8 INTRODUCTION Therefore its differential, dE{yfi):T(N(M)\yfi)^Rm, given by dE(y^(v9w) = v + w, is surjective. Thus, by the inverse function theorem, E is a diffeomorphism of a neighborhood of (y, 0) into Rm. Of course, E will not, in general, be a diffeomorphism of N(M) Rm. Indeed a critical value of E is called a focal point ofM. Thus x E Rm is wctf a focal point if, for all (y, w) such that E(y, w) = x, the rank of dE is m. Now the differential of E with respect to w always has rank m I Thus x is a critical value if and only if there is some (y, w) with E(y, w) = x and such that the differential of E with respect to y has rank less than I Now consider the function \px given as above by r(y) = \\y-x\2 = \{y,y-2x) + i\x\\ Then dxpx = (fy,.y - x) so that d^x = 0 if and only if j x is normal to TMy, and then x = E(y,y x). The figure below suggests the following: (i) d2\px is singular if and only if x is a focal point. (ii) Let L be the line through y and x. If there is no focal point between y and x then ^ x has a minimum at j, and, indeed, d2\px is positive definite. (iii) If there is a focal point of the form E(y,v), v E L, where (, i) E N(M) is a singular point, and y lies between 0 and x y then \px is not a minimum, and the index of d \px is related to the number of such focal points. (In the figure, i//* and \px have a minimum at y while xpx has a maximum.) *3 \xl^-"y- x * Indeed, we shall prove the Hessian d \^x(y) is nondegenerate if and only if x = E(y,v) and rank dE/ \ = m, i.e., x is not a focal point of a neighborhood of
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