LEVI UMBILICAL SURFACES IN COMPLEX SPACE

ROBERTO MONTI AND DANIELE MORBIDELLI

Abstract. We define a complex connection on a real hypersurface of ℂ^{n+1} which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in ℂ^{n+1}, n ≥ 2, which are Levi umbilical and have non zero constant Levi curvature. It turns out that such surfaces are contained either in a sphere or in the boundary of a complex tube domain with spherical section.

1. Introduction

Let M be a (2n + 1)-dimensional real surface embedded in ℂ^{n+1}, denote by h the ℂ-linear extension of the second fundamental form of M and by g be the restriction to the complexified tangent bundle ℂT M of the standard hermitian product of ℂ^{n+1}. The surface M is Levi umbilical if h(Z, W̄) = Hg(Z, W̄) for some scalar function H (the Levi curvature) and for all holomorphic tangent vector fields Z and W. Levi umbilicality is weaker than Euclidean umbilicality because it contains no information on terms of the form h(Z, W) with holomorphic Z and W. In particular, it is easy to construct Levi umbilical surfaces which are neither spheres nor hyperplanes. Indeed, any surface which is the zero set F = 0 of a smooth defining function F (z, z̄) = |z|^{2} + Φ(z, z̄), where Φ is any polyharmonic function in ℂ^{n+1}, is Levi umbilical (see Example 2.3).

In view of these examples, a natural question is whether there is any version of the classical Darboux theorem for usual umbilical surfaces. In this paper we classify Levi umbilical surfaces with constant non zero Levi curvature. An example of such surfaces are, of course, the spheres {z ∈ ℂ^{n+1} : |z| = r}, r > 0. A less trivial example is the boundary of spherical tubes, i.e. surfaces of the form (see Example 2.2)

{z ∈ ℂ^{n+1} : ∑_{h=1}^{n+1}(z_{h} + z̄_{h})^{2} = r^{2}}, r > 0.  (1.1)

Our main result states that there are no other examples. More precisely, we prove that any (2n + 1)-dimensional oriented connected surface embedded in ℂ^{n+1}, n ≥ 2, which is Levi umbilical and has non zero constant Levi curvature is necessarily contained either in a sphere or, up to complex isometries of ℂ^{n+1}, in a spherical cylinder of the form (1.1). This is proved in Theorem 5.1. It is interesting to observe the appearance of tube domains, which are relevant objects in several complex variables, see [Kr].

Date: 8th July 2005.

2 ROBERTO MONTI AND DANIELE MORBIDELLI

This classification follows from the analysis of a system of Codazzi equations for h, where covariant derivatives are computed with respect to a suitable complex connection ∇ on M. Though very natural, this connection and the corresponding Codazzi equations do not seem to be studied in the literature. The main features of ∇ are:

(a) both the holomorphic and the antiholomorphic bundles are parallel;

(b) the restriction g to ℂT M of the hermitian product in ℂ^{n+1} satisfies ∇g = 0. Briefly, the connection is constructed in the following way. Let ν be a real unit normal to M and consider N = 2^{−1/2} (ν − iT), the holomorphic unit normal to M. Here, T = J(ν) where J is the standard complex structure of ℂ^{n+1}. Then, given a holomorphic tangent vector field Z and a tangent vector U, we define

∇_{U}Z = D_{U}Z − g(D_{U}Z, N̄)N,

where D is the standard connection in ℂ^{n+1}. Then, this definition, along with ∇T = 0, is extended to the whole tangent bundle, giving rise to a connection satisfying (a) and (b) (see Section 3).

Properties (a) and (b) are similar to the ones of the Tanaka-Webster connection on strictly pseudoconvex Cauchy-Riemann manifolds (see [T] and [W]). Whereas for this connection the Levi form −idϑ associated with a contact form ϑ plays the role of the metric and is required to be parallel, in our case the metric inherited from ℂ^{n+1} is required to be parallel. See also the discussion in Remark 3.2. This produces a connection which seems to be more suitable for our purposes. A different connection is introduced by Klingenberg in [Kl]. It arises as orthogonal projection of the standard connection in the space and, in general, does not satisfy property (a).

A typical example of Codazzi equation for h, written in components with respect to a holomorphic frame Z_{1}, ..., Z_{n}, is (see Remark 4.2)

∇_{α}h_{βγ̄} − ∇_{β}h_{αγ̄} = ih_{βγ̄}h_{α0} − ih_{αγ̄}h_{β0},  (1.2)

where h_{αβ̄} = h(Z_{α}, Z̄_{β}) for α, β = 1, ..., n and index 0 refers to T. In Theorem 4.1, we compute the system of equations needed in the classification theorem. In these equations, as in (1.2), there is a non vanishing right hand side, reflecting both the non vanishing of Tor_{∇} and the non vanishing of g(D_{Z}N, N̄).

Concerning the restriction n ≥ 2 in the classification theorem, note that for n = 1 the umbilicality property is satisfied by any hypersurface of ℂ^{2}. Moreover, by the existence and regularity results proved by Slodkowski and Tomassini [ST] and Citti, Lanconelli and Montanari [CLM] for the Levi equation, there are smooth graphs in ℂ^{2} with prescribed boundary and with constant Levi curvature which do not belong to the classes described above. Then, the natural question is whether a compact surface in ℂ^{2} having constant Levi curvature is necessarily a sphere. This question has been recently addressed in [HL] by Hounie and Lanconelli, who give an affirmative answer in the class of Reinhardt domains.

Another result implied by our Codazzi equations is the classification of connected pseudoconvex surfaces with non zero constant Levi curvature and vanishing h_{αβ} (the symmetric part of the second fundamental form). Up to complex isometry, such surfaces are contained in a sphere or in a spherical cylinder of the form

{z ∈ ℂ^{n+1} : ∑_{i=m}^{n+1} |z_{i}|^{2} = r^{2}}, r > 0, 1 ≤ m ≤ n.

This is established in Theorem 5.2, which improves Theorem 5.2 in [Kl], where the result is proved by a global argument under compactness and strict pseudoconvexity assumptions (see Remark 5.3).

The notion of Levi curvature was introduced by Bedford and Gaveau in [BG] and it has been recently generalized by Montanari and Lanconelli in [ML]. There is an increasing interest on problems concerning this curvature, mainly from the point of view of partial differential equations. Other significant references are Citti and Montanari [CM], Huisken and Klingenberg [HK] and Montanari and Lascialfari [MLa]. The tools developed in this work could be useful in the study also of other problems concerning real hypersurfaces in complex space.

Concerning terminology, we call “Levi form” the hermitian map (Z, W) ↦ h(Z, W̄), with holomorphic Z and W. This is justified by the fact that h(Z, W̄) coincides with the Levi form associated with a natural pseudohermitian structure (see [W] or [JL] for this notion) inherited by M from the ambient (see the discussion in Section 2).

Notation. Greek indices α, β etc. run from 1 to n, Latin indices h, k run from 1 to n + 1. We let ∂_{h} = \frac{∂}{∂z_{h}}, ∂h̄ = \frac{∂}{∂z̄_{h}} and F_{h} = ∂_{h}F. J is the standard complex structure and D is the usual connection in ℂ^{n+1}. The standard hermitian product g in ℂ^{n+1} is normalized by g(∂_{h}, ∂k̄) = g(∂h̄, ∂_{k}) = \frac{1}{2}δ_{hk}, g(∂_{h}, ∂_{k}) = g(∂h̄, ∂k̄) = 0, where δ_{hk} is the Kronecker symbol. The metric tensors g_{αβ̄} and g^{αβ̄}, which are related by g^{αβ̄}g_{γβ̄} = δ_{αγ}, are used to lower and raise indices, e.g. h_{α}^{β} = g^{βγ̄}h_{αγ̄}. If h is symmetric, we equivalently write h_{α}^{β} = h_{α}^{β}. We adopt the summation convention. If E is a bundle we denote by Γ(E) the sections of E. Finally, [U, V] denotes the Lie bracket of vector fields and Tor_{∇}(U, V) = ∇_{U}V − ∇_{V} U − [U, V] is the torsion of the connection ∇.

Acknowledgments. We are indebted to Ermanno Lanconelli and Annamaria Montanari for several fruitful conversations on Levi curvature.

2. Levi form and examples

Let M ⊂ ℂ^{n+1} be a real hypersurface oriented by a real unit normal ν. We denote by H = T^{1,0}M (resp. H̄ = T^{0,1}M) the holomorphic (resp. antiholomorphic) tangent bundle of M. We restrict the complex structure J to H⊕H̄ and the metric g to ℂT M. The vector field T = J(ν) is tangent to M. Then, the complexified tangent bundle ℂT M can be decomposed as a direct sum H ⊕ H̄ ⊕ ℂT and the decomposition is orthogonal with respect to g. The holomorphic unit normal to M is the holomorphic vector field

N = \frac{1}{\sqrt{2}} (ν − iT) .  (2.1)

Up to orientation, N is defined uniquely on M by |N| = 1 and g(N, U) = 0 for all U ∈ H ⊕ H̄. Here and in the following, |V |^{2} = g(V, V̄). We have the relations

T = \frac{i}{\sqrt{2}}(N − N̄), ν = \frac{1}{\sqrt{2}}(N + N̄).  (2.2)

There is a unique real 1-form η on M such that

η(T) = 1 and η(Z) = 0 for all Z ∈ H ⊕ H̄.  (2.3)

Precisely, η(Z) = g(Z, T) for any Z ∈ ℂT M. The Levi form on M associated with η is the hermitian form on H defined by

L_{η}(Z, W̄) = \frac{1}{2i}dη(Z, W̄), Z, W ∈ H.  (2.4)

Denote by h the ℂ-linear extension to ℂT M × ℂT M of the second fundamental form of M. For Z, W ∈ ℂT M let

h(Z, W) = g(Z, D_{W} ν).  (2.5)

Note that h(Z, W) = h(W, Z) and \overline{h(Z, W}) = h(Z̄, W̄).

The Levi form associated with η coincides with the hermitian part of the second fundamental form, i.e. L_{η}(Z, W̄) = h(Z, W̄) for all Z, W ∈ H. Indeed, by (2.3) and (2.2),

\begin{aligned}dη(Z, W̄) & = Zη(W̄) − W̄ η(Z) − η([Z, W̄]) = −η([Z, W̄]) \\ & = g([W̄ , Z], T) = \frac{i}{\sqrt{2}}g(DW̄ Z − D_{Z} W̄ , N − N̄). \end{aligned}  (2.6)

Since g(DW̄ Z, N) = g(D_{Z} W̄ , N̄) = 0, we find

dη(Z, W̄) = −\frac{i}{\sqrt{2}}(g(DW̄ Z, N + N̄) + g(D_{Z} W̄ , N + N̄)) = −2ig(DW̄ Z, ν). (2.7)

The claim follows.

The Levi curvature H of M is the trace of the Levi form. The surface M is Levi umbilical if h(Z, W̄) = Hg(Z, W̄) for all Z, W ∈ H. In order to express these definitions in components, fix a frame Z_{1}, ..., Z_{n} of holomorphic tangent vector fields. Let h_{αβ̄} = h(Z_{α}, Zβ̄) and g_{αβ̄} = g(Z_{α}, Zβ̄). The Levi curvature of M is

H = \frac{1}{n}h_{α}^{α}.  (2.8)

The surface M is Levi umbilical if h_{αβ̄} = Hg_{αβ̄}. Observe that the relation between the Levi curvature H = H_{ℂ} and the standard mean curvature H_{ℝ} is (2n + 1)H_{ℝ} = 2nH_{ℂ} + h(T, T).

It is useful to compute the Levi curvature by means of a defining function. Let M = {z ∈ ℂ^{n+1} : F (z) = 0} for some smooth function F : ℂ^{n+1} → ℝ. The holomorphic unit normal is

N = \sqrt{2}\frac{Fh̄}{|∂F |}∂_{h},  where |∂F |^{2} = F_{h}Fh̄.  (2.9)

LEVI UMBILICAL SURFACES IN COMPLEX SPACE 5

The complex Hessian D^{2}F induces a hermitian form on holomorphic vector fields of ℂ^{n+1} by letting D^{2}F (U, V̄) = U^{h}V k̄F_{hk̄}, where U = U^{h}∂_{h} and V̄ = V k̄∂k̄. As observed in [ML], the Levi form can be written as

h(U, V̄) = \frac{1}{2|∂F |}D^{2}F (U, V̄), U, V ∈ H.  (2.10)

Moreover, the Levi curvature of M is

H = \frac{1}{n|∂F |} (F_{hh̄} − \frac{F_{k}Fh̄F_{hk̄}}{|∂F |^{2}} ) .  (2.11)

We briefly check (2.10). By (2.2) and (2.9), we have

h(U, V̄) = g(U, DV̄ ν) = \frac{1}{\sqrt{2}}g(U, DV̄ (N + N̄)) = g (U, DV̄ (\frac{F_{h}}{|∂F |}∂h̄)) .

As g(F_{h}∂h̄, U) = 0, we get

g (U, DV̄ (\frac{F_{h}}{|∂F |}∂h̄)) = \frac{1}{|∂F |}g(U, DV̄ (F_{h}∂h̄)) = \frac{1}{2|∂F |}D^{2}F (U, V̄).

In order to prove (2.11), assume, for instance, F_{n+1} ≠ 0 near a point P ∈ M and consider the local holomorphic frame near P

Z_{α} = ∂_{α} − \frac{F_{α}}{F_{n+1}} ∂_{n+1}, α = 1, ..., n.  (2.12)

The application of (2.10) to the Z_{α}’s gives

h_{αβ̄} = \frac{1}{2|∂F |}\{F_{αβ̄} − \frac{Fβ̄}{F\overline{n+1}} F_{α\overline{n+1}} − \frac{F_{α}}{F_{n+1}} F_{n+1,β̄} + \frac{F_{α}Fβ̄}{|F_{n+1}|^{2}} F_{n+1,\overline{n+1}}\}.

The metric tensor and its inverse are respectively

g_{αβ̄} = \frac{1}{2} (δ_{αβ} + \frac{F_{α}Fβ̄}{|F_{n+1}|^{2}} ) and g^{αβ̄} = 2 (δ_{αβ} − \frac{F_{β}Fᾱ}{|∂F |^{2}} ) .

Then, a short computation gives

H = \frac{1}{n}g^{αβ̄}h_{αβ̄} = \frac{1}{n|∂F |} (F_{hh̄} − \frac{F_{k}Fh̄F_{hk̄}}{|∂F |^{2}} ) .

In the next proposition we collect some useful identities.

Proposition 2.1. Let M ⊂ ℂ^{n+1} be an oriented surface with real unit normal ν, T = J(ν) and holomorphic unit normal N. Then:

i) g(D_{Z}N, N̄) = g([T, Z], T) for all Z ∈ Γ(H); ii) g(D_{Z}N, N̄) = ih(T, Z) for all Z ∈ Γ(ℂT M); iii) g([Z, W̄], T) = −2ih(Z, W̄) for all Z, W ∈ Γ(H).

Proof. Note that g(D_{Z}T, T) = 0, because T is real. Moreover, by (2.2), we have for any Z ∈ Γ(H)

g([T, Z], T) = g(D_{T} Z − D_{Z}T, T) = \frac{1}{2}g(D_{N−N̄}Z, N̄⟩.

6 ROBERTO MONTI AND DANIELE MORBIDELLI

We used the orthogonality g(D_{N−N̄}Z, N̄) = 0, which holds because Z is holomorphic. Thus

2g([T, Z], T) = g(D_{Z}N + [N, Z], N̄) − g(D_{Z} N̄ + [N̄, Z], N̄)

= g([N − N̄, Z], N̄) + g(D_{Z}N, N̄)

= \frac{1}{i}g([T, Z], ν + iT) + g(D_{Z}N, N̄) = g([T, Z], T) + g(D_{Z}N, N̄).

We used again (2.2) and g([T, Z], ν) = 0. This proves i).

In order to check ii), note that

\begin{aligned}g(D_{Z}N, N̄) & = g(D_{Z}N, N + N̄) = \sqrt{2}g(D_{Z}N, ν) = g(D_{Z}ν, ν) − ig(D_{Z}T, ν) \\ & = −ig(D_{Z}T, ν) = ih(T, Z). \end{aligned}

Identity iii) is proved in (2.6)–(2.7).

Now we discuss a couple of examples showing the existence of non trivial Levi umbilical surfaces.

Example 2.2 (Boundary of spherical tubes). The surface M = {z ∈ ℂ^{n+1} : F (z) = 0}, where

F (z) = \frac{1}{2} ∑_{h=1}^{n+1}(z_{h} + z̄_{h})^{2} − 1,

is a Levi umbilical cylinder with spherical section having constant Levi curvature. Indeed, the complex derivatives of F are F_{h} = Fh̄ = z_{h} + z̄_{h} and F_{hk̄} = δ_{hk}. Then |∂F | = \sqrt{2} and, by (2.11), the Levi curvature is H = 1/\sqrt{2}. The complex Hessian of F is the identity and, by (2.10), the condition h_{αβ̄} = \frac{1}{\sqrt{2}}g_{αβ̄} is identically satisfied on M.

Example 2.3. It is possible to construct compact Levi umbilical surfaces by polyharmonic perturbations of the sphere. Consider

M = {z ∈ ℂ^{n+1} : |z|^{2} + λΦ(z) = 1} ,  (2.13)

where λ is a real parameter and

Φ(z) = \frac{1}{2} ∑_{h=1}^{n+1}(z_{h}^{2} + z̄_{h}^{2}).

The derivative of the defining function F (z) = |z|^{2} + λΦ(z) − 1 are F_{h} = z̄_{h} + λz_{h} and F_{hk̄} = δ_{hk}. On the set M we have |∂F (z)|^{2} = 2 − (1 − λ^{2})|z|^{2}. Then, |∂F | is constant on M if and only if λ = 0, 1, −1. If |λ| < 1, M is a smooth compact surface bounding the region {z ∈ ℂ^{n+1} : F (z) < 0}. Indeed, M is an ellipsoid: letting z = x + iy, we have F (z) = (1 + λ)|x|^{2} + (1 − λ)|y|^{2} − 1. Moreover, on M |∂F (z)|^{2} = 2 − (1 + λ)(1 − 2λ|x|^{2}) ≥ 1 − λ > 0. By formula (2.11), the Levi curvature of M is H = |∂F |^{−1}. The complex Hessian of F is the identity and, by (2.10), the surface M is Levi umbilical and

h_{αβ̄} = \frac{1}{|∂F |}g_{αβ̄}.  (2.14)

LEVI UMBILICAL SURFACES IN COMPLEX SPACE 7

Many other examples of compact Levi umbilical surfaces can be constructed, taking as Φ in (2.13) any polyharmonic function, i.e. any smooth function satisfying Φ_{hk̄} = 0. In fact, the complex Hessian of the corresponding defining function is the identity. Therefore condition (2.14) is satisfied.

3. The connection and its properties

In this section, we define the covariant derivative ∇ on an oriented, smooth hypersurface M ⊂ ℂ^{n+1} starting from the standard connection D in ℂ^{n+1}. A vector field V ∈ Γ(ℂT M) can be uniquely decomposed as

V = Z + W̄ + fT,  (3.1)

where Z, W ∈ Γ(H) and f ∈ C^{∞}(M) is a complex valued function. We define ∇ : Γ(ℂT M) × Γ(ℂT M) → Γ(ℂT M) by letting, for U, V ∈ Γ(ℂT M) with V as in (3.1),

∇_{U}V = D_{U}Z − g(D_{U}Z, N̄)N + D_{U} W̄ − g(D_{U} W̄ , N)N̄ + (Uf)T.  (3.2)

Here, N is the holomorphic unit normal. Equivalently, let for U ∈ Γ(ℂT M) and Z, W ∈ Γ(H)

\begin{matrix}∇_{U}Z = D_{U}Z − g(D_{U}Z, N̄)N, \\ ∇_{U} W̄ = D_{U} W̄ − g(D_{U} W̄ , N)N̄, \\ ∇_{U}T = 0. \end{matrix}  (3.3)

We have the following

Theorem 3.1. ∇ is a complex connection on M and satisfies the following properties: (C1) \overline{∇_{U}V} = ∇Ū V̄ for all U, V ∈ Γ(ℂT M); (C2) ∇_{U}(J(V)) = J(∇_{U}V) for all U, V ∈ Γ(ℂT M); (C3) The bundles H and H̄ are parallel; (C4) ∇g = 0; (C5) Tor_{∇}(U, V) = 0 for all U, V ∈ Γ(H); (C6) Tor_{∇}(U, V̄) = −g([U, V̄], T)T for all U, V ∈ Γ(H).

Proof. Properties (C1), (C2) and the fact that ∇ is a connection are easy and we omit their proof.

Property (C3) amounts to say that the covariant derivative of a holomorphic (resp. antiholomorphic) vector field is still a holomorphic (resp. antiholomorphic) vector field. But this is an immediate consequence of definition (3.2) and of the orthogonal decomposition T_{P}^{1,0}ℂ^{n+1} = H_{P} ⊕ ℂN_{P} , at any point P ∈ M.

In order to prove property (C4), let

V_{1} = Z_{1} + W̄_{1} + f_{1}T, V_{2} = Z_{2} + W̄_{2} + f_{2}T,

where Z_{1}, Z_{2}, W_{1}, W_{2} ∈ Γ(H) and f_{1}, f_{2} are complex valued functions. By the metric property of the standard connection D in ℂ^{n+1}, we have

\begin{aligned}Ug(V_{1}, V̄_{2}) & = Ug(Z_{1}, Z̄_{2}) + Ug(W̄_{1}, W_{2}) + Ug(f_{1}T, f̄_{2}T) \\ & = g(D_{U}Z_{1}, Z̄_{2}) + g(Z_{1}, D_{U} Z̄_{2}) + g(D_{U} W̄_{1}, W_{2}) + g(W̄_{1}, D_{U}W_{2}) \\ & + g(D_{U}(f_{1}T), f̄_{2}T) + g(f_{1}T, D_{U}(f̄_{2}T)). \end{aligned} (3.4)

We claim that the following identities hold

\begin{matrix}g(D_{U}Z_{1}, Z̄_{2}) = g(∇_{U}Z_{1}, V̄_{2}), & g(D_{U} W̄_{1}, W_{2}) = g(∇_{U} W̄_{1}, V̄_{2}), \\ g(Z_{1}, D_{U} Z̄_{2}) = g(V_{1}, ∇_{U} Z̄_{2}), & g(W̄_{1}, D_{U}W_{2}) = g(V_{1}, ∇_{U}W_{2}). \end{matrix}  (3.5)

We check the first one only. Since g(N, Z̄_{2}) = 0, we have g(D_{U}Z_{1}, Z̄_{2}) = g(∇_{U}Z_{1}, Z̄_{2}) and property (C3) gives g(∇_{U}Z_{1}, Z̄_{2}) = g(∇_{U}Z_{1}, V̄_{2}). The following identities also hold

g(D_{U}(f_{1}T), f̄_{2}T) = g((Uf_{1})T, V̄_{2}),  g(f_{1}T, D_{U}(f̄_{2}T)) = g(V_{1}, (Uf̄_{2})T). (3.6)

We check the first one. Since g(D_{U}T, T) = 0, then g(D_{U}(f_{1}T), f̄_{2}T) = g((Uf_{1})T, f̄_{2}T)+ g(f_{1}D_{U}T, f̄_{2}T) = g((Uf_{1})T, f̄_{2}T). But T is orthogonal to Z_{2} and W̄_{2}. Thus we get the claim. Replacing (3.5) and (3.6) into (3.4) we get Ug(V_{1}, V̄_{2}) = g(∇_{U}V_{1}, V̄_{2}) + g(V_{1}, ∇_{U} V̄_{2}), which means ∇g = 0.

Statement (C5), Tor_{∇}(U, V) = 0 for U, V ∈ Γ(H), follows from Tor_{D}(U, V) = 0 and [U, V] ∈ Γ(H). Concerning property (C6), observe that a connection leaving H and H̄ parallel cannot be, in general, torsion free, because the horizontal distribution needs not be integrable (in other words, it may be [H, H̄] ⊈ H ⊕ H̄). Take W ∈ Γ(H ⊕ H̄). Then

g(Tor_{∇}(U, V̄), W̄) = g(Tor_{D}(U, V̄) + g(DV̄ U, N̄)N − g(D_{U} V̄ , N)N̄, W̄ ) = 0,

because Tor_{D}(U, V̄) = 0 and g(N, W̄) = g(N̄, W̄) = 0. Then Tor_{∇}(U, V̄) = λT for some function λ and, by (C3), λ = g(Tor_{∇}(U, V̄), T) = −g([U, V̄], T).

Recall that the restriction of the hermitian product in ℂ^{n+1} to ℂT M induces the orthogonal decomposition

ℂT M = H ⊕ H̄ ⊕ ℂT.  (3.7)

Denote by Π_{H} : ℂT M → H the projection onto H and by ΠH̄ the projection onto H̄. Then it is easy to check that for U, V ∈ Γ(H), we have

∇ŪV = Π_{H}([Ū, V]) and ∇_{U} V̄ = ΠH̄([U, V̄]).  (3.8)

This follows from (C3) and (C6).

Remark 3.2. If M is a strictly pseudoconvex CR manifold, there is a natural connection associated with a given contact form ϑ, which was introduced by Tanaka and Webster in [T] and [W]. Although it was designed for different scopes from ours, we highlight some analogies and differences between our connection ∇ and the Tanaka-Webster one.

LEVI UMBILICAL SURFACES IN COMPLEX SPACE 9

The Levi form (Z, W) ↦ −idϑ(Z, W̄) is a non degenerate hermitian form on H. Then, ϑ induces a decomposition of ℂT M similar to (3.7). The vector field T is replaced in this construction by the characteristic vector field T′, defined by ϑ(T′) = 1 and dϑ(T, Z) = 0 for all Z ∈ H. In the Tanaka-Webster connection, the Levi form dϑ essentially plays the role of the metric and is required to be parallel. Covariant derivatives of holomorphic vector fields along antiholomorphic ones are defined by relations analogous to (3.8) (see [T, Lemma 3.2, p. 31]), but with the projections Π′_{H} and Π′ ; H̄ induced by dϑ. The characteristic vector field T′ of ϑ is in general different from T for any choice of the contact form ϑ.

Similarly to the Tanaka-Webster connection, the property ∇T = 0 is forced by (C4) and (C5). Indeed, the one dimensional bundle generated by T is the orthogonal complement with respect to the parallel metric g of the parallel bundle H ⊕ H. Then ∇_{U}T = λT for some function λ and U ∈ ℂT M. But, since T is real, 0 = Ug(T, T) = 2g(∇_{U}T, T = 2λ. Therefore ∇T = 0.

Remark 3.3. The connection ∇ is not uniquely determined on the whole tangent bundle Γ(ℂT M) by properties (C1)–(C6). In particular, ∇_{T} U with U ∈ Γ(H) is not uniquely determined. In (3.3), we let ∇_{T} U = D_{T} U − g(D_{T} U, N̄)N. An alternative possibility, consistent with (3.8), is to set

∇′_{T} U = Π_{H}([T, U]) and ∇′_{T} Ū = ΠH̄([T, Ū]).

The resulting connection ∇′ still satisfies (C1)–(C6). Our choice ∇, however, seems to be more suitable than ∇′ to work with Codazzi equations.

Remark 3.4. The real tangent bundle has the orthogonal decomposition T M = Re(H ⊕ H̄) ⊕ ℝT. Then, for Y ∈ Γ(Re(H ⊕ H̄)), V ∈ Γ(T M) and f real function, we have

∇_{V} (Y + fT) = D_{V} Y − g(D_{V} Y, ν)ν − g(D_{V} Y, T)T + (V f)T.  (3.9)

Indeed, taking X = Z + Z̄ with holomorphic Z, we have

∇_{V} (Z + Z̄) = ∇_{V} Z + ∇_{V} Z̄ = D_{V} Z − g(D_{V} Z, N̄)N + D_{V} Z̄ − g(D_{V} Z̄, N)N̄
= D_{V} X − g(D_{V} Z, ν)(ν − iT) − g(D_{V} Z̄, ν)(ν + iT)
= D_{V} X − g(D_{V} X, ν)ν + ig(D_{V} (Z − Z̄), ν)T
= D_{V} X − g(D_{V} X, ν)ν + g(D_{V} (J(X)), −J(T))T
= D_{V} X − g(D_{V} X, ν)ν − g(D_{V} X, T)T.

We used ν = −J(T) and the property D_{V} ∘ J = J ∘ D_{V}.

4. Codazzi equations

In this section we compute the system of Codazzi equations.

Theorem 4.1. The Levi form h on a hypersurface M ⊂ ℂ^{n+1} satisfies the following Codazzi equations

∇_{β}h_{αγ̄} − ∇γ̄h_{αβ} = ih_{αβ}h_{γ̄0} − ih_{αγ̄}h_{β0} − 2ih_{βγ̄}h_{α0},  (4.1a)
∇β̄h_{α0} − ∇_{0}h_{αβ̄} = ih_{αλ}hλ ; β̄ − ih_{αλ̄}hλ̄ ; β̄ + ih_{αβ̄}h_{00},  (4.1b)
∇_{β}h_{α0} − ∇_{0}h_{αβ} = ih_{βλ̄}hλ̄ ; α − ih_{βλ}h_{α}^{λ} + ih_{αβ}h_{00} − 2ih_{α0}h_{β0},  (4.1c)
∇_{α}h_{00} − ∇_{0}h_{α0} = 2ih_{α}^{λ}h_{λ0} − 2ihλ̄ ; αh_{λ̄0} − ih_{αλ}h_{0}^{λ} + ih_{αλ̄}hλ̄ ; 0 − ih_{α0}h_{00}. (4.1d)

Proof. The proof relies on the fact that the standard connection D in ℂ^{n+1} has vanishing curvature. We shall also use several times the formula

D_{Z}U = ∇_{Z}U − \sqrt{2}h(U, Z)N,  U ∈ Γ(H), Z ∈ Γ(ℂT M).  (4.2)

Let Z, W ∈ Γ(ℂT M) and U ∈ Γ(H). Denote by R_{D} and R_{∇} the standard curvature endomorphisms of D and ∇. Using (4.2), we have

\begin{aligned}0 & = R_{D}(Z, W)U = D_{Z}D_{W} U − D_{W} D_{Z}U − D_{[Z,W]}U \\ & = D_{Z}(∇_{W} U − \sqrt{2}h(U, W)N) − D_{W} (∇_{Z}U − \sqrt{2}h(U, Z)N) − D_{[Z,W]}U \\ & = R_{∇}(Z, W)U − \sqrt{2}h(U, W)D_{Z}N + \sqrt{2}h(U, Z)D_{W} N \\ & − \sqrt{2}(Zh(U, W) − h(∇_{Z}U, W) − W h(U, Z) + h(∇_{W} U, Z) − h(U, [Z, W]))N. \end{aligned} (4.3)

Multiplying by N̄ and using g(R_{∇}(Z, W)U, N̄) = 0, we get the equation for h

\begin{aligned}∇_{Z}h(U, W) − ∇_{W} h(U, Z) = h(U, Tor_{∇}(W, Z)) & − h(U, W)g(D_{Z}N, N̄) \\ & + h(U, Z)g(D_{W} N, N̄), \end{aligned}  (4.4)

where ∇_{Z}h(U, W) = Zh(U, W)−h(∇_{Z}U, W)−h(U, ∇_{Z}W) is the covariant derivative of h. Note that by Proposition 2.1, we have g(D_{Z}N, N̄) = ih(Z, T) for any Z ∈ Γ(ℂT M).

In order to prove (4.1a), take Z, W, U ∈ Γ(H) and write (4.4) with W̄ instead of W. By Theorem 3.1, the torsion satisfies Tor_{∇}(Z, W̄) = −g([Z, W̄], T)T. Moreover, by Proposition 2.1 we have g([Z, W̄], T) = −2ih(Z, W̄). Thus, equation (4.4) becomes

∇_{Z}h(U, W̄) − ∇W̄ h(U, Z) = ih(U, Z)h(W̄ , T) − ih(U, W̄)h(Z, T) − 2ih(Z, W̄)h(U, T). (4.5)

This is formula (4.1a).

In order to prove (4.1b), we take Z, U ∈ Γ(H). By (4.4), we have

\begin{aligned}∇Z̄h(U, T) − ∇_{T} h(U, Z̄) = h(U, ∇_{T} Z̄ + [Z̄, T]) & − ih(U, T)h(Z̄, T) \\ & + ih(U, Z̄)h(T, T), \end{aligned}  (4.6)

because ∇T = 0.

We analyze the right hand side of (4.6). By Tor_{D}(Z̄, T) = 0 and the second equation of (3.3), we have

\begin{aligned}∇_{T} Z̄ + [Z̄, T] & = DZ̄T − g(D_{T} Z̄, N)N̄ \\ & = DZ̄T − g(DZ̄T, ν)ν − ig(D_{T} Z̄, ν)T. \end{aligned}  (4.7)

We also used g([Z̄, T], ν) = 0, which implies g(DZ̄T, ν) = g(D_{T} Z̄, ν). The vector field V = DZ̄T − g(DZ̄T, ν)ν is tangent to M and g(V, T) = 0. Therefore, for any holomorphic frame Z_{1}, ..., Z_{n}, we have

\begin{aligned}V & = g^{λμ̄}g(DZ̄T, Zμ̄)Z_{λ} + g^{μλ̄}g(DZ̄T, Z_{μ})Zλ̄ \\ & = ig^{λμ̄}g(DZ̄ν, Zμ̄)Z_{λ} − ig^{μλ̄}g(DZ̄ν, Z_{μ})Zλ̄ \\ & = ig^{λμ̄}h(Zμ̄, Z̄)Z_{λ} − ig^{μλ̄}h(Z_{μ}, Z̄)Zλ̄. \end{aligned}  (4.8)

In order to get the second equality in (4.8), we used the isometry J and the relations T = J(ν), J(DZ̄T) = DZ̄(J(T)), J(Z_{μ}) = iZ_{μ} and J(Zμ̄) = −iZμ̄. Thus, (4.7)–(4.8) give

\begin{aligned}h(U, ∇_{T} Z̄ + [Z̄, T]) = ih(T, Z̄)h(U, T) & + ig^{λμ̄}h(Zμ̄, Z̄)h(U, Z_{λ}) \\ & − ig^{μλ̄}h(Z_{μ}, Z̄)h(U, Zλ̄). \end{aligned}  (4.9)

Replacing (4.9) into (4.6), we finally find

\begin{aligned}∇Z̄h(U, T) − ∇_{T} h(U, Z̄) = ih(U, Z̄)h(T, T) & + ig^{λμ̄}h(Zμ̄, Z̄)h(U, Z_{λ}) \\ & − ig^{μλ̄}h(Z_{μ}, Z̄)h(U, Zλ̄), \end{aligned}

which is identity (4.1b).

In order to prove (4.1c), take Z, U ∈ Γ(H). By (4.4), we have

\begin{aligned}∇_{Z}h(U, T) − ∇_{T} h(U, Z) & = ih(U, Z)h(T, T) − ih(U, T)h(Z, T) \\ & + h(U, ∇_{T} Z + [Z, T]). \end{aligned}  (4.10)

On conjugating (4.7), we find ∇_{T} Z + [Z, T] = V̄ − ih(Z, T)T, where the vector field V̄ = D_{Z}T − g(D_{Z}T, ν)ν is, by (4.8),

V̄ = ig^{λμ̄}h(Z, Zμ̄)Z_{λ} − ig^{μλ̄}h(Z, Z_{μ})Zλ̄.

Thus, equation (4.10) reads

\begin{matrix}∇_{Z}h(U, T) − ∇_{T} h(U, Z) = & ih(U, Z)h(T, T) − 2ih(U, T)h(Z, T) \\ + ig^{λμ̄}h(U, Z_{λ})h(Z, Zμ̄) − ig^{μλ̄}h(Z, Z_{μ})h(U, Zλ̄). \end{matrix}

The proof of identity (4.1c) is accomplished.

In order to prove (4.1d), take Z ∈ Γ(H) and start from the identity

D_{T} D_{Z}T − D_{Z}D_{T} T − D_{[T,Z]}T = 0.  (4.11)

Observe that D_{Z}T = U − h(Z, T)ν for some U ∈ Γ(H ⊕ H̄), because g(D_{Z}T, T) = 0. Precisely, as in (4.8), we have

U = ig^{λμ̄}h(Z, Zμ̄)Z_{λ} − ig^{μλ̄}h(Z, Z_{μ})Zλ̄.  (4.12)

Then D_{T} D_{Z}T = D_{T} U − T h(Z, T)ν − h(Z, T)D_{T} ν, and multiplying by ν,

g(D_{T} D_{Z}T, ν) = −h(U, T) − T h(Z, T),  (4.13)

because g(D_{T} ν, ν) = 0.

We analyze the second term in the left hand side of (4.11). A computation similar to (4.8) furnishes

D_{T} T − g(D_{T} T, ν)ν = ig^{λμ̄}h(Zμ̄, T)Z_{λ} − ig^{μλ̄}h(Z_{μ}, T)Zλ̄ = W,  (4.14)

where W ∈ Γ(H ⊕ H̄) is defined by the last equality. Thus,

g(D_{Z}D_{T} T, ν) = −h(Z, W) − Zh(T, T).  (4.15)

Finally, we study the third term in the left hand side of (4.11). We have

\begin{aligned}[T, Z] & = D_{T} Z − D_{Z}T = ∇_{T} Z + g(D_{T} Z, N̄)N − D_{Z}T \\ & = ∇_{T} Z − h(Z, T)(ν − iT) − D_{Z}T = ∇_{T} Z − U + ih(Z, T)T, \end{aligned}

where U is defined after (4.11). This yields

g(D_{[T,Z]}T, ν) = −h([T, Z], T) = −h(∇_{T} Z, T) + h(U, T) − ih(Z, T)h(T, T). (4.16)

Multiplying (4.11) by ν and using (4.13), (4.15) and (4.16), we obtain

∇_{Z}h(T, T) − ∇_{T} h(Z, T) = 2h(U, T) − h(Z, W) − ih(Z, T)h(T, T).

Replacing the expressions for U and W in (4.12) and (4.14), we get formula (4.1d).

Remark 4.2. The second fundamental form h satisfies also other Codazzi equations. For instance, we have

∇_{α}h_{βγ̄} − ∇_{β}h_{αγ̄} = ih_{βγ̄}h_{α0} − ih_{αγ̄}h_{β0},  (4.17a)
∇_{α}h_{βγ} − ∇_{γ}h_{βα} = ih_{βα}h_{γ0} − ih_{βγ}h_{α0},  (4.17b)

Identity (4.17a) can be obtained interchanging α and β in identity (4.1a) and taking the difference of the two equations. Identity (4.17b) follows from (4.4) on choosing Z, U, W ∈ Γ(H) and using Tor_{∇}(W, Z) = 0.

Notice also that, letting Z, U, V, W ∈ Γ(H) and multiplying identity (4.3) by V, we get the Gauss-type equation

g(R_{∇}(Z, W̄)U, V̄) = 2{h(U, W̄)h(V̄ , Z) − h(U, Z)h(V̄ , W̄)}.

5. Classification results

In this section we prove the following results:

Theorem 5.1. Let M ⊂ ℂ^{n+1}, n ≥ 2, be a (2n + 1)-dimensional, connected Levi umbilical surface with constant Levi curvature H ≠ 0. Then M is contained either in a sphere or in the boundary of a spherical tube.

LEVI UMBILICAL SURFACES IN COMPLEX SPACE 13

Theorem 5.2. Let M be a connected pseudovonvex hypersurface in ℂ^{n+1}, n ≥ 1, with constant Levi curvature H ≠ 0 and h_{αβ} = 0. Then, up to a complex isometry, M is a contained in a sphere or in a cylinder of the form

{z ∈ ℂ^{n+1} : ∑_{i=m}^{n+1} |z_{i}|^{2} = r^{2}}, r > 0, 1 ≤ m ≤ n.  (5.1)

Remark 5.3. The only compact surface among the ones defined in (5.1) is the sphere. Theorem 5.2 improves [Kl, Theorem 5.2], because we assume neither compactness nor strict pseudoconvexity of M.

A slight modification of the argument also shows that if strict pseudoconvexity (but not compactness) is added as hypothesis in Theorem 5.2, then the surface M must be contained in a sphere.

Proof of Theorem 5.1. Possibly changing the orientation of M, assume H > 0. Observe preliminarily that, given an orthonormal frame Z_{α}, by Proposition 2.1 part iii), we have

∑_{α=1}^{n} g([Z_{α}, Zᾱ], T) = −2inH ≠ 0,  (5.2)

provided that H ≠ 0. Then at least one term in the sum is non zero and the distribution Re(H ⊕ H̄) is bracket generating.

We accomplish the proof in several steps. Step 1. We claim that

h_{α0} = 0.  (5.3)

Indeed, contracting the indices α and γ̄ in the Codazzi equation (4.17a), we get

∇_{α}h_{β}^{α} − ∇_{β}h_{α}^{α} = ih_{β}^{α}h_{α0} − ih_{α}^{α}h_{β0}.  (5.4)

The fundamental form satisfies h_{αβ̄} = Hg_{αβ̄} and thus h_{α}^{β} = Hδ_{α}^{β} and h_{α}^{α} = nH. Then the left hand side in (5.4) vanishes. Therefore (n − 1)Hh_{β0} = 0. The claim follows.

As a consequence of (5.3), it turns out that h satisfies the identities

hβ̄ ; αhμ ; β̄ = (H^{2} − h_{00}H)δ_{α}^{μ},  (5.5a)
∇_{0}h_{αβ} + ih_{00}h_{αβ} = 0,  (5.5b)
∇_{α}h_{00} = 0.  (5.5c)

To show (5.5a), observe that, since (5.3) holds and h_{αβ̄} = Hg_{αβ̄}, the left hand side of identity (4.1b) vanishes. Thus, using again h_{αβ̄} = Hg_{αβ̄} in the right-hand side, we find the equation

h_{αλ}hλ ; β̄ = (H^{2} − h_{00}H)g_{αβ̄}.

Contracting with g^{μβ̄} yields (5.5a). Equations (5.5b) and (5.5c) follow from (4.1c) and (4.1d), letting h_{α0} = 0 and h_{αβ̄} = Hg_{αβ̄}.

14 ROBERTO MONTI AND DANIELE MORBIDELLI

Notice also that equation (5.5a) gives

|h_{αβ}|^{2} := hβ̄ ; αhα ; β̄ = nH(H − h_{00}),  (5.6)

which implies h_{00} ≤ H. Moreover, equation (5.5c) and ∇T = 0 give Zh(T, T) = ∇_{Z}h(T, T) = 0 on M for any Z ∈ H. On conjugating, the equation is satisfied also for all Z ∈ H̄. Since M is connected, from (5.2) it follows that

h(T, T) = constant = h_{00} on M.  (5.7)

Take P ∈ M and denote by L the shape operator, L(X) = D_{X}ν, X ∈ T_{P} M. Step 2. If X ∈ T_{P} M is an eigenvector of L with |X| = 1 and g(X, T) = 0, then Y = J(X) is an eigenvector of L with |Y | = 1.

Indeed, assume that L(X) = λX for some λ ∈ ℝ and let Z = X − iJ(X) = X − iY ∈ H_{P}. By (5.3), since L(X) is orthogonal to T,

0 = h(Z, T) = g(L(X) − iL(Y), T) = −ig(L(Y), T).

Therefore L(Y) is orthogonal to T. Moreover, by the symmetry of L, g(L(Y), X) = g(L(X), Y) = λg(X, Y) = λg(X, J(X)) = 0. Finally, if W ∈ H_{P} satisfies g(Z, W̄) = 0, it must be also g(X, W̄) = 0 and thus g(L(X), W̄) = λg(X, W̄) = 0. Since M is Levi umbilical, we also have g(L(Z), W̄) = Hg(Z, W̄). Eventually, we get

g(L(Y), W̄) = ig(L(X) − iL(Y), W̄) = ig(L(Z), W̄) = iHg(Z, W̄) = 0.

Taking the conjugate we also find g(L(Y), W) = 0. Ultimately, we showed that L(Y) has no component orthogonal to Y and our claim is proved.

Step 3. At any point P ∈ M there exists an orthonormal basis {X_{α}, Y_{α} = J(X_{α}), T : α = 1, …, n} of T_{P} M such that

\begin{matrix}L(X_{α}) = (H + \sqrt{H^{2} − h_{00}H})X_{α}, \\ L(Y_{α}) = (H − \sqrt{H^{2} − h_{00}H})Y_{α}, \\ L(T) = h_{00}T. \end{matrix}  (5.8)

Note first that, by (5.3), h(T, X) = 0 for any X ∈ H_{P} ⊕ H̄_{P}. Then we have L(T) = h_{00}T, by (5.7), and the orthogonal complement of T at any point P ∈ M is an invariant subspace for L. We diagonalize L restricted to this invariant subspace. By Step 1, for any eigenvector X_{α} with eigenvalue λ_{α}, there is an eigenvector Y_{α} = J(X_{α}) with eigenvalue μ_{α}. Thus we get an orthonormal basis {T, X_{α}, Y_{α}, α = 1, …, n} of T_{P} M. We may assume λ_{α} ≥ μ_{α}.

The values of λ_{α} and μ_{α} are determined by (5.5a) and by Levi umbilicality. Indeed, letting Z_{α} = X_{α} − iY_{α}, we have g_{αβ̄} = 2δ_{αβ}. Since M is Levi umbilical,

2H = Hg(Z_{α}, Zᾱ) = g(L(Z_{α}), Zᾱ) = g(λ_{α}X_{α} − iμ_{α}Y_{α}, X_{α} + iY_{α}) = λ_{α} + μ_{α}. (5.9)

Moreover, since h_{αβ} = g(Z_{α}, L(Z_{β})) = (λ_{α} − μ_{α})δ_{αβ}, it is hβ̄ ; α = \frac{1}{2}(λ_{α} − μ_{α})δ_{α}^{β}. Thus,

(H^{2} − h_{00}H)δ_{α}^{γ} = hβ̄ ; αhγ ; β̄ = \frac{1}{4}(λ_{α} − μ_{α})^{2}δ_{α}^{λ}.  (5.10)

The solutions to equations (5.9) and (5.10) are λ_{α} = H + \sqrt{H^{2} − h_{00}H} and μ_{α} = H − \sqrt{H^{2} − h_{00}H}. The proof of Step 3 is concluded.

In Step 3, we established that the principal curvatures of M are the constant numbers (5.8). By a classical result going back to Segre [S], if a connected hypersurface in ℝ^{N+1} has constant principal curvatures, then it must be a plane, a sphere or a cylinder, i.e. a Cartesian product 𝕊^{p} × ℝ^{N−p}, where 𝕊^{p} is a p-dimensional sphere and 0 ≤ p ≤ N. In particular, a surface with constant curvatures can have at most two different ones. The numbers in (5.8) are not pairwise different only in the following two cases:

Case A: h_{00} = H and

Case B: h_{00} = 0.

In Case A, all the principal curvatures are equal to H and the surface M must be contained in a sphere of radius \frac{1}{H}. In Case B the surface must be a cylinder. In the latter case, equations (5.8) become

L(X_{α}) = 2HX_{α}, L(Y_{α}) = 0 and L(T) = 0.  (5.11)

Fix a point P. After a complex rotation, we may assume that the vectors at P satisfying (5.11) are X_{α} = ∂_{x_{α}}, Y_{α} = ∂_{y_{α}} and T = ∂_{y_{n+1}}. This means that ker(L) = span{∂_{y_{h}} : h = 1, …, n + 1}. For a cylinder, ker(L) is the same at any point (after the trivial identification between different tangent spaces of ℝ^{2n}). Moreover, the remaining n principal curvatures are all equal to 2H. Then the surface is contained in a cylinder of equation

∑_{k=1}^{n+1}(x_{k} − b_{k})^{2} = \frac{1}{4H^{2}} ,

for suitable constants b_{k}. The proof is concluded.

Proof of Theorem 5.2. Without loss of generality we can assume H > 0. Step A. First we prove that h_{α0} = 0. Since h_{αβ} = 0, (4.1a) becomes ∇_{β}h_{αγ̄}+ih_{αγ̄}h_{β0}+ 2ih_{α0}h_{βγ̄} = 0. Contracting with g^{γ̄α} gives ∇_{β}h_{α}^{α} + ih_{α}^{α}h_{β0} + 2ih_{α0}h_{β}^{α} = 0. The Levi curvature is constant and then

nHh_{β0} + 2h_{α0}h_{β}^{α} = 0.  (5.12)

Denote by k_{(λ)}, λ = 1, …, n, the principal Levi curvatures of M at a point P. This means that there is an orthonormal family of holomorphic vectors V_{(λ)} = V_{(λ)}^{β}Z_{β} ∈ H_{P}, λ = 1, …, n, such that h_{β}^{α}V_{(λ)}^{β} = k_{(λ)}V_{(λ)}^{α}. Contracting (5.12) with V_{(λ)}^{β} yields

(nH + 2k_{(λ)})h(T, V_{(λ)}) = 0.

By pseudoconvexity, it is k_{(λ)} ≥ 0 for all λ = 1, …, n. Since H > 0, this implies h(T, V_{(λ)}) = 0 for any λ = 1, …, n, which ensures h_{α0} = 0.

Inserting h_{αβ} = 0, h_{α0} = 0 and h_{α}^{α} = nH = constant in equations (4.1a), (4.1b) and (4.1d), we find

∇_{β}h_{αγ̄} = 0,  (5.13a)
∇_{0}h_{αβ̄} = ih_{αλ̄}hλ̄ ; β̄ − ih_{αβ̄}h_{00},  (5.13b)
∇_{α}h_{00} = 0.  (5.13c)

Equation (5.13c) and ∇T = 0 imply that Zh_{00} = 0 for all holomorphic Z. Since the horizontal distribution is bracket generating, we conclude that h_{00} is constant on M. Contracting α and β̄ in (5.13b) and using H = constant, we find h_{αλ̄}h^{αλ̄} = nHh_{00}. If h_{00} = 0, it follows that h_{αλ̄} = 0 and thus H = 0. This is not possible and h_{00} must be a non zero constant. Since h_{α0} = 0, by Step A we also have L(T) = h_{00}T. Step B. If X ∈ T_{P} M is a real tangent vector orthogonal to T and such that L(X) = λX, then the vector Y = J(X) satisfies L(Y) = μY. This follows from h_{αβ} = 0 and can be proved as in Step 2 of the proof of Theorem 5.1. Moreover, letting Z = X −iY we have

0 = h(Z, Z) = g(L(Z), Z) = g(λX − iμY, X − iY) = λ − μ.

Therefore λ = μ.

Iterating this process n times, we find an orthonormal basis {X_{α}, Y_{α} = J(X_{α}), T : α = 1, …, n} of T_{P} M such that

L(X_{α}) = λ_{α}X_{α}, L(Y_{α}) = λ_{α}Y_{α}.  (5.14)

Notice that L sends H into H, because h_{αβ} = 0 and h_{α0} = 0. Moreover, letting Z_{α} = X_{α} − iY_{α} we have L(Z_{α}) = λ_{α}Z_{α}. The numbers λ_{1}, …, λ_{n} are the eigenvalues of the Levi form at the point P, i.e. h(Z_{α}, Zβ̄) = λ_{α}g(Z_{α}, Zβ̄). Step C. We claim that the eigenvalues of L are constant. First observe that any pair of points in M can be connected by a horizontal path γ : [0, 1] → M, i.e. a piecewise C^{1} curve such that g(γ̇, T) = 0. This follows from the rank condition (5.2). Take P, Q ∈ M and connect them by a horizontal curve γ with γ(0) = P and γ(1) = Q. Let {X_{α}^{P} , Y_{α}^{P} = J(X_{α}^{P}), T : α = 1, ..., n} be an orthonormal basis of T_{P} M satisfying (5.14). Let Z_{α}^{P} = X_{α}^{P} − iY_{α}^{P} and let Z_{α} be the parallel extension of Z_{α}^{P} along γ, that is

∇γ̇Z_{α} = 0 along γ and Z_{α}(P) = Z_{α}^{P} .  (5.15)

The vector field Z_{α} is holomorphic Equation (5.13a) and its conjugate imply

∇γ̇h(Z, W̄) = 0, for all holomorphic Z, W.  (5.16)

Then, from (5.16) and (5.15) it follows that

\frac{d}{dt}h(Z_{α}, Zβ̄) = ∇γ̇h(Z_{α}, Zβ̄) + h(∇γ̇Z_{α}, Zβ̄) + h(Z_{α}, ∇γ̇Zβ̄) = 0.

Thus h(Z_{α}, Zβ̄) is constant along γ and

h(Z_{α}, Zβ̄) = h(Z_{α}^{P} , ZP ; β̄) = 2λ_{α}δ_{αβ},

where the λ_{α}’s are the Levi eigenvalues at P. Since g is parallel, we also have g(Z_{α}, Zβ̄) = g(Z_{α}^{P} , ZP ; β̄) = 2δ_{αβ}. Eventually, we get h(Z_{α}, Zβ̄) = λ_{α}g(Z_{α}, Zβ̄), where the λ_{α}’s are again the eigenvalues at P. This means that also at the point Q = γ(1) the eigenvalues of L are λ_{1}, λ_{2}, …, λ_{n} and h_{00}.

Step D. The shape operator L has constant eigenvalues λ_{1}, ..., λ_{n}, h_{00}. Each eigenvalue λ_{α} has multiplicity 2 and the corresponding eigenspace is a complex subspace of ℂ^{n+1}. By Segre’s theorem on hypersurfaces with constant curvatures, M can have no more than two different constant curvatures and it is contained either in a sphere or in a cylinder with spherical section. We may assume λ_{1} = ⋯ = λ_{m} = 0 and λ_{m+1} = ⋯ = λ_{n} = h_{00} for some 0 ≤ m ≤ n − 1. In case m = 0 we have a sphere. In case 1 ≤ m ≤ n − 1 we have a cylinder of the form (5.1). The case case m = n is excluded, because we have a cylinder of the form ℂ^{n} × 𝕊^{1} which has H = 0.

The proof is concluded.

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Roberto Monti: Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni, 7, 35100 Padova, Italy

E-mail address: monti@math.unipd.it

Daniele Morbidelli: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40127 Bologna, Italy

E-mail address: morbidel@dm.unibo.it