Riemannian Geometry

By Prof. Ved Vivek Datar | IISc Bangalore

Learners enrolled: 446 | Exam registration: 26

ABOUT THE COURSE:

It will be a rigorous course on Riemannian geometry. The course will begin with a brief review of smooth manifolds and the geometry of curves and surfaces. I will then introduce Riemannian metrics and cover some standard topics such as Levi-Civita connection and associated curvature, geodesics, completeness, 1^st and 2^nd variation formulae and Jacobi fields. If time permits, the course will end with more advance topics, such as comparison theorems (Myers, Bishop-Gromov etc.) and the Bochner technique.

A justification for the course: The course aims to fill a much needed gap in the differential geometry courses offered in NPTEL and will equip the students with the necessary background to branch out into various advanced topics in differential geometry and geometric analysis. At some point in the future, the instructor also plans to offer a course in geometric analysis on NPTEL, and this would be a pre-requisite for that.

INTENDED AUDIENCE: Masters and PhD students, advanced undergraduates

PREREQUISITES: Multivariable calculus and Smooth manifolds is a must (so must be familiar with calculus on manifolds, tensors, forms and Stokes’ theorem). Helpful to have had a course on curves and surfaces, but my course would be independent of that.

Summary

Course Status :	Ongoing
Course Type :	Elective
Language for course content :	English
Duration :	12 weeks
Category :	Mathematics
Credit Points :	3
Level :	Postgraduate
Start Date :	19 Jan 2026
End Date :	10 Apr 2026
Enrollment Ends :	02 Feb 2026
Exam Registration Ends :	13 Feb 2026
Exam Date :	24 Apr 2026 IST
NCrF Level	4.5 — 8.0

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.

Page Visits

Course layout

Week 1: Introduction, review of curves and surfaces, Review of smooth manifolds – tensors, differential forms, Stokes’ theorem

Week 2: Introduction to Riemannian metrics, examples and basic constructions

Week 3: Levi-Civita connection, induced connection on tensors and forms, Parallel transport

Week 4: Curvature of Levi-Civita connection, Sectional curvature, Ricci curvature and scalar curvature, geometry of sub-manifolds

Week 5: Geodesics, first variation formula

Week 6: Local behaviour of geodesics, exponential map, normal coordinates

Week 7: Metric geometry, Hopf-Rinow, regularity of distance function

Week 8: 2^nd variation formula, Jacobi fields, index form

Week 9: Jacobi fields (cont.), characterization of space forms

Week 10: Catch-up (if on schedule, then move onto comparison geometry)

Week 11: Overview of comparison geometry (Rauch, Myer and Bishop-Gromov)

Week 12: Bochner technique

Books and references

Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Riemannian geometry, Third edition., Universitext. Springer-Verlag, Berlin, 2004.
Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.
John Lee, Riemannian Geometry - An introduction to curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.
Ved Datar, Lectures on Riemannian geometry, online notes on my webpage,

Instructor bio

Prof. Ved Vivek Datar

IISc Bangalore

Prof. Ved Vivek Datar work in geometric analysis and complex differential geometry. My main research interest is to understand the relations between existence of canonical metrics and solutions to other natural PDEs on complex manifolds to topological and algebra-geometric obstructions. I also work on comparison and rigidity problems in Kahler geometry. Broadly speaking this involves understanding how curvature conditions on Kahler manifolds impose restrictions on the topology or analytic/algebraic structure of the underlying complex manifold.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: April 24, 2026 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

Registration url: Announcements will be made when the registration form is open for registrations.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Please note that assignments encompass all types (including quizzes, programming tasks, and essay submissions) available in the specific week.

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IISc Bangalore .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team

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