**R**

^{n},

**E**

^{n}and conceptual difference between them. Examples of curves and surfaces. Handle construction. Surfaces of revolution : torus.

Evaluation procedure | Maximum score |
---|---|

Mid semestral examination | 20 |

End semestral examination | 30 |

Assignments | 20 |

2 Tests | 10 each |

Seminars | 10 |

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 :

Week 1 : August 6 - 10

Wednesday

Introduction. **R**^{n},
**E**^{n} 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 (C^{1}), 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.

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

Friday

Tangent spaces on **R**^{n}. Tangent at
a point in **R**^{n} 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

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.

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

Quiz-II

Week 13: November 12-16

Wednesday

Exterior differentiation and Stokes' theorem.

End Sem Examination

November 22, 2012