Riemannian manifolds are higher dimensional generalizations of surfaces in the 3-dimensional Euclidean space. However, unlike surfaces, Riemannian manifolds are intrinsically defined and exist independently of an ambient Euclidean space. The characteristic object in Riemannian geometry is a metric tensor, an analogue of the inner product in Euclidean geometry, which renders the underlying manifold into a metric space. As in the theory of surfaces, various versions of curvature emerge as the main tool in the study of the structure and the classification of Riemannian manifolds. Since Bernhard Riemann's seminal lecture in the mid-nineteenth century, Riemannian geometry has grown into a vast subject and, besides pure mathematics, it exercises a profound influence on such varied fields as Hamiltonian mechanics, integrable systems, robotics, tomography, quantum theory and gravity.
Topics in this introductory course to Riemannian geometry include metrics, connections, geodesics and curvature, which will be studied from a number of points of view. Time permitting, Riemannian submanifolds and the Gauss-Bonnet theorem will also be discussed. As examples, the standard models for constant curvature manifolds will also be introduced.
Prerequisites: Working knowledge of the basics of differentiable manifolds. Familiarity with geometry of surfaces, Lie groups and rudiments of topology will be helpful.
Text: John M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer, 1997. (Corrigenda).
Grading: Your course grade will be based on three homework assignments and a take-home final exam. The homework assignments count 60% towards the grade and the final 40%.
The homework assignments will be due 10/12, 11/2, and 11/21.