3. Vector Bundles 4. Lie groups and Lie algebras 5. Differential forms 6. De Rham cohomology, Differential complex Integration and Stokes's Theorem will be taught in the second semester, together with Hodge theory etc. 二、指定用書(Text Books) L.Tu, An Introduction to Manifolds, UTX. (easy to read) 三、參考書籍(References) 1. D. Barden and C. Thomas, An Introduction to Differential Manifolds. (easy to read) 2. S. Morita, Geometry of Differential forms. (easy to read) 3. S. Kobayashi, Fundations of Differential Geometry I & II. 4. Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM. 5. W.Boothby, An Introduction to Differential Manifolds and Riemannian Geometry. 6. V.Guillemin and A.Pollack, Differential Topology. 7. I.M.Singer and J.A.Thorpe, Lecture notes on Elementary Topology and Geometry, UTM. 8. R.Bott and L.Tu, Differential Forms in Algebraic Topology, GTM. 9. M.Spivak, A Comprehensive Introduction to Differential Geometry (I). 四、教學方式(Teaching Method) Traditional， 有習題課(星期三第9堂)。 五、成績考核(Evaluation) 期中考50%，期末考50%。 六、 採用下列何項 AI 使用規則 (Indicate which of the following options you use to manage student use of the AI) 本課程無涉及AI使用 Not applicable。 七、其他細節會在課程網頁上公佈及更新 http://www.math.nthu.edu.tw/~nankuo/DG2026F.html

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