MTH413 : Differential Geometry

Evaluation

Evaluation procedure Maximum score
Mid semestral examination 20
End semestral examination 30
Assignments 20
2 Tests 10 each
Seminars 10

Important dates

Due date for assignment 1 : August 29, 2012
Test 1 (of 10 points) : September 6, 2012
Due date for assignment 2 : Septermber 20, 2012
Mid semestral exam :
Test 2 (of 10 points) : November 1, 2012
Final exam :

The course

Week 1 : August 6 - 10
Wednesday
Introduction. Rn, En and conceptual difference between them. Examples of curves and surfaces. Handle construction. Surfaces of revolution : torus.

Thursday (tutorial)
Two equivalent definitions of projective plane. Introduction to curves and Serret-Frenet apparatus.

Friday
Multivariable calculus: Differentiation of scalar fields. Directional and total derivatives, sufficient condition for existence of total derivatives (C1), mean value theorem.

Week 2 : August 13 - 17
Wednesday
Holiday

Thursday
Multivariable calculus: Differentiation of vector fields. Directional and total deriviatives. Jacobians. Chain rule.

Friday
Inverse function theorem: Contraction mapping theorem. Inverse function theorem.
Assignment : Implicit function theorem.

Week 3 : August 20 - 24
Wednesday
Definition of C structures and C manifolds. Some lemmas which are useful to prove that quotient spaces are manifolds.

Thursday (tutorial)
Review of the definition of a C manifolds. Examples of C manifolds : Matrices, GL(n, R), circle, torus, product of two manifolds.

Friday
Definition of C maps between manifolds. Theorems about bump functions and applications. Definition of the rank of a C map.

Week 4 : August 27 - 31
Wednesday
Submanifolds : immersed, imbedded and regular.

Thursday (tutorial)
Projective space and Grassmannians.

Friday
Imbedded iff regular. Lie groups - definition and some examples.

Week 5: September 3 - 7
Wednesday (tutorial)
Review of the assignment and some problems.

Thursday
Class test - I

Friday
Tangent spaces on Rn. Tangent at a point in Rn gives rise to derivation.

Week 6: September 10 - 14
Wednesday
Tangent spaces on a manifold. Derivative of a C map between two manifolds.

Thursday
Review of Grassmannians.

Friday
Introduction to vector fields.

Week 7: September 17 - 21
Wednesday
Holiday : Ganesh Chaturthi

Thursday
Vector fields as derivations on C(M). Vector fields on submanifolds.

Friday
Tangent bundle. Examples and exercises.

Mid-sem and mid-sem break : September 26 - October 7
The mid-sem paper.
Solution for the mid-sem paper.

Week 8: October 8 - 12
Wednesday
Discussion on the mid-sem paper.

Thursday
Tutorial and discussion on the mid-sem paper.

Friday
One parameter group actions.

Week 9: October 15-19
Wednesday
Infintesimal generators for one parameter group actions; local one parameter group actions.

Thursday
Every vector field is the infintesimal generator of a unique local one parameter group action.

Friday (tutorial)
Some examples of one parameter group actions.

Week 10: October 22-26
Wednesday
Holiday: Dusshera

Thursday
Covectors and Riemannian manifolds

Friday
Quiz-I
Examples and computation of "length of a curve" on a Riemannian manifold.

Week 11: October 29 - November 2
Wednesday
Tensors and differential forms.

Thursday
Some more on differential forms and orientable manifolds. Volume form on a Riemannian manifold.

Friday (tutorial)
Some examples and computations. Metric on a Riemannian manifold.

Week 12: November 5-9
Wednesday
Statement of partition of unity. Integration of forms on a manifold.

Thursday
Manifolds with boundary and integration.

Friday (tutorial)
Examples.
Quiz-II

Week 13: November 12-16
Wednesday
Exterior differentiation and Stokes' theorem.

End Sem Examination
November 22, 2012
2PM - 5PM
Paper.
Answer key.

Assignments

  1. Due on August 29th
  2. Due on November 16th

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