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Geometry of Curves and Surfaces

By Prof. Abhijit Pal   |   IIT Kanpur
Learners enrolled: 71
ABOUT THE COURSE:

Curves and Surfaces are basic geometric objects that appear naturally in various fields of science and engineering. It is worthwhile to understand the intrinsic geometry of curves & surfaces. This is an introductory course on differential geometry of curves and surfaces, meant for undergraduate students who have some background in multivariable calculus.

INTENDED AUDIENCE: PG & PhD Students but an UG student with background in topology, algebra and complex analysis can take this course.

PREREQUISITES: Multivariable Calculus
Summary
Course Status : Upcoming
Course Type : Elective
Language for course content : English
Duration : 12 weeks
Category :
  • Mathematics
Credit Points : 3
Level : Undergraduate/Postgraduate
Start Date : 21 Jul 2025
End Date : 10 Oct 2025
Enrollment Ends : 28 Jul 2025
Exam Registration Ends : 15 Aug 2025
Exam Date : 26 Oct 2025 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.


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Course layout

Week 1-3:  Curves in R² and R³, Regular Curves, Arc Length, Local theory of Curves: Curvature & torsion of a curve, Frenet-Serret Formulas, Fundamental Theorem of the local theory of curves, Global properties of plane curves : Isoperimetric inequality and the four vertex theorem.

Week 4-7: Review of multivariable calculus : Differentiable functions, Implicit function theorem, Inverse function theorem. Regular Surfaces in R³, Regular values of a differentiable function, Inverse image of regular values, Differentiable functions on surfaces, Tangent Plane on surfaces, Differential of a Map, First Fundamental Form, Length of curves on surfaces, Area, orientation on a surface, orientable surfaces.

Week 8-10: Gauss Map and its Fundamental Properties, Second Fundamental Form, Normal Curvature, Principal Curvature, Gaussian Curvature, Classification of point on surfaces : Elliptic, Hyperbolic, Parabolic, Planar, Gauss Map in terms of Local Coordinates

Week 11&12: Isometries and local isometries between surfaces, Christoffel symbols, Gauss’s Theorema Egregium (Remarkable Theorem), Parallel Transport, Geodesics on surfaces, Triangulation of a surface, Euler Characteristic of a surface, Gauss-Bonnet Theorem

Books and references

(1) Differential Geometry of Curves & Surfaces, Manfredo P. Do Carmo, 2nd Edition, Dover Publications
(2) Elementary Differential Geometry, Andrew Pressley, Springer
(3) An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby, Academic Press
(4) Riemannian Geometry, Manfredo P. Do Carmo, Springer

Instructor bio

Prof. Abhijit Pal

IIT Kanpur
Prof. Abhijit Pal is a faculty member in the Department of Mathematics & Statistics of IIT-Kanpur. He has done PhD from ISI-Kolkata in 2011 under the supervision of Prof. Mahan Mj. His research interest lies in Hyperbolic Geometry, Geometric Group Theory, specially in hyperbolic groups and relatively hyperbolic groups.

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: October 26, 2025 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 IIT Kanpur .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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