Geometry

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 360 Linear Algebra II for undergraduates, MAT 660 Linear Algebra for graduate students

Overview: This course provides a bridge to modern geometry. It provides a unified axiomatic approach leading to a coherent overview of the classical geometries (affine, projective, hyperbolic, spherical), culminating in a treatment of surfaces that sets the stage for future study of differential geometry.

Detailed Syllabus:

1. Affine geometry – finite planes, planes over fields, affine transformations, collineations, affine coordinates, triangles and parallelograms, classical theorems of Menelaus and others.
2. Projective geometry – finite planes, projective completion of affine planes, homogeneous coordinates, projective transformations, collineations, projective line, poles and polars.
3. Conics – affine and projective classifications, group actions.
4. Euclidean geometry – isometries, triangles, parallelograms, length minimizing curves, geometry of plane curves.
5. Hyperbolic geometry – Poincare upper half plane, Poincare metric, distance function, triangles and area, two-point homogeneity.
6. Spherical geometry – Sphere, tangent space, great circles, triangles and area, two-point homogeneity
7. Surfaces – Level surfaces, parametrized surfaces, curvature, Gauss theorem, introduction to manifolds.

References:

1. An Expedition to Geometry by S Kumaresan and G Santhanam. Hindustan Book Agency, 2005.
2. Geometry by M. Audin. Springer International Edition, Indian reprint, 2004.
3. Geometry by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray. Cambridge University Press, 2^nd edition, 2012.
4. Geometry by Roger A Fenn, Springer International Edition, 2005.