At the end of this course the students will be able to understand differential geometry, its connection and significance to other areas of mathematics. Extend many of the basic concepts and tools of multivariable calculus and linear algebra to the contexts of calculus on Riemannian geometry, develop and demonstrate a level of expertise in mathematical reasoning appropriate to a challenging upper-level mathematics course.
Historical background; Motivation and applications, Index notation and summation convention, Space curves, The tangent vector field, Parameterization, Arc length, Curvature, Principal normal, Binormal, Torsion, Osculating, Normal and Rectifying planes, Frenet-Serret Theorem, Spherical images, Sphere curves, Spherical contacts, Fundamental theorem of space curves, Line integrals and Green’s theorem, Local surface theory, Coordinate transformations, Tangent and the Normal planes, Parametric curves, First fundamental form and the metric tensor, Normal and geodesic curvatures, Gauss’s formulae, Christoffel symbols of first and second kinds, Parallel vector fields along a curve and parallelism, Second fundamental form and the Weingarten map, Principal, Gaussian, Mean and Normal curvatures, Dupin indicatrices, Conjugate and asymptotic directions, Isometries and the fundamental theorem of surfaces
