I.The Basic Information of the Course
Course Number:202012420009
The English Name of the Course：Differential Geometry
The Chinese Name of the Course：微分几何
In-class Hours and Allocation: total class hours:48, classroom teaching: 40 class hours, classroom discussion: 8 class hours
Credit(s): 3
Semester: 1
Applied Discipline(Professional Degree Category)：Mathematics
Course Object Oriented：Academic Doctor
Evaluation Mode:Process Evaluation
Teaching Method:Seminar-style Teaching, Case Teaching
Course Opening Department:College of Mathematical Sciences
Notes: Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “All Disciplines”, Mathematics general course may fill in Science and Engineering Discipline, and the other courses fill in “Name of Applicable Discipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course
Point Set Topology, Functions of Real Variable, Functional Analysis
III. The Objectives and Requirements of the Course
Sample as follows：
This course aims to introduce the basic concepts and tools of differential geometry, including submanifolds, vector bundles, vector fields, tangent maps, tensor fields, external differential forms, and integrals on manifolds. Basic ideas and basic methods, introduces the method of applying differential geometry knowledge to solve mathematical problems.
By learning this course, the students should master the basic concepts, basic ideas and methods of differential geometry, understand the geometric meaning of differential in differential geometry, and have the basic ability to apply basic knowledge of differential geometry to solve mathematical problems.
IV. The Content of the Course
Chapter 1 Introduction (6 class hours)
Introduction to the development of differential geometry, n-dimensional Euclidean space, smooth mapping, curvilinear coordinates, tensor.
Chapter 2 Differential Manifold（10 class hours）
Some basic concepts of differential geometry, examples of common differential manifolds, mapping, immersion, embedding, differential homeomorphism, submanifolds and regular submanifolds, unit decomposition
Chapter 3 Tangent Vector Field（12 class hours）
Basic concepts of vector bundles, tangent vectors, tangent spaces, tangent bundles, tangent maps and their matrices, tangent vector fields and integral curves under local coordinate bases
Chapter 4 Integral and Stokes Theorem in External Differential Form（12 class hours）
Induction of external differential forms and submanifolds, orientation of manifolds and induction of submanifolds, definition of integrals of external differential forms on manifolds, Stokes theorem and its proof, integration on manifolds and Stokes theorem;
Chapter 5 Lie Group Preliminary（8 class hours）
Lie Group, Introduction to Fiber Bundles, Dual Space and Tensor Space
Course for Ideological and Political Education: Taking matlab software banned by mathwork in our school, and a group of our school having to pay a high fee for the use of numerical simulation software every year as an example to illustrate the role of the differential manifold course content in the grid construction of numerical simulation software The basic important role, and at the same time inform students that the purpose of the country to sort out the "stuck neck" technology is to break the foreign technology monopoly, break the unfriendly foreign forces to suppress the undesirable attempts to suppress China’s development, and call on students to actively devote themselves to the motherland’s scientific research. China's rise is escorted, and for the early realization of the great Chinese dream, study hard now and work hard in the future.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Xu Shenlin, Xue Chunhua. Manifold. Higher Education Press, 1991.
2. Yang Wannian. Differential Manifolds and Their Applications. Chongqing University Press, 1992.
3. Chen Weiheng. Differential Geometry (2ndEdition), Peking University Press, 2017.
4. Chen Weiheng. Differential Manifold (2nd Edition), Higher Education Press,2001.
5.Qiu Chengtong, Sun Licha. Differential Geometry, Higher Education Press, 2018.
B. Learning Resources
http://www.icourse163.org/course/XZNU-1206700841
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Leader in charge of teaching at the College (Signature)：
Date：
中文
