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The holomorphic Gauss Parametrization

We give a local parametric description of all holomorphic hypersurfaces in complex Euclidean and projective spaces with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.


Introduction
Let M n be a complete connected complex immersed hypersurface of C n+1 whose index of relative nullity, that is, the dimension of the kernel of its second fundamental form, satisfies ν ≥ n − 1 everywhere. Equivalently, its Gauss map ϕ: M n → CP n that assigns to each point in M n its normal complex line in C n+1 , satisfies rank dϕ ≤ 1. Then, it was shown by Abe ( [1]) that the hypersurface must be an (n − 1)-cylinder.
The situation is even more restrictive for a complete hypersurface M n of the complex projective space CP n+1 . From a general result also due to Abe ([2]) it follows that if ν > 0 then M n must be a totally geodesically embedded CP n ⊂ CP n+1 .
Naturally, the situation is quite different in the local case. In fact, for any integer ν 0 > 0 there are plenty of local hypersurfaces M n in C n+1 and CP n+1 with constant index of relative nullity ν = ν 0 that are neither part of cylinders in C n+1 or totally geodesic in CP n+1 .
Our main goal in this note is to give a parametric description of all complex hypersurfaces in C n+1 and CP n+1 with constant ν > 0. As a consequence, the above global results will be immediate corollaries of our local construction achieved by imposing on the hypersurfaces the absence of singularities.
So far everything just said is an analogue to what happens for real hypersurfaces in Euclidean space R n+1 and the round sphere S n+1 . The now called Gauss parametrization was introduced by Sbrana ( [13]) as a tool to classify the locally isometrically deformable Euclidean hypersurfaces. In recent years, it has proved to be quite a useful tool, giving rise to several applications; see [3], [4], [5], [6], [7], [9] and [11].
The parametrization of hypersurfaces in C n+1 we give here works similarly for hypersurfaces in R n+1 and provides an equivalent form of the Gauss parametrization given in [5]. Our parametrization for CP n+1 is a perfect analogue of the Gauss parametrization in the sphere S n+1 . We point out that the holomorphicity hypothesis is redundant for submanifolds in CP N with ν > 0 ( [8]).

The parametrization in C N
Let f : M n → C n+p be a holomorphic isometric immersion of a Kähler Riemannian manifold M n = (M n , , ) with Levi-Civita connection ∇, normal connection ∇ ⊥ , and second fundamental form α : T M ⊕ T M → T ⊥ M. We denote by J the complex structures of both M n and C n+1 .
Recall that the relative nullity subspace ∆(x) of f at x ∈ M n is given by Along the open dense subset M 0 ⊆ M n where ν is constant, it is well known that ∆ is a smooth integrable distribution whose leaves are totally geodesic in both M n and C n+p . Locally on a saturated open subset U ⊆ M 0 , the space of leaves of this distribution, that we denote byM = U/∆, is naturally a complex manifold of dimension n−ν whose projection π: U →M n−ν is holomorphic. This space can be naturally identified with a complex submanifold of U of dimension n − ν transversal to the leaves of relative nullity. We point out that the space of leaves is well defined globally on M 0 if M n is complete; see [5]. Let i: C N * → C N * be the inversion given by i(z) = z/ z 2 . Its differential is stands for the reflection in the z direction. Notice that R z satisfies where J is the complex structure in C N . Let f : M n → C n+p be a holomorphic isometric immersion of a Kähler manifold. Decompose the position vector of the immersion as according to the orthogonal holomorphic bundle decomposition for each x ∈ M n .
If there is an open subset U ⊆ M n for which the position vector f is tangent to f , then by analyticity f is everywhere tangent and, regarded as a tangent vector field, must belong to the relative nullity. Hence, f must be a (complex) cone through the origin. By means of a generic translation f + p 0 , we assume from now on that the position vector is not tangent on an open dense subset of M n , that we continue calling M n .
Differentiating (1) and taking normal components, we get where (3) In particular, if the codimension is p = 1 we conclude that that is, g is anti-holomorphic. Moreover, observe that since f is holomorphic, by (2) and (5) we have that ker dg = ∆ at each point. In other words, g is constant along the leaves of relative nullity of f and, locally, there is an anti-holomorphic immersion f:M n−ν → C n+1 such that g =f • π. We will always consider onM n−ν the Kähler metric induced byf.
We have proved: Observe that, as a consequence, the Gauss map N: M n → CP n of f given by is anti-holomorphic; see [12].
Our purpose now is to describe f locally by means of the geometry off .
Theorem 2. Letf :M n−ν → C n+1 be an anti-holomorphic isometric immersion of a Kähler manifold with νf = 0 whose position vector is never tangent. Let L be the holomorphic vector subbundle given by Then, the map f : L → C n+1 defined as parametrizes, at regular points, a holomorphic Kähler hypersurface with constant index of relative nullity ν f = ν. Conversely, any such hypersurface can be parametrized this way.
Proof: For the direct statement, observe first that f is holomorphic by Proposition 1.
Moreover, since f,f • π = 1, we have that that is,f • π is normal to f . From the definition, it is clear that the fibers of L are contained in the relative nullity of f , and they must coincide sincef is never tangent. For the converse, we follow the arguments before Proposition 1 writing Since g is normal to f and ker dg = ∆, by dimension reasons we conclude that the leaf of ∆ through x is simply (contained in) a translation of L(π(x)) defined by (6). Therefore, we set where h ∈ L ⊥ and ξ(x) ∈ L(π(x)). Again by dimension reasons, ξ parametrizes each leaf of L when x moves along a leaf of relative nullity. Now, differentiating f, g = 1, we obtain f, dg = 0. It follows that By (6) and (8) we have that i(f) + h andf ⊥ are linearly dependent, say, i(f ) + h = λf ⊥ Since h is tangent to f , taking the inner product withf yields i(f ) + h = i(f ⊥ ), as we wanted to prove.

Remark 3.
Let H + and H − denote the sets of hypersurfaces in C N without relative nullity and whose position vectors are never tangent, that are holomorphic and anti-holomorphic, respectively. Since the roles of holomorphic and anti-holomorphic submanifolds can be reversed in the above arguments, the map defined on H + ∪ H − given by is a bijection that swaps H + with H − such that (f * ) * = f . In the case of holomorphic curves it was shown in [10] that this map is conformal.
We now compute the singular set and second fundamental form of the submanifold using the parametrization (7), the latter being completely determined by Af by the holomorphicity of f . Let P : (7) is

Proposition 4. The singular set of f in the parametrization
In particular, S is also the singular set of the submanifold itself.
Proof: Take x ∈M n−ν and ξ ∈ L(x). From (7) we see that df ξ is the identity on L(x). Notice that any vector transversal to L(x) at ξ can be written as ψ * x Z = dψ x (Z) for some Z ∈ T xM and ψ ∈ Γ(L) such that ψ(x) = ξ. Sincef is always normal to f , we have where a subspace as a subindex means to take its orthogonal projection. For the last equality, first observe that and then use thatf is always normal to f to compute the functions λ j , j = 1, 2. The first claim now follows from the fact that the right-hand side of (10) depends only on Z and not on ψ.
The second part now follows since as we wanted.
Recall that first normal space off at x ∈M n−ν is the subspace of N 1 f (x) ⊆ T ⊥ xM spanned by the image of the second fundamental form off at x. Equivalently, where the orthogonal complement is taken in the normal bundle.
Proof: Assume that the conclusion does not hold. Hence, the polynomial has a complex root u + iv, associated to an eigenvector U + iV = 0 of the corresponding complexified endomorphism, that is,

But this is equivalent tô
In turn, using JÂ δ =Â Jδ = −Â δ J, we easily see that this is equivalent tô Since L is holomorphic and the leaves of relative nullity are complete, we get a contradiction with Proposition 4, because either U − JV or U + JV is non-zero.

Remark 6.
Observe that the previous result holds along each complete relative nullity leaf of f , even if the submanifold is not itself complete.
As an application of Theorem 2 we give a simple and direct proof of Abe's cylinder theorem ( [1]).
Corollary 7. Let f : M n → C n+1 be a holomorphic isometric immersion of a complete Kähler Riemannian manifold. If the index of relative nullity satisfies ν ≥ n − 1 everywhere, then f is an (n−1)-cylinder, that is, M n = M 1 1 ×C n−1 , and there is f 1 : Proof: If f is not totally geodesic, for which the result trivially holds, by the hypothesiŝ f is an anti-holomorphic curve and, by Corollary 5, we have But since L is orthogonal to the position vectorf , we conclude that the first normal space is parallel since 0 = ψ * Z,f = ψ * Z, i(f ⊥ ) for any ψ ∈ Γ(L). This parallelism implies thatf reduces codimension, that is, it is an anti-holomorphic plane curve inside some C 2 ⊂ C n+1 , and L is the orthogonal complement of this plane.

The parametrization in CP N
We show next that our parametrization in C n+1 can be used to obtain a similar parametrization for holomorphic hypersurfaces of CP n+1 . The latter is cleaner than the former since it does not have the restriction about the position vectors to be nowhere tangent, and the bundle used to parametrize is the (projectivized) normal bundle itself and not a subbundle of it.
The condition in Theorem 2 that the position vector of a complex submanifold f : M n → C n+p is never tangent is equivalent to the cone f c over f to be an immersion, where the map f c : C * × M n → C n+p is given by f c (z, x) = zf (x). Here we denote as usual C N * = C N \ {0}. Moreover, f has index of relative nullity ν if and only if f c has index of relative nullity ν + 1, and the position vector of the cone, that now is everywhere tangent, belongs to the relative nullity. Equivalently, the position vector of f is never tangent if and only if f ♯ =π • f : M n → CP n+p−1 is an immersion, wherê π: C N * → CP N −1 denotes the projection to the quotient, and f and f ♯ have the same index of relative nullity. We conclude that to understand the submanifolds with constant relative nullity ν 0 in CP N is equivalent to understand the cones in C N +1 with ν ≡ ν 0 +1.
We claim that the latter are described as in (7), but without the term i(f ⊥ ). Let f 1 : M n−1 1 → C n ⊂ C n+1 be the isometric immersion obtained as the intersection of a cone f : M n → C n+1 with constant index of relative nullity ν + 1 with a hyperplane, say, C n = {z n+1 = 1}, so that f 1 has constant index of relative nullity ν and is never tangent. By Theorem 2, we have a parametrization of f 1 in C n as Thus, f 1 = (η + ξ, 1) in C n+1 where η = i(f ⊥ 1 ). Hence, we may parametrize f as f (w, ξ) = w(η, 1) + ξ, ξ ∈ L 1 , w ∈ C * .
Settingf = (f 1 , −1), we thus parametrize f as wheref :M n−ν → C n+2 is never tangent, and This proves our claim. From the above description of the cones, we get for the immersion f ♯ a parametrization over the projectivized bundle P (L) of L, namely, f ♯ : P (L) → CP n+1 , Now, observe that L coincides with the normal space off c , once we identify the fibers of the normal space off c when translated along the lines inside the cone that pass through 0 (we are allowed to do this because these are lines of relative nullity off c ). In other words, we have a natural identification between the normal space off ♯ and L, and hence we can treat both as the same fiber bundle. In particular, the corresponding complex projectivized bundles are also identified: P (T ⊥ f ♯M ) = P (L). These are holomorphic fiber bundles of dimension n with CP ν fibers. We conclude from (11) the following. We point out that the holomorphicity hypothesis in the converse is redundant when the submanifold has relative nullity. It was shown in [8] that any isometric immersion of a Kähler manifold into CP N with positive index of relative nullity must be holomorphic.
Remark 9. Taking the Gauss map is an involution onH + ∪H − that swapsH + withH − , whereH + andH − denote the sets of holomorphic and anti-holomorphic hypersurfaces of CP n+1 with vanishing relative nullity. As opposed to the C N case, here there is no restriction on the position vectors; see Remark 3.