As an informal introduction to Riemannian manifold, geometry of submanifolds in (pseudo) Euclidean spaces is introduced.
Students are expected to learn
- Pseudo Euclidean space.
- Induced metrics on submanifolds in a (pseudo) Euclidean space.
- Covariant derivatives on submanifolds.
- Geodesics on submanifolds.
pseudo Euclidean space, submanifolds, covariant derivatives.
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
A standard lecture course.
Homeworks will be assined for each lesson.
|Course schedule||Required learning|
|Class 1||Bilinear forms||Details will be provided during each class session|
|Class 2||Metric vector spaces||Details will be provided during each class session|
|Class 3||Pseudo Euclidean space||Details will be provided during each class session|
|Class 4||Induced metrics on submanifolds||Details will be provided during each class session|
|Class 5||Vector fields on submanifolds||Details will be provided during each class session|
|Class 6||Covariant derivatives||Details will be provided during each class session|
|Class 7||Geodesics and completeness||Details will be provided during each class session|
No textbook is set.
Lecture note will be provided.
B. O'Neill, Semi-Riemannian Geometry, Academic Press, 1983; ISBN-13: 978-0-12-526740-3
Graded by homeworks
Knowledge on differential geometry of curves and surfaces (as in MTH.B211 "Introduction to Geometry I" and MTH.B212 "Introduction to
Geometry II", or Sections 1 to 10 of the text book "Differential Geometry of Curves and Surfaces" by M. Umehara and K.
Yamada), and knowledge of fundamental notions of differentable manifolds (MTH.301/MTH.302) are required.
N/A. Contact by E-mails, or at the classroom.
For details, visit the web-site of this class http://www.math.titech.ac.jp/~kotaro/class/2019/geom-a