Unit I

Definition of Manifolds, Differentiable and Analytic Manifolds, Examples of Manifolds, Product of Manifolds, Mappings between Manifolds, Submanifolds, Tangent Vectors.

Unit II

Differentials, The Differential of a Function, Infinitesimal Transformation, Tangent Space, Tangent Vector.

Unit III

Cotangent Space, Vector Fields, Smooth Curve in a Manifold. Differential Forms− k-forms, Exterior Differential, its Existence and Uniqueness.

Unit IV

Exact Differential Forms. De Rham Cohomology Group, Betti Number, Poincare’s Lemma, Inverse Function Theorem, Implicit Function Theorem and its Applications, Integral Curve of a Smooth Vector Field.

Unit V

Orientable Manifolds−Definition and Examples. Smooth Partition of Unity− Definition and Existence. Riemannian Manifolds− Definition and Examples.

Course Outcomes:

CO 1: To familiarize the concept of manifolds and learn their properties
CO 2: To understand the concept of tangent spaces and its properties
CO 3: To generalize the ideas of curves/derivatives to manifolds
CO 4: To prove the inverse /implicit function theorems in manifolds
Co 5: To understand Riemannian manifolds and their relevance

1. M.Singer., J.A. Thorpe, “Lecture Notes on Elementary Topology and Geometry”, Undergraduate Texts in Mathematics, 2008.
2. P.M.Cohn, “Lie Groups”, Cambridge University Press, 1965.
3. Claude Chevalley, “Theory of Lie Groups”, Fifteenth Reprint, Princeton University Press, 1999.