A Modern Approach to Classical Differential Geometry

Differential Geometry is the study of the differentiable properties of curves and surfaces at a point. The properties of small segments of curves and surfaces are investigated, leading to the study of these properties in the large. In Michael Spivak\u27s A Comprehensive Introduction to Differential Geometry, Volume I (Brandeis University, Lecture Notes), the author presents the groundwork for an analytical study of manifolds: tangent bundles, tensors, vector fields, differential forms, Riemannian metrics, and the equations of structure based on Lie groups. This paper will be a survey of the use of some of these structural techniques in presenting the geometric aspects of classical differential geometry