1 / 82

Foundations of Differential Geometry and General Relativity: A Comprehensive Overview

This course provides an in-depth exploration of the foundational concepts in differential geometry and general relativity. Beginning with the introduction to manifolds and privileged observers, it delves into affine and curved manifolds, homeomorphisms, and the topology of differentiable structures. Students will learn about smooth manifolds, tangent spaces, and the intuitive aspects of curving geometric realities. The course emphasizes mathematical rigor while illustrating concepts through examples, including the stereographic projection on spheres, and introduces differential forms and covariant principles.

ferrol
Télécharger la présentation

Foundations of Differential Geometry and General Relativity: A Comprehensive Overview

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Corso di Relatività GeneraleI Parte Fondamenti di Geometria Differenziale e Relatività Generale

  2. Summary of Riemanian Geometry and Vielbein formulation Manifolds

  3. Introduction to manifolds

  4. Privileged observers and affine manifolds Both Newtonian Physics and Special Relativity have privileged observers Affine Manifold

  5. Affine Manifolds

  6. Curved Manifolds and Atlases The intuitive idea of an atlas of open charts, suitably reformulated in mathematical terms,provides the very definition of a differentiable manifold

  7. Homeomorphisms

  8. Topology invariant underhomeomorphisms

  9. Homeomorphic spaces

  10. Open charts

  11. Picture of an open chart Homeomorphism

  12. Differentiable structure M1

  13. The axiom M2 Transition functions

  14. Picture of transition functions

  15. The axiom M3 Differentiable Manifolds

  16. This is a constructive definition

  17. Smooth manifolds

  18. Complex Manifolds

  19. Example the SN sphere

  20. The stereographic projection

  21. The transition function There are just two open charts and the transition function is the following one

  22. Calculus on Manifolds

  23. Functions on Manifolds

  24. Local description

  25. Gluing Rules

  26. Global Functions

  27. Germs of Smooth Functions

  28. Germs atp2 M

  29. Towards tangent spaces: Curves on a Manifold

  30. Curves

  31. Loops

  32. Tangent vectors at a point p 2 M Intuitively the tangent in p at a curve that starts from p is the curve’s initial direction

  33. Tangent spaces at p 2 M

  34. Example: tangent space at p 2 S2 Let us make this intuitive notion mathematically precise

  35. Tangent vectors and germs

  36. Composed function

  37. Derivative = tangent vector

  38. Differential operator

  39. Tangent vectors and derivations of algebras Algebra of germs

  40. Derivations of algebras

  41. Vector space of derivations

  42. Vectors as differential operators

  43. Vector controvariance We have where

  44. Controvariant versus covariant vectors

  45. Introducing differential forms DEFINITION:

  46. Cotangent space Definition

  47. Differential 1-forms at p 2 M =  dx

  48. Why named differential forms

  49. Covariance

  50. Fibre bundles

More Related