The name is absent

Define H_p(M²) = T_p(M²) П JT_p(M²) where peM² and T_p(M²) is the
real tangent space to M² at p. The vector space H_p(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 H_p(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 ⅞f^fc 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(z_l,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 + w^z) + 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.

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