Advanced Instructional School (AIS)- Differential Geometry and Lie groups

Venue: BP and Pune Univ.
Dates: 1st - 28th Dec, 2006


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name R. S. Kulkarni and S. Kumaresan

AIS-DGLG being organised in Pune in December 2006 is the Advanced Instructional School on Differential Geometry and Lie Groups being organised on behalf of NBHM.

National Coordinating Committee
  • Director: Ravi S. Kulkarni
  • Secretary: J. K. Verma
  • Members : S. D. Adhikari, Satya Deo, Shobha Madan, I. B. S. Passi, Ravi A. Rao, S. A. Katre

Local Co-ordinator

  • S. A. Katre, University of Pune


Members of the Local Organising Committee


  • Bhaskaracharya Pratishthana: C. S. Inamdar (Custodian), R. V. Gurjar (Res. Director, Hon.), R. R. Simha
  • University of Pune: B .N. Waphare (HOD, Maths.), S. A. Katre


Speakers and Syllabus



Differential Geometry, in its present-day avatar, is known as the study of connections on fiber bundles. These ``connections" are implicit in any branch of mathematics/applied mathematics/theoretical physics dealing with continuum models, and their discrete analogues are developed even in algebra and arithmetic. In building these ``connections", the Lie groups play a foundational role.

One of the aims of this AIS is to communicate  this viewpoint starting from the very concrete study of curves and  surfaces in the Euclidean n-space, an introduction to euclidean,  spherical, and hyperbolic geometries, and generalities on differentiable manifolds and fiber bundles.

The special topics include:
i) an introduction to 2-dimensional differential geometry with its connections to Riemann surfaces,
ii) an introduction to Riemannian geometry and the currently hot topic of Ricci flows, and
iii) an introduction to Lie groups.
We expect that there will be some follow-up workshops to this material in the next two years.

A special feature of the ATM programme is the inclusion of ``Unity of Mathematics"- Lectures by some distinguished scientists. In this AIS, Prof. JAYANT NARLIKAR, IUCAA, Pune,  will deliver three lectures on ``Cosmological Models and Spacetime symmetries", and Prof. M. S. RAGHUNATHAN, TIFR, Mumbai, will deliver 2 lectures.


Proposed syllabus and speakers
  • A. Mangasuli

email: anandateertha at

1) A) Multivariate Calculus, Inverse Function Theorem, Existence and Uniqueness theorem on ODE (3 lectures)
B) Differentiable Manifolds, Tangent/cotangent bundles, Associated bundles, Differential forms, Lie derivative, Stokes's theorem, Frobenius's theorem (5 lectures)


  • Ravi Kulkarni

email: punekulk at

2) Frenet-Serret theory, Euler's theory on normal curvatures, Gaussian curvature and theorema egregium (3 lectures)


  • G. Santhanam and Ravi Kulkarni

email: santhana at

3) 2-dimensional Euclidean, Spherical, and Hyperbolic geometries (4 lectures).

The groups of isometries, curves of constant geodesic curvature, the dynamical types of isometries. The alternate representations of isometry groups using complex numbers, when the above geometries are considered as 1-dimensional complex Euclidean, elliptic, or hyperbolic geometries. The disk- and upper half plane models of hyperbolic geometry. 2-dimensional affine geometry.

4) $ n$-dimensional Euclidean, Spherical, and Hyperbolic geometries, and their special cases when $ n = 3, 4, 5$.

Description of dynamical types of isometries.

Hyperboloid model of hyperbolic geometry.

Description of the groups of isometries for $ n = 3, 4, 5$ using complex numbers and quaternions.

Introduction to some special Lie groups. (3 lectures)



  • Akhil Ranjan and Harish Seshadri

email: aranjan at    e-mail:harish at

5) $ n$-dim. Riemannian Geometry

Riemannian metrics (examples), Levi-Civita connection,distance, geodesics, first-variation formula, Hopf-Rinow, curvature, second-variation formula, Jacobi fields, Rauch comparison theorem, Bonnet-Myers theorem, Cartan-Hadamard theorem.

Ricci Flows (16 lectures)

Examples, Heat-type equations and maximum principles, Existence and uniqueness, Ricci flow on surfaces.


  • S. Kumaresan

email: kumaresa at

6) Compact Lie Groups (8 lectures) ``Hands-on" introduction to $ O(n), U(n), Sp(n)$ and their representations, and the statements of general theorems on compact Lie groups without proof.


  • Ravi Kulkarni


7) Noncompact Lie Groups (2 lectures)


  • Kaushal Verma

email: kverma at

8) Harmonic maps (8 lectures)

(A) Quick review of Sobolev spaces, in particular $ W^{\{1, 2\}}$, followed by the solution to the Dirichlet Problem on domains in $ {\mathbb{R}}^n$. Regularity theory for Laplacian.

(B) Harmonic maps between Riemann surfaces: topics to be covered include

(i) definition and examples
(ii) theorem on the existence of harmonic maps
(iii) second variation of a harmonic map and some applications (e.g. the finiteness of the conformal automorphism group of a compact hyperbolic surface)
(iv) The theorem of Schoen-Yau which shows that harmonic maps are actually diffeomorphisms when the surfaces under consideration satisfy relevant topological properties.
(v) Some parts of M. Wolf's paper (J. Diff. Geom. no. 29 , 1989) titled `The Teichmüller theory of harmonic maps'.

(C) Definition of a harmonic map in higher dimensions. Examples and statement of the main existence theorem due to Eells-Sampson.


1. Glen E. Bredon, Topology and Geometry, Springer GTM 139, Indian reprint-2006.
2. Do Carmo, Riemannian Geometry, Boston, Birkhauser, 1993.
3. J. Hubbard, Teichmüller Theory, Vol. 1, Matrix Edition, 2006.
4. Jurgen Jost, Compact Riemann Surfaces, J. Jost, Springer, Universitext, 1997.
5. S. Kumaresan, A Course in Differential Geometry & Lie Groups, Hindustan Book Agency, 2002.
6. S. Kumaresan, A Course in Riemannian Geometry - Lectures notes (To be published).
7. S. Kumaresan and G. Santhanam, An Expedition to Geometry, New Delhi, Hindustan Book Agency, 2005
8. J. Milnor, Morse Theory, Princeton University Press, 1963.
9. Peter Petersen, Riemannian Geometry, Springer GTM 171.

UM Lectures

  • J. V. NARLIKAR : Cosmological Models with Spacetime Symmetries
  • M. S. RAGHUNATHAN : Arithmetic and Differential Geometry


Abstract: Cosmological Models with Spacetime Symmetries
1. Introduction: the expectations of a mathematical model in cosmology.
2. Spacetime symmetries: Groups of motions, Killing equations, concepts of homogeneity and isotropy, Roberton-Walker spacetimes, de Sitter metric.
3. Homogeneous but anisotropic spacetimes: some examples of models with spin, Godel's metric,the Heckmann-Schucking metric.
4. Evidence for symmetries in the actual universe.

1. Riemannian Geometry by Eisenhart (Princeton)
2. Gravitation and Cosmology by S. Weinberg (Wiley)
3. General Relativity and Cosmology by J.V. Narlikar (Macmillan)


Abstract: Arithmetic and Differential Geometry
The uniformisation theory of Riemann surfaces tells us the following. Let S (resp. C, resp. H) denote the sphere (resp. the complex plane, resp.the upper half plane) equipped with the Riemannian metric of constant sectional curvature 1 (resp. 0, resp. -1). Then the universal cover of any connected compact surface can be identified with one and only one of S, C or H in such a way that the deck-transformation group acts as a group of isometries of that constant curvature space. One cannot expect such a strong result in higher dimensions. A simply connected manifold of higher dimension does not in general carry a homogeneous Riemannian metric. In fact it is a nontrivial fact that there do exist compact manifolds in higher dimension that carry metrics with the universal covering being a homogeneous Riemannian manifold (of non-negative sectional curvature). The only methods of construction of such manifolds that will cover all dimensions involve arithmetical considerations. A famous result due to G. A. Margulis asserts in fact that in most cases the construction necessarily comes from arithmetic. In these lecture I will discuss some of the ideas involved in these matters.

Associate Teachers

1.Vikram Aithal email: vikram at
2. Sanjay Pant email: sanjpant at
3. Srikanth K. V. e-mail: kvsrikantha at (15-28 Dec)

Schedule of Lectures / Tutorials


  1 st Dec., Fri 2 nd, Sat 4 th, Mon 5 th, Tue 6 th, Wed 7 th, Thur 8 th, Fri 9 th, Sat
     9.30 - 11.00  AM AM AM AM AM AM AM AM
     11.00 - 11.30  Tea Tea
     11.30 - 1.00  RK RK RK RK RK GS GS GS
     1.00 - 2.15  Lunch Lunch
     2.15 - 3.15  Tutorials Tutorials
     3.15 - 3.45  Library / Tea Library / Tea
     3.45 - 5.15  Tutorials Tutorials


  11 th Dec., Mon 12 th,Tue 13 th, Wed 14 th,Thurs 15 th, Fri 16 th, Sat
     9.30 - 11.00  GS GS GS GS GS RK
     11.00 - 11.30  Tea
     11.30 - 1.00  JVN JVN JVN SK SK SK
     1.00 - 2.15  Lunch
     2.15 - 3.15  Tutorials --
     3.15 - 3.45  Library / Tea --
     3.45 - 5.15  Tutorials SK Seminar/Tutorial --


  18 th Dec., Mon 19 th,Tue 20 th, Wed 21 th, Thurs 22 th, Fri 23 th, Sat
     9.30 - 11.00  HS HS HS HS HS HS
     11.00 - 11.30  Tea
     11.30 - 1.00  SK SK SK SK RK RK
     1.00 - 2.15  Lunch
     2.15 - 3.15  Tutorials
     3.15 - 3.45  Library / Tea
     3.45 - 5.15  RK RK KV KV KV KV


  25 th, Dec. Mon th, Tue 27 th, Wed 28 th, Thur
9.30 - 11.00  HS HS HS HS
11.00 - 11.30  Tea
11.30 - 1.00  KV KV KV MSR
1.00 - 2.15 Lunch
2.15 - 3.15 Tutorials FeedBack Session
3.15 - 3.45  Library / Tea Valedictory Function
3.45 - 5.15 KV Seminar/Tutorial MSR ---




AM- Anand Mangasuli RK- Ravi Kulkarni, KV- Kaushal Verma
GS- G. Santhanam HS- Harish Seshadri SK- S. Kumaresan
UM-Lecture Series : JVN- Jayant Narlikar MSR- M. S. Raghunathan


Vikram Aithal, Sanjay Pant, Srikanth K. V.

 Selected Applicants



Sr. No. Name of Participant Accommodation H,K-> in Hostel of BP, R-5-> in BP main building, flat-> near by flat
1. Gautam Borisagar, IIT Bombay, Mumbai Flat
2. Mr. Sanjit Das, IIT Kharagpur H-4
3. Mr. Priyabrat Gochhayat, Berhampur Uni., Bhajabihar R-5
4. Mr. Sanjeev Kumar, G. B. Pant Univ., Uttaranchal R-5
5. Mr. Jaydeep Sengupta, Univ. of North Bengal, Darjeeling H-4
6. Mr. Ajay Singh Thakur, IMSc, Chennai H-7
7. Mr. Umesh V. Dubey, IMSc, Chennai H-7
8. Mr. Tarakanta Nayak, IIT-Guwahati K-2
9. Mr. Sushil Gorai, IISc, Bangalore H-4
10. Ms. Prachi Mittal, IISc, Bangalore H-1
11. Dr. T. Venkatesh, Karnatak Univ. H-7
12. Mr. Pavinder Singh, Univ. of Jammu, Jammu H-4
13. Mr. Nandakumar M. Univ. of Calicut, Kerala R-5
14. Mr. Jagmohan Tanti, BP, Pune K-2 (Local)
15. Dr. Bhimashankar Waghe, Kashibai Nawale Engg. College, Pune Local
16. Mr. Devendra Shirolkar, Dept. of Maths, Univ. Pune, Pune Local
17. Mr. Soumen Sarkar, ISI, Kolkata H-5
18. Mr. Diganta Borah, I.I.Sc. Bangalore R-5
19. Ms. Shilpa Gondhali, TIFR H-1
20. Mr. Chandrasheel Bhagwat, TIFR, Mumbai Self Accom.
21. Mr. Hemant Pawar, NDA, Pune Local
22. Mr. Vadiraja Bhatta, NITK, Surathkal R-5
23. Mr. Debasis Sen, ISI, Kolkata H-5
24. Mr. Dheeraj Kulkarni, HRI Self Accom.
25. Mr. Rabiul Islam, Univ. of Calicath, Kolkata H-5
26. Mr. Abhijit Pal, ISI, Kolkata H-5
27. Mr. Prem Pandey, IMSc, Chennai H-7
28. Mr. Niraj Prasad, Mumbai Univ. Flat
29. Mr. Angom Tiken Singh, Nehu, Shillong, Meghalaya Flat
30. Mr. Patel Jayminkumar M., Sardar Patel Univ., Gujrat Flat
31. Mr. Manoj Kumar Pandey, A.P.S. Univ., Rewa Flat
32. Ms. Tummala Vinutha, Andhra Univ., A.P H-1
33. Mr. Rajeev B., Alappuzh, Kerala Flat
34. Mr. Subrata Bhowmik, Univ. of Tripura H-4
35. Mr. Nanasaheb Phatangare, Ahmednagar, Maharashtra Flat
36. Vikas Jadhav, Wadia College, Pune Local
37. Manjusha Joshi, BP, Pune Local

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