Differential Geometry

After the introduction of coordinates, it became possible to treat figures in plane and space by analytical methods, and calculus has been the main means for the study of curved figures. For example, one attaches the tangent line to a curve at each point. One sees how tangent lines change with points of the curve and gets an invariant called the curvature. C. F. Gauss, with whom differential geometry really began, systematically studied intrinsic geometry of surfaces in Euclidean space. Surface theory of Gauss with the discovery of non-Euclidean geometry motivated B. Riemann to introduce the concept of manifold that opened a huge world of diverse geometries. Current differential geometry mainly deals with the various geometric structures on manifolds and their relation to topological and differential structures of manifolds. Results in linear algebra (Matrices, Vectors, Determinants and Linear Algebra) and Euclidean geometry (Basic Notions of Geometry and Euclidean Geometry) are assumed to be known as aids in enhancing the understanding this chapter.