This course is an exercise session for the lecture course [ZUA.B301 : Geometry I], from which the materials for exercise are chosen.
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 |
Students are given exercise problems related to what is taught in the course [Geometry I].
Course schedule | Required learning | |
---|---|---|
Class 1 | Discussion session on the following materials: the definition of manifolds, examples of manifolds (spheres) | Details will be provided during each class session. |
Class 2 | Discussion session on the following materials: examples of manifolds (examples which are not spheres), differentiable structures | Details will be provided during each class session. |
Class 3 | Discussion session on the following materials: smooth functins 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 | Discussion session on the following materials: 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 | Discussion session on the following materials: real projective spaces, curves on real projective spaces | Details will be provided during each class session. |
Class 6 | Discussion session on the following materials: complex projective spaces, the definition of tangent vectors | Details will be provided during each class session. |
Class 7 | Discussion session on the following materials: the definition of tangent spaces, vector space structure on tangent spaces | Details will be provided during each class session. |
Class 8 | Eevaluation of progress | Details will be provided during each class session. |
Class 9 | Discussion session on the following materials: the differential of a map, regular points, critical points | Details will be provided during each class session. |
Class 10 | Discussion session on the following materials: inverse function theorem, the inverse image of a regular value, Sard's theorem | Details will be provided during each class session. |
Class 11 | Discussion session on the following materials: immersion, embedding | Details will be provided during each class session. |
Class 12 | Discussion session on the following materials: relationship between submanifolds and embeddings | Details will be provided during each class session. |
Class 13 | Discussion session on the following materials: Whitney's embedding theorem, partition of unity | Details will be provided during each class session. |
Class 14 | Discussion session on the following materials: vector field, bracket, integral curves of vector fields | Details will be provided during each class session. |
Class 15 | Discussion session on the following materials: 1 parameter groups of transformations | 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, oral presentation for exercise problems. Details will be provided during class sessions.
Students are expected to have passed Set and Topology II, Advanced Calculus I.
Strongly recommended to take [ZUA.B301 : Geometry I ](if not passed yet) at the same time