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RlCE UNIVERSITY STUDIES
Define Hp(M2) = Tp(M2) П JTp(M2) where peM2 and Tp(M2) is the
real tangent space to M2 at p. The vector space Hp(M2) is called the vector
space of holomorphic tangent vectors to M2 at p. Then p is an exceptional
point in M2 if the complex dimension of Hp(M2^) is one.
As an example, if S2 denotes the two-sphere in standard position in
R3 ⊂ C2, then the exceptional points in S2 are the north and south poles.
Letting p be an exceptional point of M2, we choose differentiable co-
ordinates и, v in a neighborhood of p in M2, vanishing at p, and analytic
coordinates z1,z2 f°r a neighborhood of p in C2, vanishing at p. The
equations of M2 in a neighborhood of p can then be written as
z1 = ∕1(u,u)
z2 = ∕2(n,f)>
where∕1,∕2are complex valued functions of к and t>, defined in a neigh-
borhood of u = V = 0. (We denote by ⅞ffc the class of functions contin-
uously differentiable of order k, 1 ≤ к ≤ co.)
By properly choosing our coordinates these equations may be put in
the form
z1 = и + iι> = w
z2 = g(w),
where g is complex valued and vanishes to second order at и = v = 0.
Then p is exceptional if and only if the determinant of the Jacobian
J = J(zl,z2∕ι∣,v) vanishes at p = 0.
Expanding g(w) about p = 0 we have
g(w) = αw2 + yιvψ + βw2 + A(ιv),
where λ vanishes to third order at w = 0. Assuming у ≠ 0 and using co-
ordinate changes (see [7]) we obtain g(vv) = β(w2 + ιv2) + w + λ(w),
where β ≥ 0. If we assume ∖β∖ ≠ ⅛∣y∣, we find β ≠ -2-.
Definition 2.1. If iβ<⅛, then β(w2 + wz) + wiv = 2∕3(n2 — t>2) +
ιt2 + v2 = c, for a positive constant c, is the equation of an ellipse, and p
is called a point of elliptic type.
Definition 2.2. If ∕J>⅜, then β(yv2 + ιi>2) + w = c is the equation
of a hyperbola and p is called a point of hyperbolic type.
Definition 2.3. If β = ∣, we say that p is a point of parabolic type.
Note that if β < ɪ. then the point p is a minimum point for the function
Reg, while if β > ⅜, the point p is a saddle point for the function Reg.