Submanifolds in Euclidean Space
In this section we study the general case. Let F" be a submanifold of dimension n in E 171 • The number p = m - n is called the codimension ofF" . Consider the normal space N< at each point of some neighborhood of x0 . In each N, choose the orthonormal basis n1, ••• , np in such a way that all na would be regular vector-functions depending on u1, • •• , u 11 • With respect to each of the normals we define the second fundamental form as
II" = (nu r,/uJ ) du1 dui, f7 = I, .. . ,p . Denote (ncr, r u'ui ) by Lij.