Special Mathematics Lecture

Contact:
Serge Richard (richard@math.nagoya-u.ac.jp), Rm. 247 in Sci. Bldg. A

Differential geometry (Fall 2024)

Registration code : 0063611
Schedule : Wednesday (18.30 - 20.00) in room 207 of Science building A and on Zoom

• Syllabus : see here

• SML official rule : See here

• NU EMI : Link to the website

• PRESS : One article in the Chunichi Shimbun related to NUEMI and this course, and its translation in The Japan Times

• Class dates :
  October 2, 9, 16, 23, 30
  November 6, 13, 20, 27
  December 4, 11, 18, 25
  January 8, 15
  

• Program (tentative) :
  I) Differentiable manifolds
  II) Tensors, tensor fields and differential forms
  III) Integration on manifolds
  IV) Riemannian manifolds
  V) Curvature
  VI) General relativity
  

• Lecture notes : (updated on a regular basis)
  The cumulative notes

• Weekly summaries : 1, 2, 3, 4, 5, 6, Summary, 7, 8, 9, 10, 11, 12, 13, 14

• Study sessions : Will be organized on an individual basis by some students.

• 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, contact me or Fujii Moe.

• Works submitted by the students :
  Homomorphisms of algebras and tangent spaces, by Rafi Rizqy Firdaus
  Topological spaces and Hausdorff property, by Yamamoto Mei
  Homomorphisms of algebras and tangent spaces, by Koichi Kato
  Proving that ℝⁿ is a topological manifold, by Hadiko Rifqi Aufa Sholih
  Topological spaces, by Elisabeth Bérat
  Topological spaces and topological manifolds, by Théo Lesieur
  Hausdorff property and subspace topology, by Elisabeth Bérat
  On the dual of a finite dimensional real vector space, by Koichi Kato
  Antisymmetry, wedge product, and exterior algebra, by Luk Yat Ming
  Change of coordinate frames, isomorphism of tangent spaces, Lie algebra, by Théo Lesieur
  Tensors, symmetric tensors, alternating tensors, by Park Mingyu
  On the exterior derivative, by Oleh Dmytruk
  Compatibility of the topological definitions with the 'usual' definitions on ℝⁿ , by Adam Matefy
  About tensors, by Elisabeth Bérat
  Tangent and cotangent bundles, by Nguyen Tue Tai
  Mathematics behind cartographic projections, by Ding Muyan
  About De Rham cohomology, by Théo Lesieur
  Proving the existence of a Riemannian metric on any smooth manifold, by Hadiko Rifqi Aufa Sholih
  A few concrete computations on tensors and tensor fields, by Liu Zhe
  The unit circle is a manifold, by Ma Tianyang
  Properties of the Lie bracket, by Hadiko Rifqi Aufa Sholih
  Commutators and vector fields, by Yamamoto Mei and Ding Muyan
  Existence and uniqueness of a Riemannian connection on any Riemannian manifold, by Hadiko Rifqi Aufa Sholih
  Surface Areas, by Ding Muyan
  Stokes' theorem on m-dimensional manifold, by Rafi Rizqy Firdaus
  Smooth vector fields and Lie algebra, by Rafi Rizqy Firdaus and Jongruk Pukdee
  Applications of the generalized Stokes' theorem, by Ding Muyan
  Examples of manifolds not homeomorphic to open subsets of Euclidean spaces, by Iftekhar Alam Jewel
  Open sets define a topology on ℝⁿ, by Harumi Ozaki
  On Koszul formula, by Ding Muyan
  Stokes' Theorem, by Oleh Dmytruk
  Properties of timelike vectors, by Ding Muyan
  Smooth vector fields and Lie algebra, by Harumi Ozaki
  On future and past, by Oleh Dmytruk
  Bianchi identities, with an appendix on the covariant derivative of tensor fields, by Nguyen Tue Tai
  Wedge product and orientation, by Park Mingyu
  On Cartan structure equations, by Hadiko Rifqi Aufa Sholih
  Covariant derivative and a few concrete computations, by Liu Zhe
  Functional: the dual vector space, by Hevidu Samarakoon and Riichi Yatagai
  On Lie bracket, by Ma Tianyang
  Bianchi identity, by Sirawich Saranakomkoop
  

• 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
  [P] Techniques of differential topology in relativity, written by R. Penrose. A very classical book
  [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