The aim of this course is to familiarize the students with basic notions and properties on diferentiable manifolds, which are important not only in mahematics but in related areas such as theoretical physics. It is not easy for beginners to comprehend these abstract notions without suitable training. We supply many concrete examples in each lecture.
The contents of this course is as follows: definition and examples of manifolds, smooth functions on manifolds, smooth maps between manifolds, constructing manifolds by using the inverse images of regular values, definition of tangent vectors and tangent spaces. Each lecture will be accompanied by a problem solving class. This course will be succeeded by [MTH. B302 : Geometry II] in the second quater.
Students are expected to
・understand the definition of manifolds.
・know more than 5 examples of manifolds.
・understand the definitions of smooth functions on manifolds, and smooth maps between manifolds.
・be familiar with the method of constructing manifolds by using the inverse images of regular values.
・understand the definition of tangent vectors and tangent spaces.
Manifolds, differentiable structures, smooth function, smooth map, regular value, projective space, tangent vector, tangent space
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course accompanied by discussion sessions
Course schedule | Required learning | |
---|---|---|
Class 1 | The definition of manifolds, examples of manifolds (spheres) | Details will be provided during each class session. |
Class 2 | Discussion session | Details will be provided during each class session. |
Class 3 | Examples of manifolds (examples which are not spheres), differentiable structures | Details will be provided during each class session. |
Class 4 | Discussion session | Details will be provided during each class session. |
Class 5 | Smooth functions and maps, construction of manifolds as the inverse image of a regular value of a map | Details will be provided during each class session. |
Class 6 | Discussion session | Details will be provided during each class session. |
Class 7 | Proof of the fact that the inverse image of a regular value is a manifold | Details will be provided during each class session. |
Class 8 | Discussion session | Details will be provided during each class session. |
Class 9 | Real projective spaces, curves on real projective spaces | Details will be provided during each class session. |
Class 10 | Discussion session | Details will be provided during each class session. |
Class 11 | Complex projective spaces, the definition of tangent vectors | Details will be provided during each class session. |
Class 12 | Discussion sesssion | Details will be provided during each class session. |
Class 13 | The definition of tangent spaces, vector space structure on tangent spaces | Details will be provided during each class session. |
Class 14 | Discussion session | Details will be provided during each class session. |
Class 15 | Evaluation of progress | Details will be provided during each class session. |
None required
Yozo Matsushima, Differentiable Manifolds (Translated by E.T. Kobayashi), Marcel Dekker, Inc., 1972
Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, 1983
Final exam and discussion session. Details will be provided during class sessions.
Students are expected to have passed [Introduction to Topology Ⅲ, Ⅳ], and [Introduction to Analysis Ⅰ, Ⅱ].