Surfaces in Rn

This chapter shows how the theory of curvature at a point p in a twodimensional surface S extends from R³ to Rⁿ. As before, choose orthonormal coordinates on Rⁿ with the origin at p and S tangent to the x_l, x₂-plane at p. The tangent plane T_pS to S at p is now the x₁,x₂-plane; the orthogonal complement T_pS^⊥ is the x₃, . . . , x_n,-plane; and locally S is the graph of a function