THE ENVELOPE OF HOLOMORPHY OF A TWO-MANIFOLD IN C2 53
not exist a point such that the tangent space to the manifold at this point
contains a nonzero complex subspace of C".
In [13] it was proven that a totally real oriented two-manifold embedded
in C2 must have Euler number zero and thus be a torus. In [lð] it was
shown that a totally real submanifold is not extendible.
Denote by ʃ the set of embeddings of the torus in C2 such that the
embedded tori are not totally real. In Section 4 we shall prove the following
theorem.
Theorem 1.2. There is an open dense set of embeddings in ST such
that each two-torus embedded by an element in this dense set has the
property that its envelope of holomorphy contains a real three-dimensional
fβa manifold (1 ≤ n < ∞).
There is an example of a torus which is embedded in C2 and is not totally
real. Let T2 be the standard torus in R3 ⊂ C2 (R3 = {(z∣,z2)eC⅛z2=0})
with its axis of symmetry as the Rez1 axis. There are exactly four points
on the torus T2 which have tangent spaces parallel to the plane z2 = 0.
If we consider an oriented two-manifold of positive genus, we have the
following theorem.
Theorem 1.3. There is an open set of embeddings of an oriented
compact two-manifold of positive genus in C2 with the following property:
each manifold embedded by an element in this open set has an envelope
of holomorphy which contains a three-manifold.
In Section 2 we discuss elliptic and hyperbolic points on a two-manifold
in C2 and state a local theorem on extendibility.
Thom Transversality Theory is the topic of Section 3. We shall give
definitions and theorems and apply the theory to a two-manifold in C2.
Let M2 be an oriented two-manifold embedded in C2 such that M2
has only a finite number of exceptional points, and each exceptional point
Isofeitherellipticorhyperbolictype. In Section 4 we discuss Bishop’s proof
that the number of elliptic points on M2 minus the number of hyperbolic
points on M2 equals the Euler number of M2.
2. Elliptic and hyperbolic points on a two-manifold in C2.
Let M2 denote a real two-dimensional differentiable manifold embedded
in C2. A point p in M2 will be called exceptional if the tangent space to
M2 at p is a complex-linear subspace of C2 of complex dimension one.
If TfC2 ) denotes the real tangent space to C2 at x ∈ C2, we have an
almost complex tensor J : TfC2) → TfC2) given by the complex structure
on C2. J is given by multiplication by i.