Unit-I

Curves in the plane and in space, arc-length, re-parametrization, level curves Vs parametrized curves.

Unit-II

Curvature, plane curves, space curve-global properties of curves, the isoperimetric inequality, the four-vertex theorem.

Unit-III

Surfaces in the three dimension, smooth surfaces, tangents, normals and orientability, quadratic surfaces, triply orthogonal systems, applications of inverse function theorem, the first fundamental form, lengths of curves on surfaces, isometric surfaces.

Unit-IV

Conformal mapping of surfaces, surfaces area, equiareal mapss and a theorem of Archimedes, curvature of surfaces, the second fundamental form, the curvature of curves on a surface, the normal and principal curvatures, geometric interpretation of principal curvatures.

Unit-V

The Gaussian curvature and the mean curvatures, the pseudoshere, flat surfaces, surfaces of a constant mean curvature, Gaussian curvature of a compance surfaces, the Gauss map.

Course Outcomes
CO1: Students should acquire knowledge about the application of methods of differential calculus to the study of geometry with emphasis on the differential geometry of surfaces
CO2: Students should acquire knowledge about the application of methods of integral calculus to the study of geometry with emphasis on the differential geometry of surfaces.
CO3: Students should be able to apply this knowledge independently to analyze and solve mathematical problems in contexts where methods of differential geometry are relevant.

Text Book:

• Andreq Pressley, Elementary Differential Geometry, Springer, 2006.

Reference Books:

1. Thierry Aubin, A Course in Differential Geometry, American Mathematical Society, 2007.
2. K. L. Wardle, Differential Geometry, Dover, 2008.