Axioms of Planes on Real Hypersurfaces in Complex Space Forms

Miguel Ortega^{∗}

A Riemannian manifold M satisfies the axiom of 2-planes if at each point p ∈ M and each tangent 2-plane V at the point, there exists a totally geodesic submanifold L with p ∈ L and T_{p}L = V. On the other hand, real hypersurfaces in complex space forms have nice families of tangent 2-planes. If we restrict the above definition to those planes, several definitions of axioms of 2-planes arise naturally. We classify real hypersurfaces in non-flat complex space forms satisfying these axioms.

Mathematics Subject Classification(1991): 53C42, 53C55.

Introduction

Totally geodesic submanifolds often help to describe the Riemannian geometry of the ambient manifold. One point of view is the existence of sufficiently many totally geodesic submanifolds passing trough each point. That is to say, given a point and a tangent plane at such point, we wonder if there exists a totally geodesic submanifold such that the point lies in such submanifold and the tangent space to the submanifold at that point is the chosen tangent plane. This property is known as axiom of planes, and we can define different axioms of planes by choosing different types of tangent planes. These definitions are closely related to the sectional curvature of the chosen planes. Indeed, a classical result of E. Cartan says that a Riemannian manifold has constant sectional curvature if and only if it satisfies the axiom of planes. Among Kaehler manifolds, the usual types of tangent planes are holomorphic and totally real. The complex dimension have to be at least three. It is not surprising that a Kaehler manifold satisfying the axiom of holomorphic (respectively, totally real) planes has constant holomorphic (respectively, totally real) sectional curvature. Besides, a Kaehler manifold has constant holomorphic sectional curvature if and only if it has constant totally real sectional curvature. Therefore, a Kaehler manifold satisfies the axiom of holomorphic planes if and only if it satisfies the axiom of totally real planes. (See [3], [7], etc.) Now we turn our attention to real hypersurfaces in non-flat complex space forms. The definitions of axioms of planes must be adapted to them by choosing tangent planes that are orthogonal to the structure vector field ξ of the almost contact structure (see Section 1). We classify real hypersurfaces in non-flat complex space forms satisfying the axiom of holomorphic (respectively, totally real) 2-planes in Theorem 1 (respectively, Theorem 2). But the relationship between these axioms of planes on real hypersurfaces is not the same as on Kaehler manifolds, because the lists of real hypersurfaces appearing in Theorem 1 and 2 are not equal. Therefore, the axiom of holomorphic 2-planes is not equivalent to the axiom of totally real 2-planes on real hypersurfaces in non-flat complex space forms. The author would like to thank Professors M. Kimura and S. Maeda, and the referee for several suggestions and opinions that improved this paper. Also, the referee kindly simplified the proof of Lemma 3.

∗This research was supported by a grant of the Ministry of Education, Science, Culture and Sports of Japan.

1 Preliminaries

Let ℂM^{m}(c), m ≥ 2, be a non-flat complex space form endowed with the metric of g of constant holomorphic sectional curvature c ≠ 0. For sake of simplicity, if c > 0, we will use c = 4, and the space form will be called the complex projective space, ℂP^{m}. And if c < 0, we will use c = −4, so that we will call it the complex hyperbolic space, ℂH^{m}. Let J be the almost complex structure. The expression of the curvature tensor R̄ of ℂM^{m}(c) is well known,

\begin{aligned}R̄(X, Y)Z & = & (c/4)\{g(Y, Z)X − g(X, Z)Y \\ & & +g(JY, Z)JX − g(JX, Z)JY − 2g(JX, Y)JZ\} \end{aligned}  (1)

for any Z, Y, Z tangent to ℂM^{m}(c). Let L be a connected submanifold in ℂM^{m}(c) without boundary, but we suppose L is not a real hypersurface. We will denote by ¯∇ and ∇^{L} the Levi-Civita connection of ℂM^{m}(c) and L respectively. Let h be the second fundamental form of L, and ∇^{⊥} the normal connection of L. If ν is a locally defined unit vector field normal to L, we have the Gauss and Weingarten formulae

\begin{aligned}¯∇_{X}Y & = & ∇_{X}^{L}Y + h(X, Y) \\ ¯∇_{X}ν & = & −A_{ν}X + ∇_{X}^{⊥}ν \end{aligned}  (2)

for any X, Y tangent to L, where A_{ν} is the Weingarten endomorphism associated to ν. Moreover, if ν_{1} and ν_{2} are two locally defined unit vector fields normal to L, we have the Ricci equation,

g(R̄(X, Y)ν_{1}, ν_{2}) = g(R^{⊥}(X, Y)ν_{1}, ν_{2}) − g([A_{ν_{1}}, A_{ν_{2}}]X, Y)  (3)

for any X, Y tangent to L, where R^{⊥} is the curvature operator associated to the normal connection ∇^{⊥}.

Let M be a connected real hypersurface without boundary in ℂM^{m}(c). We will fix special notations for the geometric tools of M. The restriction of the metric to M will also be denoted by g. Choose N a locally defined unit normal vector field on M. The Levi-Civita connection of M will be denoted by ∇. Let A be the Weingarten endomorphism associated to N. In this case, the Gauss and Weingarten formulae are as follows:

\begin{aligned}¯∇_{X}Y & = & ∇_{X}Y + g(AX, Y)N \\ ¯∇_{X}N & = & −AX \end{aligned}  (4)

for any X, Y ∈ T M. A local tangent vector field X is called principal if it is an eigenvector of A, and its associated eigenfunction is called principal curvature. Given a point p ∈ M and a principal curvature λ, we write down T_{λ}(p) = {X ∈ T_{p}M : A_{p}X = λ(p)X}. This linear subspace is called the principal distribution associated to λ at p. The dimension of the principal distribution is known as the multiplicity of the principal curvature. The multiplicity of a principal distribution depends on the point, although there exists a dense open subset on which it is locally constant.

Given a vector field X tangent to M on a neighbourhood of a point p ∈ M, we put JX = ϕX + η(X)N and JN = −ξ, where ϕX and η(X)N are the tangential and the normal component of JX respectively. Thus, ϕ is a skew-symmetric tensor of type (1,1) and η is a 1-form on M. Besides, ξ is a locally defined vector field tangent to M. The set (ϕ, ξ, η, g) is called an almost contact metric structure on M, whose elementary properties are the following

\begin{matrix}η(X) = g(X, ξ), ϕ^{2} = −I + η ⊗ ξ, ϕξ = 0, η(ξ) = 1, ∇_{X}ξ = ϕAX \\ g(ϕX, Y) + g(X, ϕY) = 0, g(ϕX, ϕY) = g(X, Y) − η(X)η(Y) \end{matrix}  (5)

for any X, Y ∈ T M, where I denotes the identity transformation on T M. The Gauss equation of M is

\begin{aligned}R(X, Y)Z & = & (c/4)\{g(Y, Z)X − g(X, Z)Y + g(ϕY, Z)ϕX − g(ϕX, Z)ϕY \\ & & −2g(ϕX, Y)ϕZ\} + g(AY, Z)AX − g(AX, Y)AZ \end{aligned}  (6)

for any X, Y, Z ∈ T M. The Codazzi equation of M is

(∇_{X}A)Y − (∇_{Y} A)X = (c/4){η(X)ϕY − η(Y)ϕX − 2g(ϕX, Y)ξ}  (7)

for any X, Y ∈ T M. Given a point p ∈ M, we define 𝔻_{p} = {X ∈ T_{p}M : g_{p}(X, ξ) = 0}. We also denote U𝔻_{p} = {X ∈ 𝔻_{p} :∥ X ∥= 1}. It is easy to check that the distribution defined by p ∈ M ↦ 𝔻_{p} ⊂ T_{p}M is a vector bundle on M. We define the element B ∈ hom (𝔻, 𝔻) by the following. Given a point p ∈ M, and X ∈ 𝔻_{p}, we define B_{p}X as the orthogonal component of A_{p}X on 𝔻_{p}. Clearly, B is symmetric, so it is diagonalizable. The eigenvectors of B will be called secondary vectors, and their corresponding eigenfunctions, secondary curvatures.

Clearly, any principal vector lying in 𝔻 is also secondary vector, and in that case, its principal curvature is also a secondary curvature.

A real hypersurface in ℂM^{m}(c) is called Hopf hypersurface if the vector field ξ is everywhere principal. The corresponding principal curvature will be called μ. Y. Maeda ([5]) and S. Montiel ([6]) proved that the principal curvature μ is locally constant. Cecil and Ryan ([4]), and Montiel ([6]), classified real hypersurfaces in ℂM^{m}(c), m ≥ 3, with at most two distinct principal curvatures at each point. We summarize their results in the following theorem

Theorem A Let M be a real hypersurface in ℂM^{m}(c), m ≥ 3, c = ±4, with at most two distinct principal curvatures at each point. Then M is locally congruent to one of the following:

1. In ℂP^{m},

A_{1}) a tube of radius 0 < r < π/2 over a totally geodesic hyperplane ℂP^{m−1},

2. In ℂH^{m},

A_{0}) a horosphere,

A_{1}) a tube of radius r > 0 over a totally geodesic ℂH^{k}, where k = 0, m − 1,

B) a tube of radius log((1 + \sqrt{3})/\sqrt{2}) over a totally geodesic ℝH^{m}.

All these real hypersurfaces admit a principal curvature ρ such that T_{ρ}(p) = 𝔻_{p} at each point p ∈ M, except the tube of radius log((1 + \sqrt{3})/\sqrt{2}) over a totally geodesic ℝH^{m}. As Hopf hypersurfaces satisfy A𝔻 ⊂ 𝔻, we denote by Σ_{p} the spectrum of A|𝔻(p), for each p ∈ M. The relationship between principal curvatures in Hopf hypersurfaces is shown in the following proposition. (See [2])

Proposition A Let M be a connected Hopf real hypersurface in ℂM^{m}(c), c ≠ 0, m ≥ 2 Choose a point p ∈ M.

1. If μ^{2} + c = 0, then μ/2 ∈ Σ_{p} and if λ lies in Σ_{p} \ {μ/2}, then ϕT_{λ}(p) ⊂ T_{μ/2}(p). Therefore, the multiplicity of μ/2 is at least m − 1, and T_{λ}(p) is totally real.

2. If μ^{2}+c ≠ 0, for each λ ∈ Σ_{p}, there exists exactly one τ ∈ Σ_{p} such that ϕT_{λ}(p) = T_{τ}(p), and (2λ − μ)(2τ − μ) = μ^{2} + c.

We recall that a real hypersurface in ℂM^{m}(c) is said to be ruled if the distribution 𝔻 is integrable and its leaves are open subsets of totally geodesic hyperplanes ℂM^{m−1}. The rank of A at a point p ∈ M is called the type number of M at p, and we will denote it by t(p). The following proposition is also a summary of two results that appeared in [8] and [9]. It states a characterization of ruled real hypersurfaces in ℂM^{m}(c) when the type number is small enough.

Proposition B Let M be a real hypersurface in ℂM^{m}(c), m ≥ 3, c ≠ 0, such that t(p) ≤ 2 for any point p ∈ M. Then M is ruled.

Finally, choose a point p ∈ M, and a 2-plane π ⊂ T_{p}M. If π = Span{X, Y } ⊂ 𝔻_{p}, where {X, Y } is a basis of π, we denote by K(π) = K(X, Y) the sectional curvature of the plane π. The plane π is called holomorphic (respectively, totally real), if π ⊂ 𝔻_{p} and ϕ_{p}π = π (respectively, ϕ_{p}π ⊥ π). If π is totally real, we will write down K(π) = T (X, Y). On the other hand, if π is holomorphic, we can choose an orthonormal basis {X, ϕ_{p}X} of π, so we will write down K(π) = H(X) = H(ϕ_{p}X). The holomorphic (respectively, totally real) sectional curvature is the sectional curvature of holomorphic (respectively, totally real) 2-planes.

2 Real Hypersurfaces in Complex Space Forms

We begin by making suitable definitions of axioms of planes on real hypersurfaces in non-flat complex space forms.

Definition 1 Let M be a real hypersurface in ℂM^{m}(c), m ≥ 3, c ≠ 0. We say that M satisfies the axiom of holomorphic (respectively, totally real) 2-planes if for each point p ∈ M and each holomorphic (respectively, totally real) tangent 2-plane π ⊆ 𝔻_{p} there exists a totally geodesic submanifold L in M such that p ∈ L and T_{p}L = π.

Lemma 1 Let M be a real hypersurface in ℂM^{m}(c), m ≥ 2, c ≠ 0. The following statements are pairwise equivalent:

1. M is ruled.

2. g(AX, Y) = 0 for any X, Y ∈ 𝔻.

3. B ≡ 0.

Proof : Statements 2 and 3 are equivalent. Let us proof statement 1 implies statement 2. If we take X, Y ∈ 𝔻, as M is ruled, the leaves are totally geodesic in ℂM^{m}(c), and therefore ¯∇_{X}Y ∈ 𝔻. Then, by (4), 0 = −g( ¯∇_{X}Y, N) = g(AX, Y). Conversely, if X, Y ∈ 𝔻, the same argument shows N ⊥ ¯∇_{X}Y. Besides, by (4) and (5), g(ξ, ¯∇_{X}Y) = −g( ¯∇_{X}ξ, Y) = −g(∇_{X}ξ, Y) = −g(ϕAX, Y) = 0. This shows ¯∇_{X}Y ∈ Span{N, ξ}^{⊥} = 𝔻, that is to say, 𝔻 is integrable and the leaves are totally geodesic.

Lemma 2 Ruled real hypersurfaces in ℂM^{m}(c), c ≠ 0, m ≥ 3, satisfy both axiom of holomorphic 2-planes and axiom of totally real 2-planes.

Proof : Let M be a ruled real hypersurface in ℂM^{m}(c), c ≠ 0, m ≥ 3. Given p ∈ M, we choose π ⊂ 𝔻_{p} a holomorphic (respectively, totally real) tangent 2-plane. As M is ruled, 𝔻 is integrable. Let L̃ be a leaf of 𝔻 such that p ∈ L̃ and T_{p}L̃ = 𝔻_{p}. As L̃ is locally congruent to ℂM^{m−1}, L̃ satisfies the axiom of holomorphic (respectively, totally real) 2-planes. Moreover, as π ⊂ 𝔻_{p} = T_{p}L̃, let L be a totally geodesic submanifold in L̃ such that p ∈ L and T_{p}L = π. Clearly, L is also a submanifold in M such that T L ⊂ 𝔻. We only have to proof that L is totally geodesic in M. Indeed, let ˜∇ and ∇^{L} the Levi-Civita connection of L̃ and L respectively. Given X, Y ∈ T L, as L is totally geodesic in L̃ and L̃ is also totally geodesic in ℂM^{m}(c), by (4), ∇_{X}^{L}Y = ˜∇_{X}Y = ¯∇_{X}Y = ∇_{X}Y + g(AX, Y)N = ∇_{X}Y because of Lemma 1.

Lemma 3 Real hypersurfaces of type A_{0} and A_{1} in ℂM^{m}(c), m ≥ 3, c ≠ 0, do not satisfy the axiom of holomorphic 2-planes.

Proof : Let M be a real hypersurface in ℂM^{m}(c), m ≥ 3, c ≠ 0, of type A_{0} or A_{1}. First, we should point out that there exist two real constants a, b such that

AX = aX + bη(X)ξ for any X ∈ T M  (8)

(See [4] and [6]). Let us suppose the chosen real hypersurface satisfies the axiom of holomorphic 2-planes. Choose a point p ∈ M and e ∈ U𝔻_{p}. Set π = Span{e, ϕ_{p}e}. Let L be a totally geodesic submanifold in M such that p ∈ L and T_{p}L = π. Now, it follows from (8) and the Gauss formula that L is totally umbilical in ℂM^{m}(c). It is well-known that a totally umbilical submanifold with dimension ≥2 in a non-flat complex space form is totally real (this fact is a simple consequence of the Codazzi equation). This readily implies the statement in Lemma 3.

Theorem 1 The only real hypersurfaces in ℂM^{m}(c), m ≥ 3, c ≠ 0, satisfying the axiom of holomorphic 2-planes are ruled real hypersurfaces.

Proof : By Lemma 2, we already know that ruled real hypersurfaces satisfy the axiom of holomorphic 2-planes. Conversely, let M be a connected real hypersurface in ℂM^{m}(c), m ≥ 3, c ≠ 0, satisfying the axiom of holomorphic 2-planes.

Chose p ∈ M and G a connected open neighbourhood of p ∈ M, on which ξ is globally defined. First, if B ≡ 0 on G, by Lemma 1, G is a ruled real hypersurface. Next, let us suppose that rankB ≥ 1 on G. Given q ∈ G, X ∈ U𝔻_{q}, there exists a totally geodesic submanifold L in M such that q ∈ L and T_{q}L = Span{X, ϕ_{q}X}. If R^{L} is the curvature operator of L, R_{q}(X, ϕX)ϕX = R_{q}^{L}(X, ϕX)ϕX ∈ T_{q}L = Span{X, ϕ_{p}X}. From now on, we will not write down the point to simplify the notation. By the elementary properties of curvature tensor and (6)

H(X)X = cX + g(AϕX, ϕX)AX − g(AX, ϕX)AϕX  (9)

for any X ∈ U𝔻 on G. As the rank of B is at least one, by restricting G if necessary, there exists X ∈ U𝔻 such that BX = aX for a certain non-vanishing smooth function a : G → ℝ.

If we introduce ϕX in (9), then H(X)ϕX = cϕX+aAϕX, and as a ≠ 0, then ϕX is principal. In particular, {X, ϕX} are secondary, and the orthogonal complement of Span{X, ϕX} in 𝔻 is invariant by ϕ. Therefore, by restricting G if necessary, we can find an orthonormal basis of secondary vectors {e_{1}, ϕe_{1}, …, e_{m−1}, ϕe_{m−1}} of 𝔻 defined on G such that Be_{i} = a_{i}e_{i}, Bϕe_{i} = b_{i}ϕe_{i}, where a_{i}, b_{i} : G → ℝ are smooth functions, i ∈ {1, …, m − 1}. We also define c_{i} = g(Ae_{i}, ξ) and d_{i} = g(Aϕe_{i}, ξ), c_{i}, d_{i} : G → ℝ, i ∈ {1, …, m − 1}. Now, we can write

\begin{aligned}Ae_{i} & = & a_{i}e_{i} + c_{i}ξ & i ∈ \{1, …, m − 1\} on G \\ Aϕe_{i} & = & b_{i}ϕe_{i} + d_{i}ξ \end{aligned}  (10)

and as rankB ≥ 1, we suppose that at least one of the functions a_{i} or b_{i}, i ∈ {1, …, m − 1}, does not vanish at any point of G. We can suppose without losing generality a_{i} ≠ 0 on G for a certain i ∈ {1, …, m−1}. This i will be fixed until the end of the proof. By taking the inner product of ξ and (9), we obtain the formula g(AϕX, ϕX)g(AX, ξ) = g(AϕX, X)g(AϕX, ξ), but we change X for ϕX in this last formula,

g(AX, X)g(AϕX, ξ) = g(AX, ϕX)g(AX, ξ)  (11)

for any X ∈ 𝔻 on G. Now we choose α, β ∈ ℝ such that α^{2} + β^{2} = 1. Given j, k ∈ {1, …, m − 1}, we introduce X = αe_{j} + βϕe_{k} in (11), and bearing in mind (10), we obtain

(α^{2}a_{j} + β^{2}b_{k})(αd_{j} − βc_{k}) = δ_{jk}αβ(b_{k} − a_{j})(αc_{j} + βd_{k})  (12)

where δ_{jk} is the usual Kronecker’s delta. If we take α = 0 in (12), then

b_{j}c_{j} = 0, j ∈ {1, …, m − 1}  (13)

Again, if we put β = 0 in (12), then

a_{j}d_{j} = 0, j ∈ {1, …, m − 1}  (14)

Now, if we choose α, β such that αβ ≠ 0, β^{2} − α^{2} ≠ 0, j, k ∈ {1, …, m − 1} such that j ≠ k and we introduce them in (12), then (α^{2}a_{j} + β^{2}b_{k})(αd_{j} − βc_{k}) = 0, and by (13) and (14), we obtain −αa_{j}c_{k} + βb_{k}d_{j} = 0. Now, we exchange α and β in this last equation, so we get

−βa_{j}c_{k} + αb_{k}d_{j} = 0, −αa_{j}c_{k} + βb_{k}d_{j} = 0

Now, the condition β^{2} − α^{2} ≠ 0 yields that this linear system has the following unique solution

a_{j}c_{k} = 0, b_{k}d_{j} = 0, j, k ∈ {1, …, m − 1}, j ≠ k  (15)

Now, we take j, k ∈ {1, …, m − 1} but j ≠ k. Similarly, if we set X = αe_{j} + βe_{k}, and X = αϕe_{j} + βϕe_{k}, by introducing them in (11), and bearing in mind (13) and (14), we obtain

a_{j}d_{k} = 0, b_{k}c_{j} = 0, j, k ∈ {1, …, m − 1}, j ≠ k  (16)

We should remember a_{i} ≠ 0 on G. So, from (14), (15) and (16) we see

\begin{matrix}d_{k} = 0, k = 1, …, m − 1 \\ c_{k} = 0, k = 1, …, m − 1 but k ≠ i \end{matrix}  (17)

Suppose that there exists a point q ∈ G such that c_{i}(q) ≠ 0. Thus, we can choose an open neighbourhood U ⊆ G on which c_{i}(q) ≠ 0 for any point q ∈ U. Next, from (13), (15) and (16), a_{k} = 0 for any k = 1, …, m − 1 but k ≠ i on U and b_{k} = 0 for any k = 1, …, m − 1, so that this and (17) imply the type number t(q) ≤ 2 for any q ∈ U. By Proposition B, U is a ruled real hypersurface in ℂM^{m}(c), but by Lemma 1, rankB = 0, which contradicts a_{i} ≠ 0. Therefore, we also assume c_{i} = 0 on G. This fact and equations (10) and (17) show that G is a Hopf real hypersurface. Note that (9) is equivalent to

g(AϕX, ϕX)AX − g(AϕX, X)AϕX ∈ Span{X}  (18)

for any X ∈ 𝔻 on G. Bearing in mind Lemma A, there are two cases.

\underline{Case A} μ^{2} + c = 0. We do not lose generality if we suppose a_{1} = … = a_{m−1} = μ/2. Take k, j ∈ {1, …, m − 1} such that k ≠ j. Choose two non-zero real numbers α and β. Let us consider X = αe_{k} + βϕe_{j}. By (10) and (18), (β^{2}μ/2 + α^{2}b_{k})(αAe_{k} + βAϕe_{j}) = ρ(αe_{k} + βϕe_{k}), where ρ : G → ℝ is a smooth function. From here, (β^{2}μ/2 + α^{2}b_{k})αμ/2 = αρ and (β^{2}μ/2 + α^{2}b_{k})βb_{j} = βρ, and from these two equations we see (β^{2}μ/2 + α^{2}b_{k})(μ/2 − b_{j}) = 0. Given a point q ∈ G, if b_{j}(q) ≠ μ/2, then β^{2}μ/2 + α^{2}b_{k} = 0 for any non-zero α, β, so that μ/2 = b_{k}(q) = 0. Therefore, a_{1} = … = a_{m−1} = b_{1} = … = b_{m−1} = μ/2 on G.

\underline{Case B} μ^{2} + c ≠ 0. Take k, j ∈ \{1, …, m − 1\} such that k ≠ j. Choose two non-zero real numbers α and β. Let us consider X = αe_{k} + ϕe_{j}. Similarly, from (10) and (18), (α^{2}b_{k} +β^{2}b_{j})(a_{k} −a_{j}) = 0. Suppose there exists a point q ∈ G such that a_{k}(q) ≠ a_{j}(q). Then, (α^{2}b_{k} + β^{2}b_{j})(q) = 0, and similarly, b_{k}(q) = b_{j}(q) = 0. Now we introduce these values in the formula on Lemma A, so we obtain (2a_{k}(q)−μ)μ = −μ^{2}−c ≠ 0 and (2a_{j}(q)−μ)μ = −μ^{2}−c ≠ 0, and dividing these equations we get \frac{2a_{k}(q)−μ}{2a_{j}(q)−μ)} = 1, so that a_{k}(q) = a_{j}(q). Therefore, a_{1} = … = a_{m−1}. If k, j ∈ {1, …, m − 1} such that k ≠ j, we resort to the same formula in Lemma A again, and (2a_{k}−μ)(2b_{k}−μ) = μ^{2}+c, (2a_{j} −μ)(2b_{j} −μ) = μ^{2}+c. If we divide these two equations, we obtain b_{j} = b_{k}. We will denote a = a_{1} and b = b_{1} to simplify the notations. Suppose there exists a point q ∈ G such that a(q) ≠ b(q). Then, there exists a open neighbourhood U of q in G on which a ≠ b. We will make all the computations on U until we get to contradiction. Now we consider X = e_{1} +e_{2} +ϕe_{1}. By introducing X in (18), there exists a function ρ : U → ℝ such that (a+2b)(ae_{1}+ae_{2}+bϕe_{1})−(b−a)(−ae_{1}+bϕe_{1}+bϕe_{2}) = ρ(e_{1} + e_{2} + ϕe_{1}). From here we select

(a + 2b)a = ρ, (a + 2b)b − (b − a)b = ρ, (b − a)b = 0

From here, as b ≠ a, easy computations show b = 0 and a = 0, which is a contradiction. This implies AX = aX for any X ∈ 𝔻. By Theorem A, G is a real hypersurface of type A_{0} or A_{1}. However, this is impossible due to Lemma 3. This finishes the proof.

Remark As we have pointed out, ruled real hypersurfaces satisfy the axiom of holomorphic 2-planes. The corresponding totally geodesic submanifolds in these real hypersurfaces are open subsets of totally geodesic ℂM^{1}(c) in ℂM^{m}(c).

Theorem 2 Let M be a connected real hypersurface in ℂM^{m}(c), m ≥ 3, c ≠ 0, satisfying the axiom of totally real 2-planes. Then M is one of the following:

1. A real hypersurface of type A_{0} or A_{1},

2. ruled.

Proof : Let M be a real hypersurface in ℂM^{m}(c), m ≥ 3, c ≠ 0, satisfying the axiom of totally real 2-planes. Given a point p ∈ M, choose {X, Y } ⊂ 𝔻_{p} an orthonormal pair of vectors which span a totally real 2-plane. Let L be a totally geodesic submanifold in M such that p ∈ L and Span{X, Y } = T_{p}L. As in Theorem 1, R_{p}(X, Y)Y ∈ T_{p}L = Span{X, Y }. In the sequel, we will not write down the point to simplify the notation. Now, R(X, Y)Y = T (X, Y)X and by (6),

T (X, Y)X = (c/4)X + g(AY, Y)AX − g(AX, Y)AY  (19)

for any orthonormal X, Y ∈ 𝔻_{p} such that Span{X, Y } is a totally real 2-plane, and any p ∈ M. Choose p ∈ M and G a connected open neighbourhood of p in G on which ξ is globally defined.

If B ≡ 0 on G, by Lemma 1 then G is a ruled real hypersurface. Lemma 2 finishes this case.

Next, we suppose that at p ∈ G, rankB_{p} ≥ 1. Then, by restricting G if necessary, we can find a unit secondary vector field E tangent to M on G such that its associated secondary curvature a : G → ℝ never vanishes on G. Let Y ∈ 𝔻 be unit tangent to M on G such that Span{E, Y } is totally real. By (19), T (E, Y)Y = (c/4)Y + aAY and as a does not vanish, then Y is principal on G, that is to say, all orthogonal vectors to {E, ϕE, ξ} are principal, and therefore there exists a smooth function b : G → ℝ such that

AY = bY, for any Y ⊥ {E, ϕE, ξ} on G  (20)

Let us define U = {q ∈ G : b(q) ≠ 0}, which is an open subset of M. Let us call Γ the set of interior points of G \ U, and suppose Γ is non-empty. Given a unit Z ⊥ {E, ϕE, ξ} tangent to Γ, and two non-zero real numbers α, β such that α^{2} + β^{2} = 1, we set X = αϕE + βZ, Y = βE + αϕZ, which span a totally real 2-plane. So, by introducing them in (19), by (20), we see (T (X, Y)− \frac{c}{4})(αϕE +βZ) = β^{2}aA(αϕE +βZ) = β^{2}aαAϕE and taking scalar product with Z, (T (X, Y) − \frac{c}{4})β = 0, and then T (X, Y) = c/4. Coming back we see 0 = β^{2}aαAϕE, but as a ≠ 0 and α, β are non-zero, then AϕE = 0. This and (20) show the type number satisfies t(p) ≤ 2 at any point p ∈ Γ, and by Proposition B, this subset is a ruled real hypersurface, but Lemma 1 implies B ≡ 0, which contradicts a ≠ 0. Therefore, U is dense in G. Choose Y ∈ 𝔻 tangent to M on U in the conditions of (20). We repeat the reasoning we made with E, but in this moment we use Y and we make the computations on U. There are two cases.

\underline{Case A)} m = 3, and we can define an orthonormal basis \{e_{1}, ϕe_{1}, e_{2}, ϕe_{2}, ξ\} of T M on U (or in a smaller open subset), and two smooth functions b̄, μ : U → ℝ such that μ is locally constant,

\begin{matrix}Ae_{1} = be_{1}, & Aϕe_{1} = bϕe_{1}, & b ≠ b̄ on U \\ Ae_{2} = b̄e_{2}, & Aϕe_{2} = b̄ϕe_{2}, & Aξ = μξ \end{matrix}  (21)

Choose two real numbers α, β such that α^{2} + β^{2} = 1, αβ ≠ 0, α^{4} ≠ β^{4}. With these numbers we consider X = αe_{1} + βe_{2} and Y = βϕe_{1} − αϕe_{2}. Clearly, {X, Y } span a totally real 2-plane, so that we can introduce them in (19), T (X, Y)(αe_{1} + βe_{2}) = (c/4)(αe_{1} + βe_{2}) + (β^{2}b + α^{2}b̄)(αbe_{1} + βb̄e_{2}), from which αT (X, Y) = (c/4)α + (β^{2}b + α^{2}b̄)αb and βT (X, Y) = (c/4)β + (β^{2}b + α^{2}b̄)βb̄. From here, by cancelling α and β, and subtracting the resulting equations, we obtain 0 = (β^{2}b + α^{2}b̄)(b − b̄) and thus, 0 = α^{2}b + β^{2}b̄. By exchanging α and β, and using β^{4} − α^{4} ≠ 0, we see b = b̄ = 0. This is a contradiction.

\underline{Case B)} There exists a function b : U → ℝ such that AX = bX for any X ∈ 𝔻 tangent to M on U. Therefore, U is a Hopf real hypersurface. Let μ be the principal curvature associated to ξ, which is locally constant on U. So, we only have to resort to Theorem A to know that U is locally congruent to a real hypersurface of type A_{0} or A_{1}.

The arising situation is we have a connected open neighbourhood G in M, on which we can define an orthonormal basis of T M, {e_{1}, ϕe_{1}, …, e_{m−1}, ϕe_{m−1}, ξ}, whose vectors satisfy

\begin{matrix}Ae_{1} = a_{1}e_{1} + x_{1}ξ, Aϕe_{1} = a_{2}ϕe_{1} + x_{2}ξ, Aξ = μξ + x_{1}e_{1} + x_{2}ϕe_{1} \\ Ae_{i} = be_{i}, Aϕe_{i} = bϕe_{i}, i = 2, …, m − 1 \end{matrix}

where a_{1}, a_{2}, b, x_{1}, x_{2} are smooth functions on G. Moreover, U = {q ∈ G : b(q) ≠ 0} is dense in G, and on the set U, all these functions have the following behaviour: x_{1}, x_{2} vanish, a_{1} = a_{2} = b, and b is locally constant. As G is connected, and all these functions are continuous, we obtain x_{1} = x_{2} = 0 and a_{1} = a_{2} = b =constant on G, so that G is a Hopf real hypersurface which has already been classified.

The rest of the proof is to check that all real hypersurfaces of type A_{0} or A_{1} satisfy the axiom of totally real 2-planes. We will only study the geodesic hypersphere in ℂP^{m}, as the rest of cases are quite similar. Given a natural number t and s > 0, we write down 𝕊^{t}(s) = {(x_{0}, …, x_{t}) ∈ ℝ^{t+1} : ∑_{k=0}^{t} x_{k}^{2} = s^{2}}, the standard t-sphere of radius s > 0. In case t = 1, 𝕊^{1} will denote the group of unit complex numbers. The complex unit will be denoted by i, and if z ∈ ℂ, the complex conjugate will be denoted by z̄. The tangent space at a given point will be identified by the parallel displacement in the usual way. We recall the well-known identification 𝕊^{2m+1}(1) = {(z_{0}, …, z_{m}) ∈ ℂ^{m+1} : ∑_{k=0}^{m} z_{k}z̄_{k} = 1}. Let f : 𝕊^{2m+1}(1) → ℂP^{m}(4) the Hopf’s fibration. The vertical component of the fibration at a point p ∈ 𝕊^{2m+1} is spanned by {ip}. Given 0 < r < π/2, define M̄^{2m}(r) = {(z_{0}, …, z_{m}) ∈ 𝕊^{2m+1}(1) : ∑_{k=0}^{m−1} z_{k}z̄_{k} = cos^{2}(r)}. This hypersurface is invariant under the action of the group 𝕊^{1}, and its image under f is the well-known geodesic hypersphere in ℂP^{m}. (For details, see [4]). This real hypersurface is homogeneous, so we will only proof our assertion at a point and for a certain totally real 2-plane. Given X, Y ∈ ℂ^{m+1} with coordinates X = (X_{0}, …, X_{m}), Y = (Y_{0}, …, Y_{m}), we define the metric ⟨X, Y ⟩ = ∑_{k=0}^{m} ℜ(X_{k}Ȳ_{k}). Given a point p = (p_{0}, …, p_{m}) ∈ M̄^{2m}(r), it is easy to check T_{p}M̄^{2m}(r) = {X ∈ ℂ^{m+1} : ⟨X, p⟩ = 0, ∑_{k=0}^{m−1} ℜ(X_{k}p̄_{k}) = 0}. A unit normal vector to M̄^{2m}(r) at p in 𝕊^{2m+1} is N̄_{p} = (− tan(r)p_{0}, …, − tan(r)p_{m−1}, cot(r)p_{m}). Therefore, a horizontal lift of ξ at p is ξ̄_{p} = (tan(r)ip_{0}, …, tan(r)ip_{m−1}, − cot(r)ip_{m}). Moreover, the horizontal lift of 𝔻_{f(p)} at p is 𝔻̄_{p} = {X ∈ ℂ^{m+1} : ⟨X, p⟩ = ⟨X, ip⟩ = ⟨X, N̄_{p}⟩ = ⟨X, ξ̄_{p}⟩ = 0}. By considering a real number as a complex number without imaginary part, we define the embedding ψ : 𝕊^{2}(cos(r)) → M̄^{2m}(r) given by ψ(x, y, z) = (0, …, 0, x, y, z, sin(r)). At the point q = (0, 0, cos(r)) ∈ 𝕊^{2}, we see T_{q}𝕊^{2}(cos(r)) = Span{(1, 0, 0), (0, 1, 0)}, and (ψ)_{∗q}{(1, 0, 0), (0, 1, 0)} = {(0, …, 0, 1, 0, 0, 0), (0, …, 0, 1, 0, 0)}. This means that the embedding ψ is horizontal, in the sense that every tangent space of 𝕊^{2}(cos(r)) is mapped by ψ onto a horizontal subspace. Besides, it is totally geodesic in M̄^{2m}(r), so that the surface S = f(ψ(𝕊^{2}(cos(r)))) is totally geodesic in the geodesic hypersphere f(M̄^{2m}(r)). Finally, the surface S is totally real as a submanifold in ℂP^{m}.

Remark The totally geodesic submanifolds in a real hypersurface satisfying the axiom of totally real 2-planes are open subsets of real space forms immersed in ℂM^{m}(c) either as totally geodesic if the real hypersurface is ruled, or as extrinsic 2-spheres if the real hypersurface is of type A_{0} or A_{1}.

References

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Departamento de Geometría y Topología Facultad de Ciencias, Universidad de Granada 18071 Granada (Spain)

Eingegangen am 19. July 1999