The aim of this course is the same as the one of ``Geometry I'': it is to familiarize the students with basic notions and properties on differentiable manifolds.
The contents of this course is as follows: differentials of maps, regular values, critical points, inverse function theorem, Sard's theorem, immersions and embeddings, submanifold, partition of unity, vector fileds. Each lecture will be accompanied by a problem solving class. This course is a continuation of ``Geometry I" in the first quarter and will be succeeded by ``Geometry Ⅲ" in the third quater.
Students are expected to
・understand the definition of defferentials 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.
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 accompanied by discussion sessions
Course schedule | Required learning | |
---|---|---|
Class 1 | the differential of a map, regular points, critical points | Details willbe provided during each class |
Class 2 | dicussion session | Details willbe provided during each class |
Class 3 | inverse function theorem, the inverse image of a regular value, Sard's theorem | Details willbe provided during each class |
Class 4 | discussion session | Details willbe provided during each class |
Class 5 | immersion, embedding | Details willbe provided during each class |
Class 6 | discussion session | Details willbe provided during each class |
Class 7 | relationship between submanifolds and embeddings | Details willbe provided during each class |
Class 8 | discussion session | Details willbe provided during each class |
Class 9 | Whitney's embedding theorem, partition of unity | Details willbe provided during each class |
Class 10 | discussion session | Details willbe provided during each class |
Class 11 | vector field, bracket, integral curves of vector fields | Details willbe provided during each class |
Class 12 | discussion session | Details willbe provided during each class |
Class 13 | 1 parameter groups of transformations | Details willbe provided during each class |
Class 14 | discussion session | Details willbe provided during each class |
Class 15 | evaluation of progress | Details willbe provided during each class |
None required
Yozo Matsushima, Differentiable Manifolds (Translated by E.T. Kobayashi), Marcel Dekker, Inc., 1972
Farnk 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 passes ``Geometry I".