# An Introduction to Smooth Manifolds

- Institution: IISc Bangalore
- Department: Mathematics
- Teacher: Prof. Harish Seshadri
- Level: Graduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111108134)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111108134)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

The goal of this course is to introduce the student to the basics of smooth manifold theory. The course will start with a brief outline of the prerequisites from topology and multi-variable calculus.

After the introducion of differentiable manifolds, a large class of examples, including Lie groups, will be presented. The course will culminate with a proof of Stokes' theorem on manifolds.

Intended audience: Masters and PhD students in mathematics, physics, robotics and control theory, information theory and climate sciences.

Prerequisites: Real analysis, linear algebra and multi-variable calculus, topology.

## Lectures

1. [Basic linear algebra](https://invisiblenetwork.ai/learn/nptel/111108134/a41b7dd58c13) (YouTube)
2. [Multivariable calculus - 1](https://invisiblenetwork.ai/learn/nptel/111108134/96e42ebf5408) (YouTube)
3. [Multivariable calculus - 2](https://invisiblenetwork.ai/learn/nptel/111108134/3df278b3ba95) (YouTube)
4. [The derivative map](https://invisiblenetwork.ai/learn/nptel/111108134/92b630a994a7) (YouTube)
5. [Inverse Function Theorem](https://invisiblenetwork.ai/learn/nptel/111108134/2930c320113f) (YouTube)
6. [Constant Rank Theorem](https://invisiblenetwork.ai/learn/nptel/111108134/0265bd4d50ca) (YouTube)
7. [Smooth functions with compact support](https://invisiblenetwork.ai/learn/nptel/111108134/ff554ed32e30) (YouTube)
8. [Smooth manifold](https://invisiblenetwork.ai/learn/nptel/111108134/c9469fbcfcf1) (YouTube)
9. [Examples of smooth manifolds](https://invisiblenetwork.ai/learn/nptel/111108134/a72067b87fe6) (YouTube)
10. [Higher dimensional spheres as smooth manifolds](https://invisiblenetwork.ai/learn/nptel/111108134/34de002643fa) (YouTube)
11. [Smooth maps](https://invisiblenetwork.ai/learn/nptel/111108134/b331dae9f866) (YouTube)
12. [Examples of smooth maps](https://invisiblenetwork.ai/learn/nptel/111108134/e395136f049f) (YouTube)
13. [Tangent spaces - I](https://invisiblenetwork.ai/learn/nptel/111108134/7bf1170127c7) (YouTube)
14. [Tangent spaces - 2](https://invisiblenetwork.ai/learn/nptel/111108134/86b23446495d) (YouTube)
15. [Derivatives of smooth maps](https://invisiblenetwork.ai/learn/nptel/111108134/b4b4d9925990) (YouTube)
16. [Chain rule on manifolds](https://invisiblenetwork.ai/learn/nptel/111108134/6a6c28eb1060) (YouTube)
17. [Dimension of tangent space - 1](https://invisiblenetwork.ai/learn/nptel/111108134/7f0ced5c0b42) (YouTube)
18. [Dimension of tangent space - 2](https://invisiblenetwork.ai/learn/nptel/111108134/b4ad69570d66) (YouTube)
19. [Derivative of inclusion map](https://invisiblenetwork.ai/learn/nptel/111108134/fe0525c4d46c) (YouTube)
20. [Basis of tangent space](https://invisiblenetwork.ai/learn/nptel/111108134/a9424067ceea) (YouTube)
21. [Inverse Function Theorem for manifolds](https://invisiblenetwork.ai/learn/nptel/111108134/a81ed4108643) (YouTube)
22. [Submanifolds](https://invisiblenetwork.ai/learn/nptel/111108134/5b76cf380ff5) (YouTube)
23. [Tangent space of a submanifold](https://invisiblenetwork.ai/learn/nptel/111108134/6fb3bb2b968a) (YouTube)
24. [Regular Value Theorem](https://invisiblenetwork.ai/learn/nptel/111108134/0de14f444117) (YouTube)
25. [Special linear group as a submanifold of the set of all square matrices](https://invisiblenetwork.ai/learn/nptel/111108134/b084a407de15) (YouTube)
26. [Hypersurfaces](https://invisiblenetwork.ai/learn/nptel/111108134/e7d24bf0fbea) (YouTube)
27. [Tangent spaces to level sets](https://invisiblenetwork.ai/learn/nptel/111108134/bf5c1d7627ad) (YouTube)
28. [Vector fields - 1](https://invisiblenetwork.ai/learn/nptel/111108134/cea8a88fc120) (YouTube)
29. [Vector fields - 2](https://invisiblenetwork.ai/learn/nptel/111108134/9a50f663aaee) (YouTube)
30. [Vector fields - 3](https://invisiblenetwork.ai/learn/nptel/111108134/6db45225140f) (YouTube)
31. [Lie groups - 1](https://invisiblenetwork.ai/learn/nptel/111108134/2974f734475e) (YouTube)
32. [Lie groups - 2](https://invisiblenetwork.ai/learn/nptel/111108134/bd45ebb97640) (YouTube)
33. [Integral curve and flows - 1](https://invisiblenetwork.ai/learn/nptel/111108134/88143ccdd7f2) (YouTube)
34. [Integral curve and flows - 2](https://invisiblenetwork.ai/learn/nptel/111108134/607bcddfdd90) (YouTube)
35. [Integral curve and flows - 3](https://invisiblenetwork.ai/learn/nptel/111108134/7e1c0f7b3682) (YouTube)
36. [Complete vector fields](https://invisiblenetwork.ai/learn/nptel/111108134/09ada6436a0d) (YouTube)
37. [Vector fields and smooth maps](https://invisiblenetwork.ai/learn/nptel/111108134/ceeff6a57d73) (YouTube)
38. [Lie Brackets - 1](https://invisiblenetwork.ai/learn/nptel/111108134/d7b3dc520b71) (YouTube)
39. [Lie brackets - 2](https://invisiblenetwork.ai/learn/nptel/111108134/5486daab98a1) (YouTube)
40. [Lie brackets - 3](https://invisiblenetwork.ai/learn/nptel/111108134/d526b11c8d30) (YouTube)
41. [Lie algebras of matrix groups - 1](https://invisiblenetwork.ai/learn/nptel/111108134/2dadf4b7b713) (YouTube)
42. [Lie algebras of matrix groups - 2](https://invisiblenetwork.ai/learn/nptel/111108134/2781d8d05539) (YouTube)
43. [Exponential map](https://invisiblenetwork.ai/learn/nptel/111108134/ddc255ddfab5) (YouTube)
44. [Frobenius theorems](https://invisiblenetwork.ai/learn/nptel/111108134/a6bfc4a55e14) (YouTube)
45. [Tensors and differential forms](https://invisiblenetwork.ai/learn/nptel/111108134/ca6e91637225) (YouTube)
46. [Tensors and differential forms - 2](https://invisiblenetwork.ai/learn/nptel/111108134/96cb9c41a26d) (YouTube)
47. [Pull-back form](https://invisiblenetwork.ai/learn/nptel/111108134/1ac0cdd1cb9a) (YouTube)
48. [Symmetric Tensors](https://invisiblenetwork.ai/learn/nptel/111108134/36b0dd3eea61) (YouTube)
49. [Alternating Tensors - 1](https://invisiblenetwork.ai/learn/nptel/111108134/0b3b76870081) (YouTube)
50. [Alternating Tensors - 2](https://invisiblenetwork.ai/learn/nptel/111108134/4946299b1310) (YouTube)
51. [Alternating Tensors -3](https://invisiblenetwork.ai/learn/nptel/111108134/94301cb41c6a) (YouTube)
52. [Alternating Tensors - 4](https://invisiblenetwork.ai/learn/nptel/111108134/e8b4d85ab3cd) (YouTube)
53. [Alternating Tensors - 5](https://invisiblenetwork.ai/learn/nptel/111108134/19a89e6a4c0b) (YouTube)
54. [Alternating Tensors - 6](https://invisiblenetwork.ai/learn/nptel/111108134/205d2474790e) (YouTube)
55. [Alternating Tensors - 7](https://invisiblenetwork.ai/learn/nptel/111108134/2b1b611cc697) (YouTube)
56. [Alternating Tensors - 8](https://invisiblenetwork.ai/learn/nptel/111108134/eea90a17ad03) (YouTube)
57. [Alternating Tensors - 9](https://invisiblenetwork.ai/learn/nptel/111108134/694d3abfe292) (YouTube)
58. [Differential forms on manifolds-1](https://invisiblenetwork.ai/learn/nptel/111108134/f81785a740c4) (YouTube)
59. [Differential forms on manifolds-2](https://invisiblenetwork.ai/learn/nptel/111108134/0d9f718e691b) (YouTube)
60. [The Exterior derivative - 1](https://invisiblenetwork.ai/learn/nptel/111108134/5db1342ced7e) (YouTube)
61. [The Exterior derivative - 2](https://invisiblenetwork.ai/learn/nptel/111108134/4e1d73e1c5ba) (YouTube)
62. [The Exterior derivative - 3](https://invisiblenetwork.ai/learn/nptel/111108134/96e6f4bb1ef1) (YouTube)
63. [The Exterior derivative - 4](https://invisiblenetwork.ai/learn/nptel/111108134/17be7356fa69) (YouTube)
64. [The Exterior derivative - 5](https://invisiblenetwork.ai/learn/nptel/111108134/830a25b49303) (YouTube)
65. [Special classes of forms](https://invisiblenetwork.ai/learn/nptel/111108134/925552af8a31) (YouTube)
66. [Orientation on manifolds-1](https://invisiblenetwork.ai/learn/nptel/111108134/77740718870f) (YouTube)
67. [Orientation on manifolds-2](https://invisiblenetwork.ai/learn/nptel/111108134/7171eb4dd12f) (YouTube)
68. [Orientation on manifolds-3](https://invisiblenetwork.ai/learn/nptel/111108134/cdf4a8f680f4) (YouTube)