Special Mathematics Lecture

Contact:
Serge Richard (richard@math.nagoya-u.ac.jp), Rm. 237 in Sci. Bldg. A

Differential geometry (Fall 2018)

Registration code : 0063611
Schedule : Wednesday (18:30 - 20.00) in room 207 of the Science Building A
Registration: See page 15 of this document
Additional support for Japanese students: some information

• Syllabus : the pdf file

• Class dates :
  October 3, 10, 17, 24, 31
  November 7, 14, 21, 28
  December 12, 19
  January 9, 16, 23
  

• Program :
  I) Differentiable manifolds: Lec. 1 App. 1, Lec. 2 App. 2, Lec. 3 App. 3, Lec. 4 App. 4
  II) Tensors, tensor fields and differential forms: Lec. 5 App. 5, Lec. 6 App. 6, Lec. 7 App. 7, Lec. 8 App. 8
  III) Integration on manifolds: Lec. 8, Lec. 9, App. 9
  IV) Riemannian manifolds: Lec. 9, Lec. 10 App. 10, Lec. 11 App. 11 Lec. 12
  V) Curvature: Lec. 12 App. 12, Lec. 13, App. 13, Lec. 14
  VI) General relativity Lec. 14

• Study sessions     /     trailer for prospective Japanese students

  Thursday at 6.30 in room 207 of Science Building A, with Liyang and Chang
  Monday at 6.30 in room 207 of Science Building A, with Tsuzu-san and Quang
  Tuesday at 6.30 in room 207 of Science Building A, with Song Ha and Cong Bui
  

• The cumulative notes : Notes taken by Liyang Zhang, with the table of content and the appendices: Differential geometry (this file is 35 MB)

• For the evaluation, you need to submit the solutions of some exercises and/or the proofs of some statements. These submissions can take place at any time during the semester. If you have any question, please contact me or Syoji Toyota.

• Works submitted by the students :
  Examples of smooth manifolds, by Liyang Zhang
  A homomorphism between tangent spaces, by Bui Dang Cong and Arata Suzuki
  Homomorphisms of vector space and algebra, Song Ha Nguyen, Yuki Katsurada, and Duc Truyen Dam
  On the dual of a vector space, by Koichi Kato
  On the dual of a vector space, by Yoshiyuki Endo
  On two definitions of continuity, by Yoshiyuki Endo
  A tangent vector in 2 coordinate systems, by Liyang Zhang
  On the dual of a vector space, by Shihab Fadda
  On the dual of a vector space, Song Ha Nguyen, Arata Suzuki, and Duc Truyen Dam
  Equivalence of 2 definitions of continuity, by Tomoya Tatsuno
  Commutators of vector fields, by Sparsh Mishra
  On the wedge product, Song Ha Nguyen, Arata Suzuki, and Bui Dang Cong
  On curvature and torsion, by Liyang Zhang, Chang Sun, Tomoya Tatsuno, Arata Suzuki, Koichi Kato
  On curvature and torsion, by Sparsh Mishra
  On de Rham cohomology, by Takuya Ishibashi
  On the Lie algebra of smooth vector fields, by Yuki Katsurada and Arata Suzuki
  Tensors, Duc Truyen Dam
  On curvature for torsion free connection, by Yuki Katsurada, Arata Suzuki, Song Ha Nguyen, Bui Dang Cong and Duc Truyen Dam
  On affine connection, by Sparsh Mishra
  An example of a topological manifold: the torus, by Yoshiyuki Endo
  On some properties of the curvature tensor, by Chang Sun
  Uniqueness of the maximal atlas, by Saki Takamatsu
  About the existence of a partition of unity, by Yoshihiko Terasawa
  The tangent bundle, by Naohiro Tsuzu
  On two definitions of continuity, and on commutator properties, by Kai Levi Turner
  On two homomorphisms, by Mao Fujiwara
  About tangent space, Lie Bracket, and alternating tensors, by Quang Nhat Nguyen
  On two homomorphisms, by James Boyle
  Riemannian metrics and immersion, by James Boyle
  Properties of tensors, by James Boyle
  On sectional curvatures, by Shihab Fadda
  

• References : (electronic version available upon request)
  [Au] A course in differential geometry, written by T. Aubin. Slightly more difficult, but still short
  [Be] Einstein manifolds, written by A. L. Besse. Extensive chapter 10 on holonomy groups
  [Bo] An introduction to differentiable manifolds and Riemannian geometry, written by W. M. Boothby. Slightly more involved but beautifully written
  [CS] Holonomy groups in Riemannian geometry, written by A. Clarke and B. Santoro. Lecture notes centred on holonomy groups
  [GN] An Introduction to Riemannian Geometry, written by L. Godinho and J. Natario. Certainly the main reference for this course
  [Kl] A course in differential geometry, written by W. Klingenberg. A pedestrian introduction to differential geometry, focussing on low dimensions
  [Pe] The Whitney embedding theorem, written by M. Persson. Looks very good for a graduation thesis
  [St] General relativity, written by N. Straumann. A classic book about general relativity
  [Th] Elementary topics in differential geometry, written by J. A. Thorpe. An accessible introduction to extrinsic differential geometry
  [Tu1] An Introduction to manifolds, written by L. W. Tu. Another excellent reference with useful appendices
  [Tu2] Differential geometry, written by L. W. Tu. Another excellent reference, more advanced
  

• Pictures taken in November :