# Riemannian Geometry

- Institution: IISc Bangalore
- Department: Mathematics
- Teacher: Prof. Ved Vivek Datar
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111108899)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111108899)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

## Lectures

1. [Angle Sum Property for Triangles](https://invisiblenetwork.ai/learn/nptel/111108899/87301c0094ed) (Video)
2. [Introduction to Curvature of Planar Curves](https://invisiblenetwork.ai/learn/nptel/111108899/362774bad045) (Video)
3. [Curvature of Embedded Surfaces in the 3-Dimensional Space](https://invisiblenetwork.ai/learn/nptel/111108899/2a3c689e2f8a) (Video)
4. [Review of Smooth Manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/0e39f622904d) (Video)
5. [Review of Tangent Spaces and Vector Fields](https://invisiblenetwork.ai/learn/nptel/111108899/77f3b5a40da5) (Video)
6. [Review of Cotangent Spaces, Smooth Forms and Tensors on Manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/c79388ce7edf) (Video)
7. [Introduction to Riemannian Manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/bc3a17554b3a) (Video)
8. [Some More Examples of Riemannian Manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/f764aeb9d428) (Video)
9. [Isometries on Riemannian Manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/81ccd00b5cc9) (Video)
10. [The Flat Torus](https://invisiblenetwork.ai/learn/nptel/111108899/27d0d1d76886) (Video)
11. [Local Orthonormal Frame and the Musical Isomorphisms](https://invisiblenetwork.ai/learn/nptel/111108899/9a669274ff98) (Video)
12. [Inner Products of Smooth Forms and the Riemannian Volume Element](https://invisiblenetwork.ai/learn/nptel/111108899/7b237b1dee9f) (Video)
13. [Introduction to Connections](https://invisiblenetwork.ai/learn/nptel/111108899/a309461ab770) (Video)
14. [Continuing Further with the Discussion on Connections](https://invisiblenetwork.ai/learn/nptel/111108899/762e75d678b9) (Video)
15. [Covariant Differentiation along Smooth Curves](https://invisiblenetwork.ai/learn/nptel/111108899/cb94945303a8) (Video)
16. [Parallel Transport Map](https://invisiblenetwork.ai/learn/nptel/111108899/a07839422e21) (Video)
17. [Levi-Civita Connection](https://invisiblenetwork.ai/learn/nptel/111108899/96d33c36e1e9) (Video)
18. [Covariant Differentiation of Tensors](https://invisiblenetwork.ai/learn/nptel/111108899/3a82feb064ba) (Video)
19. [Riemann Curvature Tensor](https://invisiblenetwork.ai/learn/nptel/111108899/dc1b47e91090) (Video)
20. [Properties of Curvature Tensor](https://invisiblenetwork.ai/learn/nptel/111108899/7741cb5fae2d) (Video)
21. [Curvature of Model Spaces](https://invisiblenetwork.ai/learn/nptel/111108899/6f6596c3ff19) (Video)
22. [Flat Riemannian Manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/094c6fe646a3) (Video)
23. [Sectional Curvature and Curvature Like Tensors](https://invisiblenetwork.ai/learn/nptel/111108899/8ea09e24ebef) (Video)
24. [Ricci Curvature](https://invisiblenetwork.ai/learn/nptel/111108899/70291de107d3) (Video)
25. [Further Discussion on Ricci Curvature, and Scalar Curvature](https://invisiblenetwork.ai/learn/nptel/111108899/1df631b4cca3) (Video)
26. [Geometry of Sub-manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/c4ee513e8731) (Video)
27. [Further Discussion on Geometry of Sub-manifolds](https://invisiblenetwork.ai/learn/nptel/111108899/6a1d21c72808) (Video)
28. [Gauss-Bonnet formulae](https://invisiblenetwork.ai/learn/nptel/111108899/39db88b8d389) (Video)
29. [More Discussion on local Gauss-Bonnet formula](https://invisiblenetwork.ai/learn/nptel/111108899/77cc3865e9bd) (Video)
30. [Continuing Further with the Discussion on local Gauss-Bonnet formula](https://invisiblenetwork.ai/learn/nptel/111108899/148026532956) (Video)
31. [Geodesics](https://invisiblenetwork.ai/learn/nptel/111108899/f30f543ced16) (Video)
32. [Further Discussion on Geodesics](https://invisiblenetwork.ai/learn/nptel/111108899/4e872c222bd6) (Video)
33. [Length and Energy Functionals, and the Variation Vector Field](https://invisiblenetwork.ai/learn/nptel/111108899/3a067599d958) (Video)
34. [First Variation Formula](https://invisiblenetwork.ai/learn/nptel/111108899/8c408d30a187) (Video)
35. [Discussion on Critical Points of the Energy and Length Functionals](https://invisiblenetwork.ai/learn/nptel/111108899/3ac1f1523a43) (Video)
36. [The Exponential Map](https://invisiblenetwork.ai/learn/nptel/111108899/334f4ad19163) (Video)
37. [Geodesic Normal Coordinates](https://invisiblenetwork.ai/learn/nptel/111108899/3449b94ea319) (Video)
38. [The Gauss Lemma](https://invisiblenetwork.ai/learn/nptel/111108899/d9af8e9161c0) (Video)
39. [Local Behaviour of Geodesics](https://invisiblenetwork.ai/learn/nptel/111108899/2fe4743fe31a) (Video)
40. [The Induced Metric Space Structure on a Riemannian Manifold](https://invisiblenetwork.ai/learn/nptel/111108899/b4f278a6aee0) (Video)
41. [The Hopf-Rinow Theorem](https://invisiblenetwork.ai/learn/nptel/111108899/df91045bb8cb) (Video)
42. [The Hopf-Rinow Theorem (Contd.)](https://invisiblenetwork.ai/learn/nptel/111108899/c640bae9abfe) (Video)
43. [The Second Variation Formula](https://invisiblenetwork.ai/learn/nptel/111108899/62c0bc30e0f2) (Video)
44. [Jacobi Fields](https://invisiblenetwork.ai/learn/nptel/111108899/46fe1be219ef) (Video)
45. [Jacobi Fields (Contd.)](https://invisiblenetwork.ai/learn/nptel/111108899/67add3bb673d) (Video)
46. [Taylor Expansion of the Metric in Terms of Geodesic Normal Coordinates](https://invisiblenetwork.ai/learn/nptel/111108899/2b799a5b8fb1) (Video)
47. [Geometric Characterization of the Various Curvature Notions](https://invisiblenetwork.ai/learn/nptel/111108899/8ca5c665dcbc) (Video)
48. [Conjugate Points and the Null Space of a Geodesic](https://invisiblenetwork.ai/learn/nptel/111108899/031361c01c40) (Video)
49. [Obstructions for Geodesics to be Not Minimizing](https://invisiblenetwork.ai/learn/nptel/111108899/f75fd022a733) (Video)
50. [The Segment Domain and the Cut-Locus](https://invisiblenetwork.ai/learn/nptel/111108899/c49f92b259ec) (Video)
51. [Fundamental Groups](https://invisiblenetwork.ai/learn/nptel/111108899/4afb2548950e) (Video)
52. [Covering Spaces](https://invisiblenetwork.ai/learn/nptel/111108899/d2265a8d0bc6) (Video)
53. [Cartan-Hadamard Theorem](https://invisiblenetwork.ai/learn/nptel/111108899/7a167cdf38dd) (Video)
54. [Ambrose Proposition](https://invisiblenetwork.ai/learn/nptel/111108899/c49f8721411d) (Video)
55. [Uniformization of Space Forms](https://invisiblenetwork.ai/learn/nptel/111108899/7acd61909444) (Video)
56. [Jacobi Fields in Space Forms](https://invisiblenetwork.ai/learn/nptel/111108899/926cafed6e01) (Video)
57. [Jacobi Fields Comparison](https://invisiblenetwork.ai/learn/nptel/111108899/4e214e4ce785) (Video)
58. [Proof of Uniformization Theorem for Space Forms](https://invisiblenetwork.ai/learn/nptel/111108899/4848f7866356) (Video)
59. [Myers Diameter Comparison Theorem](https://invisiblenetwork.ai/learn/nptel/111108899/2344ea9fa9eb) (Video)
60. [Schur Theorem for Sectional Curvature and Ricci Curvature](https://invisiblenetwork.ai/learn/nptel/111108899/85d54e6557a6) (Video)