10 INTRODUCTION Thus the (/, y')th entry of the matrix can be written as a2. Kdu^duj) \dUidu/V) duj\du/V) since v = x y 'dui duj' This shows that corank dE,xv\ = nullity d \px. In particular, if (y,v) is a regular point of E then d2\px(y) is non-degenerate, establishing the first part of the proposition. Let The quadratic form on Re whose matrix is (gy) is called the first fundamentalform. It is just the Euclidean scalar product on TM considered as a bilinear form on o f R when we identify R with TM. It is positive definite. The bilinear form whose matrix is M")) = ( 3 ^ . ' " ) neN(M),, is called the second fundamental form in the direction n. We let I and II denote the first and second fundamental forms at y. We have shown that ^ 7 ^ 7 ^ ^ - ^ W where I/ = J/I. Let us compute the index of the right hand side. By a linear change of coordinates in the w's, we may assume that (g.) = (5-) this amounts to introducing an orthogonal basis for the first fundamental form. Then, (/ ••) = (r-j(n)) is a symmetric matrix with eigenvalues /Xj,...,jLtm and pairwise orthogonal eigenvectors. By a further orthogonal change of the u variables we can diagonalize the matrix (f}j(n)). Thus we get ay dulduJ By-wAj = 0 - J ft-)V Thus the index of d $x is the number of /x, such that s^ 1. This is the same as 2 (number of ^ with st^ = 1) = 2 corank (/ - fO//)) 0 / l 0 / 5 = 2 corank dE tv , which proves the second part of the proposition.
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