丘成桐中国大学生数学竞赛大纲

2024年4月6日发(作者：云南小学毕业数学试卷真题)

2010年中国大学生数学竞赛（丘成桐教授发起）竞赛大纲

一．Syllabuses for Geometry and Topology

Geometry:

Curves and surfaces

1) Plane curves and space curves

2) The fundamental theorem of curves

3) Concept and examples of surfaces

4) The first and second fundamental forms

5) Normal curvature, principal curvature and the Gauss curvature

6) Orthogonal moving frames and structure equations of surfaces

7) Existence and uniqueness of surfaces

8) Isometric transformation of surfaces

9) Covariant derivatives on surfaces

10) Geodesic curvatures and geodesics, Geodesic coordinates

11) The Gauss-Bonnet formula

12) Laplacian operator on surfaces

Geometry on manifolds

1) Manifolds

2) Vector fields and differentials

3) Tensors and differential forms

4) Stokes formula

5) De Rham theorem

6) Lie derivatives

7) Lie algebras

8) Maurer-Cartan equations

9) Vector bundles