國立中央大學一百零三學年度下學期幾何學ㄧ課程網頁


宣佈事項：

 6/24(三)上課時間舉行期末考！ 考試範圍 ：Section 2.3, 2.4 and 3.1

5/13上課時間舉行第二次段考！
考試範圍：Section 1.1, Section 1.2, Section 1.3(Fundamental theorem of curves, Hopf's rotation index them, rotation index), Section 2.1 and 2.2

段考二跟期末考上課講義： Differential Geometry: A First Course in Curves and Surfaces, Lectures notes by Professor Theodore Shifrin, Link

考試 : 段考一(含解答), 段考二(含解答), 期末考

作業：作業1 (due 3/12), 作業2 (due 3/19), 作業3 (due 3/26), 作業4 (due 4/16), 作業5 (due 4/23), 作業6 (due 4/30), 作業7 (due 5/7), 作業8 (due 5/28), 作業9 (due 6/10), 作業10 (due 6/17)
 
課程進度 :
 
Part I : Analytic Geometry:
Week1:
2/25: Introduction, conic sections, focus-directrix definition of the non-degenerate conics (parabola ,ellipse)
2/26: Focus-directrix definition of the non-degenerate conics (hyperbola), Recognizing conics
Week2:
3/4: Recognizing conics (continued), Euclidean geometry (Geometry and transformation)
3/5: Euclidean geometry (continued)
Week3:
3/11: Affine geometry
3/12: Projective plane
Week4:
3/18: Projective transformation
3/19: Fundamental theorem of projective  geometry
Week5:
3/25: Cross-ratio, projective conics
3/26: Kleinian view of geometry, Review for midterm 1
Week6:
4/1: 段考一

Part II : Differential Geometry:
4/2: Curves, examples
Week7:
4/8: Arclength parametrization, Frenet frame for arclength-parametrized curves
4/9: Frenet frame for non-arclength-parametrized curves
Week8:
4/15: Some theoretic consequences of the Frenet formulas, Fundamental theorem of curve
4/16: Hopf Umlaufsatz (Hopf rotation theorem)
Week9:
4/22: Parametrization surfaces
4/23: First fundamental form
Week10:
4/29: The second fundamental form, Euler's formula
4/30: Meusnier's formula, examples
Week11:
5/6: Review for midterm II
5/7: Review for midterm II
Week12:
5/13: 段考二
5/14: Christoffel symbols
Week13:
5/20: Codazzi and Gauss equations, Gauss's Theorema Egregium, Fundamental theorem of surface theory
5/21: Covariant differentiation, parallel translation
Week14:
5/27: Parallel translation (continued)
5/28: Geodescis
Week15:
6/3: Geodesics(continued), Clairaut's relation
6/4: Holonomy
Week16:
6/10: Local and global Gauss-Bonnet theorem
6/11: Local and global Gauss-Bonnet theorem (continued)
Week17:
6/17: Review for final
6/18: Review for final
Week18:
6/24: 期末考


Office hour and Office:

黃榮宗老師：星期三下午1點至2點, 星期四下午1:30至2:30或預約, 鴻經館418, 分機：65152

林京民助教：預約，鴻經館201, 分機：65145

課本：

1. 段考一： Geometry by D. A. Brannan, M. F. Esplen, J. J. Gray, 2nd edition), 上課講義請至LMS系統下載
2. 段考二及期末考： Differential Geometry: A First Course in Curves and Surfaces, Lectures notes by Professor Theodore Shifrin, Link

參考書：

1. 邱鴻麟老師講義  
2. Elementary Differential Geometry, by Andrew Pressley, 2nd edition. Springer
授課大綱及建議習題：

Part I: Analytic Geometry

1. Conic Section: 建議習題1, 建議習題2(講義2中的Problem 1,2,3), 建議習題3
2. Affine Geometry: 建議習題(講義中所有的Problems)
3. Projective Geometry 建議習題(講義中所有的Problems)

Part II: Differential Geometry (Lecture notes by Professor Theodore Shifrin)

Chapter 1. Curves:
    1. Examples, Arclength Parametrization: 1, 2, 4, 5, 6, 7, 8, 12
    2. Local Theory: Frenet Frame: 1, 2, 3, 4, 5, 6, 8, 10, 11
    3. Some Global Results : 1, 3, 4
Chapter 2. Surfaces: Local theory
    1. Parametrized Surfaces and the First Fundamental Form: 1, 2, 3, 4, 5, 6, 7, 8, 11(a), 15, 16(a)
    2. The Gauss Map and the Second Fundamental Form: 1, 2, 3, 4(Find the principal curvatures and the principal directions only), 6, 7, 10(b), 13
    3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory: 1, 2, 3, 4, 5, 12, 18(a)(c)(d), 19
    4. Covariant Differentiation, Parallel Translation, and Geodesics: 1, 2, 3, 4, 6, 9, 10, 19
3. Surfaces: Global theory
    1. Holonomy and the Gauss-Bonnet Theorem:1, 2, 3, 6, 8(a)(b)(c)(d), 9(a)(b)(c)(d), 11


評量配分比重：
  
作業20%+第一次期中考25%+第二次期中考25%+期末考30%