MAT 511E - Differential Geometry I

Course Objectives

1.To introduce the notions of differentaiable manifold, tangent and cotangent spaces, and smooth maps;
2.To introduce differential geometric structures on hypersurfaces of Euclidean space to study their geometric properties;
3.To introduce tensors and differential forms to to develop Riemannian manifolds and submanifolds.

Course Description

Topological Spaces, product topology, metric topology, quotient topology, connectedness, compactness. Differentiable manifolds. Tangent space, vector fields. Lie Bracket., Diffeomorphism, inverse function theorem. Submanifolds. Hypersurfaces, standart connection of Euclidean Spaces. Weingarten and Gauss maps. Tensors and differential forms. Lie Derivative. Riemannian connection, Riemannian manifolds and Riemannian submanifolds.