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Differential Geometry

Instructor: Guanghua Ji
Textbook: R.S. Millman, G.D. Parker, Elements of Differential Geometr, 1977.
Grading: Be based on attendance(10%), homework(20%) and final exam(70%).
Schedule: TUE 3-4, THU 3-4; 知新楼 B115, B119.

1, 陈维桓 编著, 微分几何, 高等教育出版社, 2008.
2, 彭家贵, 陈卿 编著, 微分几何, 高等教育出版社, 2010..

3, C. Bar, Elementary Differential Geometry, Cambridge Univ Press, 2010.

4, A.N. Pressley, Elementary Differential Geometry, Springer, 2012.


I, The Local Theory of Curves

9/23, §1.1, Plane curves: Homework: p13, 2,3; p28, 1,2. §1.2, Space curves: I, Def: tangent vector, arc-length, normal, principal normal, binormal vectors, Frenet trihedron, curvature, torsion.

9/25, II, Frenet formuale and applcations; Homwwork: p28, 4,5,6. §1.3, The fundamental theorem of curves: Existence and uniqueness(up to position). HW:p28,7,13. Homework Supplement

II, The Global Theory of Curves

9/28, §2.1, Simple colsed curves: I, Rotation index theorem; II, isoperimetric inequality.

9/30, §2.2, Plane Convex curves: I Defitions: convex, vertex; II, Four vertex theorem

III, The Local Theory of Surfaces

10/9, §3.1, Regular surfaces: I, Definition, coordinate transformation. II, tangent vector, tangent space, tangent palne, normal vector. HW: p86, 1,5,9; p92, 2.

10/14, §3.2, The First fundamental forms: I, first fundamental form. II, arc length. III, Coordinate transformation. HW: p101, 1,2,7.

10/16, §3.3, Normal Curvatures, Geodesic Curvature and Gauss's formulas: I, Def. II, Gauss's formulas, Christoffel symbols.. HW: p108, 2,5,7,9.

10/21,III, Intrinsic Geometry: the geodesic curvature is intrinsic. §3.4, Geodesics: I, Definition. HW: p115, 3,5.

10/23, II, Geodesics on surface of revolution. III, "How many" geodesics there are. IV, Geodesics as shortest paths. HW: p115,9,11,12. Homework Supplement

10/28, §3.5, Parallel Vector fields I, Tangent vector field, convariant derivative, parallel. II, Parallel translate. III, Maximally straight. HW: p121,2,3,4.

10/30, §3.6, The Second fundamental forms and the Weingarten map: I, The second fund forms. II, The Weingarten map. HW, p127, 3,4,7.

11/4, §3.7, Gaussian and mean Curvatures: I,Principal, Gauss and mean curvatures, line of curvature. II, Geometric interpretation of the Gauss curvature. HW: p136, 1,4,8.

11/6, III, Dupin indicatrix, elliptic, hyperbolic, parabolic points. §3.8, Gauss' Theorema Egregium: I, Rimannian curvature tensor. II, Gauss-Codazzi-Mainardi equations. III, Gauss's Theorem Egregium. HW: p137, 6,11,13; p144, 2,3.

11/11, §3.9, The Fundamental Theorem of Surfaces: I, Isometries. II, Fundamental theorem of surfaces. HW: p152, 1,3,5,Homework Supplement.

11/13, §3.10, Some classes of Surfaces: I, Surfaces of revolution. II, Ruled surfaces. III, Developable surfaces. HW: P138, 19,37,39,46,49,52.

11/18, IV, Minimal surfaces: with smallest area. V, Surfaces of constant Gauss's curvatures: surfaces of revolution. HW: P138, 16,17.

IV, The Global Theory of Surfaces

11/20, §4.1, Compact regular surfaces: I, Gauss map is onto. II, K(p)>0 for some p. III, Meusnier theorem. HW: p176, 1,2,Homework Supplement.

11/25, §4.2, Geodesic coordinate patches: I, Geodesic coordinate patches. II, Constant Gaussian curvature: local isometric. HW: p179, 2,3,4.

11/27, §4.3, Orientability and angular variation: I, Orientable. II, Angular variation. III, Simply connected. HW: p185, 1,Homework Supplement

12/2, §4.4, The Gauss-Bonnet theorem: local version: Proof and applications. HW: p188, 2,3.

12/4, §4.5, The Gauss-Bonnet theorem: global version: I, Gauss-Bonnet thm for compact regular surface. II, Euler characteristic and genus. HW: p191, 2,3.

12/9, §4.6, Applications of Gauss-Bonnet Theorem: I, Jacobi Theorem. II, Hadamard Theorem. III, Poincare-Brouwer Theorem. HW: checking Gauss-Bonnet formula.

12/11, §4.7, Hyperbolic Geometry: hyperbolic plane, K=-1, distance, area. HW

12/16, §4.8, Differential Forms: 1-forms, moving forms, structure equations. HW

V, Differentiable Manifolds

12/18, §5.1, Manifolds: Definitions and Examples. HW: p108,3,4.

12/23, §5.2, Tangent Vectors and Tangnent Space: I, Differentiable functions on maniflds. II, Tangents vectors: algebraic def and geometric def. III, Tangent space: n-dim real vector space, basis. HW: p215, 4,5,7.

12/25, §5.3, Vector Fields and Lie Brackets: vector fields, Lie bracket, Jacobi's identity.

12/30, §5.4, The Differential of a Map and Submanifolds: I, Differential. II, Submanifolds.




    Differential geometry is a mathematical discipline that uses the techniques of differential calculusintegral calculuslinear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. ---wiki


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