Analysis on Manifolds — Fall 2025
Course at the Kerala School of Mathematics. This course develops the differential and integral calculus on differentiable manifolds, with an emphasis on differential forms, integration on manifolds, and applications to geometry. Topics include smooth manifolds, tangent and cotangent bundles, differential forms, exterior derivative, orientations and integration on manifolds, Stokes' theorem, and selected applications.
Instructor and contact
Instructor: Pranav Haridas
Office: F-5
Office hours: by appointment
Textbooks & recommended reading
- Calculus on Manifolds, Michael Spivak
- Introduction to Smooth Manifolds, John M. Lee
- An Introduction to Manifolds, Loring W. Tu
Evaluation scheme
- Quizzes: 30%
- Midterm exam: 30%
- Final examination: 40%
Lecture & assessment schedule
| Date | Type | Hours | Topic | Resources |
|---|---|---|---|---|
| Monday 11 Aug | Lecture 1 | 2 | Overview; metric topology on $\textbf{R}^n$; continuity | |
| Wednesday 13 Aug | Lecture 2 | 1 | Differentiation; directional derivatives | |
| Monday 18 Aug | Lecture 3 | 2 | Relation between differentiability and partials; chain rule; second-order partials | |
| Wednesday 20 Aug | Lecture 4 | 2 | Inverse function theorem | |
| Friday 22 Aug | Lecture 5 | 1 | Implicit function theorem | |
| Monday 25 Aug | Lecture 6 | 2 | Lebesgue measure; Integration | |
| Monday 30 Aug | Lecture 7 | 2 | Fubini's theorem | |
| Monday 01 Sep | 2 | Covers material from Weeks 1–3 | Mid-semester Examination - I | |
| Monday 08 Sep | Lecture 8 | 2 | Partition of unity | |
| Wednesday 10 Sep | Lecture 9 | 2 | Change of variable formula | |
| Friday 12 Sep | Lecture 10 | 2 | Parametrized Manifolds, Volume, and Integration | |
| Monday 15 Sep | Lecture 11 | 2 | Mainfolds in $\textbf{R}^n$ | |
| Friday 17 Sep | Lecture 12 | 2 | Mainfolds in $\textbf{R}^n$ (Contd.) | |
| Monday 20 Sep | Lecture 13 | 2 | Mainfolds with boundary | |
| Wednesday 22 Sep | Lecture 14 | 2 | Mainfolds with boundary (Contd.) | |
| Friday 24 Sep | Lecture 15 | 2 | The Volume Measure and Integration on Manifolds | |
| Monday 27 Sep | Lecture 16 | 2 | The Volume Measure and Integration on Manifolds (Contd.) | |
| Wednesday 29 Sep | 2 | Mid-semester Examination - II | ||
| Friday 01 Oct | Lecture 17 | 2 | ||
| Friday 10 October | 3 | Mid-semester Examination - III | ||
| Friday 17 October | Lecture 18 | 2 | Topological Manifolds and its properties | |
| Monday 20 October | Lecture 19 | 2 | Smooth structures | |
| Wednesday 22 October | Lecture 20 | 2 | Smooth maps between manifolds | |
| Friday 24 October | Lecture 21 | 2 | Tangent Vectors | |
| Monday 27 October | Lecture 22 | 2 | Differential of a smooth map | |
| Wednesday 5 November | Lecture 23 | 2 | Tangent Bundles | |
| Friday 7 November | Lecture 24 | 2 | Linear Algebra - I (Tensor Products) | |
| Monday 10 November | Lecture 25 | 2 | Linear Algebra - II (Alternating tensors) | |
| Wednesday 12 November | Lecture 26 | 2 | Vector Fields | |
| Friday 14 November | Lecture 27 | 2 | Cotangent Bundle | |
| Sunday 16 November | 2 | Mid-semester Examination - IV | ||
| Monday 17 November | Lecture 28 | 2 | Differential Forms | |
| Wednesday 19 November | Lecture 29 | 2 | Exterior Derivative, Introduction to Orientations | |
| Friday 21 November | Lecture 30 | 2 | Orientations and Integration on Manifolds | |
| Wednesday 26 November | 2 | Mid-semester Examination - V | ||
| Friday 05 Dec | Final exam | 3 | Final examination | Final Examination |
Course logistics & policies
- Prerequisites: Real analysis, linear algebra, basic topology.
- Collaboration: Discussion of concepts is encouraged.