1st semester *20 credits / 30 lectures*

Differential Geometry is one of the oldest mathematical disciplines. It originated from the analytic studies of geometric entities such as lines and planes, which then turned into the classical geometry of curves and surfaces. The modern ‘Differential Geometry’ focuses on different aspects of the theory of differentiable manifolds.

I strongly believe that this will be a very useful course. The contents of the course are accessible to mathematics students and are also directly relevant to topics in theoretical physics. You are familiar with the geometry and analysis in Euclidian space. Differential Geometry is basically the part of geometry that studies how to characterize spaces in terms of their curvature and study how curvature in space influence our ability to study physical laws.

We will start with a quick review of topics such as vector space, linear Algebra etc. Then we will start talking about the differentiable manifolds. This will include a comprehensive study of calculus on manifolds. Then we will talk about affine manifolds. This will include topics like affine connection, covariant derivative, parallel transport, curvature etc. Our next topic will be Riemannian manifold and some applications. Finally, we will talk about Symmetries—Lie groups and Lie algebra. If time permits, we will go over Differential forms and exterior calculus.

Tentative topics:

- Preliminaries
- Differentiable Manifold
- Affine Manifolds
- Riemannian Geometry
- Symmetries
- Differential Forms and Exterior Calculus