THE METHOD OF STATIONARY PHASE 7 critical pointy is related to the first and second fundamental forms of the surface. Let us do the computation in somewhat greater generality, as this involves only slightly more effort (cf. Milnor[3]). Let M be an ^-dimensional submanifold of Rm. Thus each point of M has a neighborhood with coordinates u = (w 1 ,..., ul) and a map y = y(u) = (yl(u),... ,ym(u)) into Rm, where the Jacobian matrix (dy/du) has rank I For simplicity we will regard M as a subset of Rm (although all that we have to say works for immersed submanifolds as well as embedded ones). We can thus consider the normal bundle, N(M), as consisting of those pairs {y,w) where y E M and w E Rm is orthogonal to the tangent space to M at y. Thus w ± TMy. Notice that from this point of view we are regarding N(M) as lying in Rm + Rm. It is easy to check that N(M) is an m-dimensional immersed submanifold. Indeed let U be a coordinate patch on M with coordinates Wj, . . . , Mg . The vectors dy dy are all tangent to M at each point of U. Extend to a basis of Rm at some p E U by adding vectors vt+l, . . . , vm. Thus is a basis of Rm at p and hence is a basis of Rm at all points of U near enough to p. By shrinking we may also call this neighborhood U. Orthonormalize this basis to get tx(w), . . . , tt(u), nt+l(u), . . . , nm(u). Then n(+l («), . . . , nm(u) span the normals to M at y(u). Thus U X R w- e gives a parametrization of the normal bundle over U where ("l w ^ m O -* (")-VHVH(W) + ••• + smnm(u)). Define the map E\ N(M) Rm by E(y,w) = y + w. Notice that T (N(M)\y,W) = ™y + NM C Rm + Rm.
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