The aim of this course is to familiarize the students with basic notions and properties on differentiable manifolds, which are important not only in mathematics 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 are 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, differentials of maps, regular values, critical points, inverse function theorem, Sard's theorem, immersions and embeddings, submanifold, partition of unity, vector fields.
We strongly recommend to take this course with [ZUA. B302 : Exercises in Geometry B I].
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.
・understand the definition of differentials of maps between manifolds.
・know more than 3 examples of submanifolds.
・be able to use "Partition of unity''.
・understand the definitions of brackets of vector fields and integral curves of vector fields.
Manifolds, differentiable structures, smooth function, smooth map, regular value, projective space, tangent vector, tangent space, differential of a map, regular value, critical point, inverse function theorem, Sard's theorem, immersion and embedding, Whitney's embedding theorem, partition of unity, vector field, bracket, integral curve, 1-parameter group of transformations
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | The definition of manifolds, examples of manifolds (spheres) | Details will be provided during each class session . |
Class 2 | Examples of manifolds (examples which are not spheres), differentiable structures | Details will be provided during each class session. |
Class 3 | 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 4 | Proof of the fact that the inverse image of a regular value is a manifold | Details will be provided during each class session. |
Class 5 | Real projective spaces, curves on real projective spaces | Details will be provided during each class session. |
Class 6 | Complex projective spaces, the definition of tangent vectors | Details will be provided during each class session. |
Class 7 | The definition of tangent spaces, vector space structure on tangent spaces | Details will be provided during each class session. |
Class 8 | The differential of a map, regular points, critical points | Details will be provided during each class session. |
Class 9 | Inverse function theorem, the inverse image of a regular value, Sard's theorem | Details will be provided during each class session. |
Class 10 | Immersion, embedding | Details will be provided during each class session. |
Class 11 | Relationship between submanifolds and embeddings | Details will be provided during each class session. |
Class 12 | Whitney's embedding theorem, partition of unity | Details will be provided during each class session. |
Class 13 | Vector field, bracket, integral curves of vector fields | Details will be provided during each class session. |
Class 14 | 1 parameter groups of transformations | Details will be provided during each class session. |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
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. Details will be provided during class sessions.
Students are expected to have passed [Set and Topology II] and [Advanced Calculus I].