Curves and Surfaces in n-Dimensional Space

Chapter 9 shows how to extend the ideas of the previous chapters to curves and surfaces in higher dimensions. In particular, we introduce the generalized Frenet frame for curves in Rn and provide formulas for the generalized curvature functions. For surfaces in ℝ ⁿ with n ≥ 4, the primary complexity comes from the fact that such surfaces have more than one dimension of normal directions. However, we can still define and work with the first fundamental form, allowing us to study all intrinsic properties: arclengths, areas, angles, geodesic curvature, geodesics, and even the Gauss-Bonnet Theorem for curves on surfaces in any Euclidean space.