Introduction. Rⁿ, Eⁿ and conceptual difference between them. Examples of curves and surfaces. Handle construction. Surfaces of revolution : torus.

Two equivalent definitions of projective plane. Introduction to curves and Serret-Frenet apparatus.

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

Holiday

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

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

Definition of C^∞ structures and C^∞ manifolds. Some lemmas which are useful to prove that quotient spaces are manifolds.

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

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

Submanifolds : immersed, imbedded and regular.

Projective space and Grassmannians.

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

Review of the assignment and some problems.

Class test - I

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

Tangent spaces on a manifold. Derivative of a C^∞ map between two manifolds.

Review of Grassmannians.

Introduction to vector fields.

Holiday : Ganesh Chaturthi

Vector fields as derivations on C^∞(M). Vector fields on submanifolds.

Tangent bundle. Examples and exercises.

The mid-sem paper.
Solution for the mid-sem paper.

Discussion on the mid-sem paper.

Tutorial and discussion on the mid-sem paper.

One parameter group actions.

Infintesimal generators for one parameter group actions; local one parameter group actions.

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

Some examples of one parameter group actions.

Holiday: Dusshera

Covectors and Riemannian manifolds

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

Tensors and differential forms.

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

Some examples and computations. Metric on a Riemannian manifold.

Statement of partition of unity. Integration of forms on a manifold.

Manifolds with boundary and integration.

Examples.
Quiz-II

Exterior differentiation and Stokes' theorem.

2PM - 5PM
Paper.
Answer key.