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Sectional curvatures of ruled real hypersurfaces in a complex hyperbolic space
Differential Geometry and its Applications 51 (2017) 1–8
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Sectional curvatures of ruled real hypersurfaces in a complex hyperbolic space Sadahiro Maeda a , Hiromasa Tanabe b,∗ a
Department of Mathematics, Saga University, 1 Honzyo, Saga, 840-8502, Japan Department of Science, National Institute of Technology, Matsue College, Matsue, Shimane 690-8518, Japan b
a r t i c l e
i n f o
Article history: Received 28 September 2016 Available online xxxx Communicated by J. Berndt Dedicated to Professor Yasunao Hattori on the occasion of his 60th birthday
a b s t r a c t n (c) (n ≥ 2) of constant A ruled real hypersurface in a nonflat complex space form M holomorphic sectional curvature c(= 0) is, in a word, a real hypersurface having n−1 (c). In this paper, we a foliation by totally geodesic complex hyperplanes M investigate the sectional curvatures K of ruled real hypersurfaces in a complex hyperbolic space and show that such hypersurfaces are classified into two types with regard to the range of K. © 2016 Elsevier B.V. All rights reserved.
MSC: primary 53B25 secondary 53C40 Keywords: Ruled real hypersurfaces Sectional curvatures Complex hyperbolic space
1. Introduction A complex n-dimensional complete and simply connected Kähler manifold of constant holomorphic secn (c). Such a space is tional curvature c(= 0) is called a nonflat complex space form, which is denoted by M n holomorphically isometric to either an n-dimensional complex projective space CP (c) or an n-dimensional complex hyperbolic space CH n (c) according as c > 0 or c < 0. n (c) is one of the most interesting fields The study of real hypersurfaces isometrically immersed into M n (c), we have the in Riemannian submanifold theory. Among typical examples of real hypersurfaces in M 2n−1 class of ruled real hypersurfaces. A real hypersurface M in Mn (c) (n ≥ 2) is said to be ruled if the holomorphic distribution T 0 M := {X ∈ TM | η(X) = 0} is integrable and each of its leaves is locally n−1 (c) of the ambient space M n (c), where η is the congruent to a totally geodesic complex hypersurface M * Corresponding author. E-mail addresses: [email protected] (S. Maeda), [email protected] (H. Tanabe). http://dx.doi.org/10.1016/j.difgeo.2016.11.007 0926-2245/© 2016 Elsevier B.V. All rights reserved.
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n (c) can be constructed in the following manner. We contact form on M . Every ruled real hypersurface in M take an arbitrary regular real curve γ : I → Mn (c) parametrized by its arclength s defined on some open n−1 (c) in such a way interval I(⊂ R). At each point γ(s) (s ∈ I) we attach a complex hyperplane Ms ∼ =M that the hyperplane Ms is orthogonal to the real plane spanned by γ(s) ˙ and J γ(s), ˙ where J denotes the n (c). Kähler structure of the ambient space. Then, we obtain a ruled real hypersurface M = s∈I Ms in M We call this M a ruled real hypersurface associated with γ. Since it may in general have singularities, we must omit such points. Every ruled real hypersurface M must not be Hopf, that is, the characteristic vector field of M is not principal on its open dense subset (see Section 2). Moreover, this class of real hypersurfaces in a complex hyperbolic space CH n (c) contains nice three examples, which are the minimal homogeneous one, the complete non-homogeneous minimal one and the non-complete non-homogeneous minimal one (for details, see [2]). Needless to say that the sectional curvature is one of the most important geometric invariants in Riemannian geometry. It is well-known that every ruled surface in 3-dimensional Euclidean space R3 has nonpositive Gaussian curvature. So, we have naturally an interest in the sectional curvature of every ruled real hypern (c). In [9], the authors showed that the sectional curvature K surface in a nonflat complex space form M n of every ruled real hypersurface in CP (c) satisfies a sharp inequality −∞ < K ≤ c. In this paper, we pay attention to ruled real hypersurfaces in CH n (c) associated with smooth regular curves and investigate the sectional curvatures K of them. We shall show that such real hypersurfaces M are classified into two types with regard to the range of K, that is, M satisfies sharp inequalities either −∞ < K ≤ c/4 or c ≤ K ≤ c/4 (Theorem in Section 4). 2. Preliminaries First of all, we set up some notations. Let M be a real hypersurface of an n (≥ 2)-dimensional nonflat n (c) endowed with Riemannian metric g and the canonical Kähler structure J through complex space form M an isometric immersion. We denote by N a unit normal local vector field on M and by A the shape operator n (c). Then the Riemannian connections ∇ of M n (c) and ∇ of M are related by Gauss and of M in M Weingarten formulas X Y = ∇X Y + g(AX, Y )N , ∇ (2.1) X N = −AX ∇ for vector fields X and Y tangent to M . On the real hypersurface M , an almost contact metric structure (φ, ξ, η, g) associated with N is naturally induced as ξ = −JN ,
η(X) = g(X, ξ) and φX = JX − η(X)N
for each tangent vector X ∈ TM . We call ξ, φ and η the characteristic vector field, the characteristic tensor field and the contact form on M , respectively. It is well-known that they satisfy
η(ξ) = 1,
φξ = 0,
φ = −id + η ⊗ ξ, 2
η ◦ φ = 0, g(φX, φY ) = g(X, Y ) − η(X)η(Y )
(2.2)
and ∇X ξ = φAX, for all vectors X, Y ∈ TM .
(2.3)
S. Maeda, H. Tanabe / Differential Geometry and its Applications 51 (2017) 1–8
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Next, we characterize ruled real hypersurfaces by the property of their shape operators. For a real hypersurface M , we define two functions μ, ν : M → R by μ := g(Aξ, ξ) and ν := Aξ − μξ .
(2.4)
Then we have n (c) the following three conditions are mutually equivalent. Lemma 1 ([8]). For a real hypersurface M in M (1) M is ruled. (2) The shape operator A of M satisfies g(AX, Y ) = 0
for ∀X, Y ∈ T 0 M = {X ∈ TM | η(X) = 0}.
(2.5)
(3) The set M1 = {p ∈ M | ν(p) = 0} is an open dense subset of M . Moreover, there exists a unit vector field U on M1 which is orthogonal to ξ and satisfies Aξ = μξ + νU,
AU = νξ
and
AX = 0
(2.6)
for any tangent vector X orthogonal to both ξ and U . The statement (3) of Lemma 1 means that the characteristic vector fields ξ of ruled real hypersurfaces are not principal, that is, they are not eigenvectors of their shape operators, at each point of M1. So, one finds that ruled real hypersurfaces are examples far from Hopf hypersurfaces. Thus, the functions μ and ν are important quantities which measure how far the characteristic vector field ξ is from being a principal curvature vector. 3. Ruled real hypersurfaces associated with smooth curves We begin this section with recalling some terminology on smooth curves in a Riemannian manifold M with Riemannian connection ∇. For a smooth curve γ = γ(s) (s ∈ I) parametrized by its arclength s, we γ˙ γ(s) call the function κ(s) := ∇ ˙ the first curvature of γ. A point γ(s0 ) is said to be an inflection point if the first curvature κ vanishes at s0 . When the curve γ does not have inflection points, we set a unit vector γ˙ γ. field Yγ along γ which is orthogonal to γ˙ by Yγ := (1/κ)∇ ˙ Then, it satisfies
γ˙ γ˙ = κYγ , ∇ γ˙ Yγ = −κγ˙ + Zγ ∇
with some vector field Zγ along γ which is orthogonal to both γ˙ and Yγ . We say γ is of proper order 2 at γ(s0 ) if Zγ(s0 ) vanishes. If all points of γ are of proper order 2, that is Zγ ≡ 0, this smooth curve γ is said to be a Frenet curve of proper order 2 with curvature κ. A Frenet curve of proper order 2 with constant γ˙ γ˙ = kYγ , ∇ γ˙ Yγ = −k γ, curvature k, that is a curve which satisfies ∇ ˙ is called a circle of curvature k(> 0). n (c). On non-inflection points of γ, We now consider a smooth curve γ in a nonflat complex space form M we define its first complex torsion by τ (s) := g(γ(s), ˙ JYγ (s)) (cf. [5]). A circle of positive curvature k in Mn (c) is said to be totally real if its first complex torsion τ vanishes. A totally real circle lies on a totally real totally geodesic surface M 2 (c) (= RP 2 (c/4) or RH 2 (c/4)). n (c). We shall investigate the Let M be a ruled real hypersurface associated with a smooth curve γ in M values of the functions μ, ν along γ.
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γ˙ γ ) n (c) (n ≥ 2) and M Lemma 2. Let γ = γ(s) (s ∈ I) be a smooth curve with first curvature κ(:= ∇ ˙ in M be a ruled real hypersurface associated with the curve γ. Then the functions μ = g(Aξ, ξ) and ν = Aξ − μξ satisfy the following. (1) If κ(s) = 0 for each s, for the first complex torsion τ of γ we have μ(γ(s)) = −κ(s)τ (s),
(3.1)
ν(γ(s)) = κ(s) (1 − τ (s) ). 2
2
2
(3.2)
(2) If κ(s0 ) = 0 for some s0 , we have μ(γ(s0 )) = ν(γ(s0 )) = 0. Proof. We should first mention that the curve γ is an integral curve of the characteristic vector field ξ of M . Since γ(s) ˙ = ξγ(s) , it follows from Gauss formula (2.1), the definition (2.4) of μ and (2.3) that γ˙ γ(s) ∇ ˙ = ∇γ˙ γ(s) ˙ + g(Aγ(s), ˙ γ(s))N ˙ γ(s)
(3.3)
= φAξγ(s) + μ(γ(s))Nγ(s) . To prove Lemma 2 we divide our arguments into four separate cases. (i) The case of ν(γ(s)) = 0. In this case, by Lemma 1(3) we can define a unit vector field U with property (2.6) in a neighborhood of the point γ(s). Then, it follows from (3.3), (2.2) and (2.3) that γ˙ γ(s) ˙ = φ μ(γ(s))ξγ(s) + ν(γ(s))Uγ(s) + μ(γ(s))Nγ(s) ∇
(3.4)
= ν(γ(s))φUγ(s) + μ(γ(s))Nγ(s) , so that the first curvature κ(s) of γ satisfies κ(s)2 = ν(γ(s))2 + μ(γ(s))2 > 0.
(3.5)
As γ(s) is not an inflection point, we have γ˙ γ(s)) τ (s) = g(γ(s), ˙ JYγ (s)) = (1/κ(s))g(ξγ(s) , J ∇ ˙ = −(1/κ(s))g(Nγ (s), ν(γ(s))φUγ(s) + μ(γ(s))Nγ(s) ) = −μ(γ(s))/κ(s). This gives (3.1) which, together with (3.5), implies (3.2). Thus, in this case we have κ(s) = 0 and the equalities (3.1) and (3.2) hold. (ii) The case that ν(γ(s0 )) = 0 and μ(γ(s0 )) > 0 for some s0 ∈ I. In this case, from the definition (2.4) of ν we see Aξγ(s0 ) = μ(γ(s0 ))ξγ(s0 ) , hence (3.3) reduces to γ˙ γ)(s (∇ ˙ 0 ) = μ(γ(s0 ))Nγ(s0 ) by use of (2.2). This means κ(s0 ) = μ(γ(s0 )) > 0 and Yγ (s0 ) = Nγ(s0 ) . Therefore τ (s0 ) can be defined and τ (s0 ) = g(γ(s ˙ 0 ), JYγ (s0 )) = g(ξγ(s0 ) , JNγ(s0 ) ) = −1. So, in this case we see κ(s0 ) = 0 and the equalities (3.1) and (3.2) hold.
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(iii) The case that ν(γ(s0 )) = 0 and μ(γ(s0 )) < 0 for some s0 ∈ I. Just as the above case (ii), one can see that κ(s0 ) = −μ(γ(s0 )) > 0, Yγ (s0 ) = −Nγ(s0 ) and τ (s0 ) = 1. In this case we also see κ(s0 ) = 0 and the equalities (3.1) and (3.2) hold. (iv) The case that ν(γ(s0 )) = 0 and μ(γ(s0 )) = 0 for some s0 ∈ I. In this case we have Aξγ(s0 ) = γ˙ γ(s μ(γ(s0 ))ξγ(s0 ) = 0, so that ∇ ˙ 0 ) = φAξγ(s0 ) + μ(γ(s0 ))Nγ(s0 ) = 0. Hence, κ(s0 ) = 0. Conversely, by our argument we can see that μ(γ(s0 )) = ν(γ(s0 )) = 0 if κ(s0 ) = 0. Altogether, we conclude the results of Lemma 2. 2 Remark 1. As a matter of course, when a curve γ has an inflection point γ(s0 ), we cannot define the complex torsion τ at s = s0 . But, in view of Lemma 2(2), we may say that both of the equalities (3.1) and (3.2) are valid at such a point in a trivial sense. By the proof of Lemma 2, it is easy to see the following: n (c) (n ≥ 2) and suppose that M has a point x0 Proposition 1. Let M be a ruled real hypersurface in M where the function ν vanishes. Let γ = γ(s) (s ∈ I) be the integral curve of the characteristic vector field ξ with γ(s0 ) = x0 . Then, the curve γ satisfies one of the following: γ˙ γ)(s (a) The point γ(s0 ) = x0 is an inflection point of the curve γ, i.e., (∇ ˙ 0 ) = 0. Moreover, in this case we have Aξx0 = 0; γ˙ γ˙ = ±κJ γ˙ holds at s = s0 , that is, orthonormal vectors γ(s (b) The equation ∇ ˙ 0 ), Yγ (s0 ) span a holomorphic plane at x0 . 4. Main result We investigate sectional curvatures of the ruled real hypersurfaces associated with smooth curves in a complex hyperbolic space. We have Theorem. Let M be a ruled real hypersurface in CH n (c) (n ≥ 2) associated with a smooth curve γ defined on an open interval I(⊂ R). Then the following two statements hold: (1) The sectional curvature K of M satisfies sharp inequalities either −∞ < K ≤ c/4 or c ≤ K ≤ c/4 on M . (2) The sectional curvature K of M satisfies c ≤ K ≤ c/4 if and only if the first curvature κ and the first complex torsion τ of γ satisfy κ(s)2 (1 − τ (s)2 ) ≤ |c|/4
(4.1)
for any s ∈ I. Here, if the curve γ has an inflection point γ(s0 ), we consider that the left hand side of (4.1) vanishes on such a point. Remark 2. In [7], the first and second authors and Kim studied ruled real hypersurfaces having the same n (c). The statement (2) of Theorem was proved in their sectional curvature as that of an ambient space M paper in case that the curve γ is a Frenet curve. That is, a Frenet curve γ of proper order d is a smooth curve that has no inflection points in a wide sense, i.e., all of its curvatures κ1 (s)(= κ(s)), · · · , κd−1 (s) are positive smooth functions along the curve γ. Our Theorem statement (2) is an improvement of theirs.
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Proof of Theorem. Let K(X, Y ) denote the sectional curvature of the plane section spanned by orthonormal tangent vectors X, Y on M . Then, the well-known equation of Gauss gives the following expression of the sectional curvature in terms of g, φ and A (for details, see [9]): K(X, Y ) = (c/4) 1 + 3g(φX, Y )2 + g(AX, X)g(AY, Y ) − g(AX, Y )2 .
(4.2)
Let p ∈ M and take an arbitrary 2-plane σ in the tangent space Tp M at the point p. Then there exist orthonormal vectors X and Y tangent to M at p, which are orthogonal to ξp , such that vectors X(t) := cos t · ξp + sin t · X and Y form the orthonormal basis of the plane σ for some real number t. First, we examine the case where ν(p) = 0. In this case, using the representation (2.6) of the shape operator A with (2.2), (2.5) and (4.2), one can find that the sectional curvature K of the plane section σ can be written as K(X(t), Y ) = (c/4) 1 + 3 sin2 t g(φX, Y )2 − ν(p)2 cos2 t g(Y, Up )2 . As c < 0, it is obvious that the maximum value of K is c/4. Moreover, we see K(X(t), Y ) ≥ (c/4)(1 + 3 sin2 t) − ν(p)2 cos2 t = (c/4) − ν(p)2 + (3c/4) + ν(p)2 sin2 t. If (3c/4) +ν(p)2 ≥ 0, the sectional curvature K takes its minimum value (c/4) −ν(p)2 at, for example, t = 0, Y = Up , X = φUp . If (3c/4) + ν(p)2 < 0, it takes the minimum value (c/4) − ν(p)2 + (3c/4) + ν(p)2 = c at, for example, t = π/2, Y = Up , X = φUp . Next, we study the case where ν(p) = 0. Since Aξp = μ(p)ξp by the definition (2.4) of ν, we see from (2.5) that K(X(t), Y ) = (c/4) 1 + 3 sin2 t g(φX, Y )2 . This implies that max Kp = c/4 and min Kp = c. Summarizing the above two cases, we have max K = c/4
(4.3)
and min K =
(c/4) − ν(p)2 c
at p ∈ M with ν(p) ≥
3|c|/2,
at p ∈ M with 0 ≤ ν(p) < 3|c|/2.
(4.4)
n−1 (c) → In [3], Lohnherr–Reckziegel studied ruled real hypersurfaces by examining the map f : I × M Mn (c) which moves a subspace Mn−1 (c) along a regular curve γ : I → Mn (c) defined on an open interval I(⊂ R). The following lemma is due to them (Corollary 3 in [3]). Lemma 3. Let M be a ruled real hypersurface M = γ : I → CH n (c), where Ms ∼ = CH n−1 (c). Then:
s∈I
Ms in CH n (c) (n ≥ 2) associated with the curve
(1) For each leaf Ms , the function ν satisfies either ν(Ms ) = 0, |c|/2 , ν(Ms ) = |c|/2 or ν(Ms ) =
|c|/2, ∞ . (2) The function ν becomes ∞ in the singularities. (3) ν has at most one zero on every leaf Ms .
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Since the initial value ν(γ(s)) of ν on each leaf Ms is given by Lemma 2, in view of (4.3), (4.4) and Lemma 3(1) we get the assertions of our theorem. 2 Remark 3. We describe the geometric meaning of the inequality (4.1). When a curve γ in CH n (c) has no γ˙ γ(s))), γ˙ γ(s)) inflection points, we see from (3.1) that μ(γ(s)) = g(J γ(s), ˙ ˙ = κ(s) cos(θ(J γ(s), ˙ ∇ ˙ where ∇ γ˙ γ(s)) γ˙ γ(s). θ(J γ(s), ˙ ∇ ˙ denotes the angle between J γ(s) ˙ and ∇ ˙ Thus, μ(γ(s)) is the curvature of γ in γ˙ γ(s))). the totally real direction of γ. ˙ Then, we can rewrite (3.2) as ν(γ(s)) = κ(s) sin(θ(J γ(s), ˙ ∇ ˙ Roughly speaking, when we construct a ruled real hypersurface M = s∈I Ms by attaching a complex hyperplane Ms γ˙ γ(s))) at each point of the curve γ as described in Introduction, if some kind of curvature κ(s) sin(θ(J γ(s), ˙ ∇ ˙ of γ is enough large, then attached hyperplanes have intersections, so that hypersurface M has singularities and the sectional curvature of M becomes −∞ (see Lemma 3(2)). Now, we give some examples. Examples 1 and 2 are related to Proposition 1. Example 1. Let γ = γ(s) (s ∈ I) be an arbitrary geodesic in CH n (c) (n ≥ 2) and consider the ruled real hypersurface M associated with γ. Then, since the first curvature κ of γ satisfies κ ≡ 0, M has the sectional curvature K with c ≤ K ≤ c/4. Moreover, Proposition 1(a) and Lemma 3(3) tell us that Aξγ(s) = 0 for any s ∈ I and that ξ is not principal at the point which does not lie on the curve γ. n (c) which satisfies ∇ γ˙ γ(s) ˙ = ±κ(s)J γ(s) ˙ for each s ∈ I is called a Kähler Example 2. A curve γ : I → M Frenet curve [6]. It is needless to say that every geodesic is a Kähler Frenet curve in a trivial sense. A Kähler 1 (c). Let M be a ruled real hypersurface in Frenet curve is lying on some totally geodesic complex line M n CH (c) associated with a Kähler Frenet curve γ. Then, we see |τ | = 1 on the curve γ and hence by Theorem hypersurface M has the sectional curvature K with c ≤ K ≤ c/4. Remark that the characteristic vector ξ is principal along the above curve γ. But, the vector ξ is not principal at the other points of M . We next give examples of minimal ruled real hypersurfaces. Example 3. Let M be a minimal ruled real hypersurface M in a complex hyperbolic space CH n (c) (n ≥ 2). Lemma 1(3) and Lemma 2 show that every integral curve γ of the characteristic vector fields ξ satisfies μ(γ(s)) = 0 (∀s ∈ I) and the first complex torsion τ of γ vanishes. In fact, it is known that every minimal ruled real hypersurface M is associated with a totally real circle γ and vice versa [3,4]. In [2], Adachi, Bao and the first author classified ruled real hypersurfaces in CH n (c) into three classes with respect to the action of its isometry group I(CH n (c)). Totally real circles in CH n (c) are classified into three classes according to their curvatures with respect to I(CH n (c)) (see [1]).
If the curvature k of a totally real circle γ is greater than |c|/2, γ is closed and bounded. If k = |c|/2, it is horocyclic, so that it is unbounded
and has a single limit point at infinity γ(∞) = γ(−∞) in the ideal boundary of CH n (c). If 0 ≤ k < |c|/2, it is unbounded and has two distinct limit points at infinity γ(∞), γ(−∞) in the ideal boundary. Corresponding to this classification of totally real circles, we call ruled real hypersurfaces associated with γ of elliptic type, parabolic type and axial type, respectively. Every minimal ruled real hypersurface is locally congruent to one of them [2]. Minimal ruled real hypersurfaces of parabolic type and axial type are complete, but those of elliptic type are not complete. Furthermore, the minimal ruled real hypersurface of parabolic type is homogeneous, i.e. this minimal ruled real hypersurface is an orbit of some subgroup of the full isometry group of the ambient space CH n (c). Now, by the criterion (4.1) one can find that minimal ruled real hypersurfaces of types axial and parabolic in CH n (c) have the sectional curvature K with c ≤ K ≤ c/4, whereas the minimal ruled real hypersurface of elliptic type satisfies −∞ < K ≤ c/4. We note that the minimal ruled real hypersurface of axial type coincides with the hypersurface in Example 1.
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Remark 4. From the aspect of Example 3, we can say that the inequality (4.1) is a generalization of a fact that a minimal ruled real hypersurface M has the sectional curvature K with c ≤ K ≤ c/4 if and only if the curvature κ(s)(≥ 0) of a generating curve γ associated to M satisfies 0 ≤ κ(s) ≤ |c| /2 for every s. n (c) we Remark 5. By the construction of ruled real hypersurfaces M in a nonflat complex space form M can see that the sectional curvature K of M satisfies a sharp inequality |c|/4 ≤ |K(X, Y )| ≤ |c| in the case of n ≥ 3 and K(X, Y ) = c in the case of n = 2 for each pair of orthonormal vectors X and Y that are orthogonal to ξp at each point p ∈ M . References [1] T. Adachi, S. Maeda, Global behaviours of circles in a complex hyperbolic space, Tsukuba J. Math. 21 (1997) 29–42. [2] T. Adachi, T. Bao, S. Maeda, Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form, Hokkaido Math. J. 43 (2014) 1–14. [3] M. Lohnherr, H. Reckziegel, On ruled real hypersurfaces in complex space forms, Geom. Dedic. 79 (1999) 267–286. [4] S. Maeda, T. Adachi, Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms, Geom. Dedic. 123 (2006) 65–72. [5] S. Maeda, Y. Ohnita, Helical geodesic immersions into complex space forms, Geom. Dedic. 30 (1989) 93–114. [6] S. Maeda, H. Tanabe, Totally geodesic immersion of Kähler manifolds and Kähler Frenet curves, Math. Z. 252 (2006) 787–795. [7] S. Maeda, H. Tanabe, Y.H. Kim, Ruled real hypersurfaces having the same sectional curvature as that of an ambient nonflat complex space form, Kodai Math. J. 39 (2016) 119–128. [8] R. Niebergall, P.J. Ryan, Real hypersurfaces in complex space forms, in: T.E. Cecil, S.S. Chern (Eds.), Tight and Taut Submanifolds, Cambridge Univ. Press, 1998, pp. 233–305. [9] H. Tanabe, S. Maeda, Sectional curvatures of ruled real hypersurfaces in a nonflat complex space form, in: Proceedings of the Workshop on Differential Geometry of Submanifolds and Its Related Topics, Saga, August 4–6, 2012, World Scientific, 2013, pp. 113–118.
Contents lists available at ScienceDirect
Differential Geometry and its Applications www.elsevier.com/locate/difgeo
Sectional curvatures of ruled real hypersurfaces in a complex hyperbolic space Sadahiro Maeda a , Hiromasa Tanabe b,∗ a
Department of Mathematics, Saga University, 1 Honzyo, Saga, 840-8502, Japan Department of Science, National Institute of Technology, Matsue College, Matsue, Shimane 690-8518, Japan b
a r t i c l e
i n f o
Article history: Received 28 September 2016 Available online xxxx Communicated by J. Berndt Dedicated to Professor Yasunao Hattori on the occasion of his 60th birthday
a b s t r a c t n (c) (n ≥ 2) of constant A ruled real hypersurface in a nonflat complex space form M holomorphic sectional curvature c(= 0) is, in a word, a real hypersurface having n−1 (c). In this paper, we a foliation by totally geodesic complex hyperplanes M investigate the sectional curvatures K of ruled real hypersurfaces in a complex hyperbolic space and show that such hypersurfaces are classified into two types with regard to the range of K. © 2016 Elsevier B.V. All rights reserved.
MSC: primary 53B25 secondary 53C40 Keywords: Ruled real hypersurfaces Sectional curvatures Complex hyperbolic space
1. Introduction A complex n-dimensional complete and simply connected Kähler manifold of constant holomorphic secn (c). Such a space is tional curvature c(= 0) is called a nonflat complex space form, which is denoted by M n holomorphically isometric to either an n-dimensional complex projective space CP (c) or an n-dimensional complex hyperbolic space CH n (c) according as c > 0 or c < 0. n (c) is one of the most interesting fields The study of real hypersurfaces isometrically immersed into M n (c), we have the in Riemannian submanifold theory. Among typical examples of real hypersurfaces in M 2n−1 class of ruled real hypersurfaces. A real hypersurface M in Mn (c) (n ≥ 2) is said to be ruled if the holomorphic distribution T 0 M := {X ∈ TM | η(X) = 0} is integrable and each of its leaves is locally n−1 (c) of the ambient space M n (c), where η is the congruent to a totally geodesic complex hypersurface M * Corresponding author. E-mail addresses: [email protected] (S. Maeda), [email protected] (H. Tanabe). http://dx.doi.org/10.1016/j.difgeo.2016.11.007 0926-2245/© 2016 Elsevier B.V. All rights reserved.
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n (c) can be constructed in the following manner. We contact form on M . Every ruled real hypersurface in M take an arbitrary regular real curve γ : I → Mn (c) parametrized by its arclength s defined on some open n−1 (c) in such a way interval I(⊂ R). At each point γ(s) (s ∈ I) we attach a complex hyperplane Ms ∼ =M that the hyperplane Ms is orthogonal to the real plane spanned by γ(s) ˙ and J γ(s), ˙ where J denotes the n (c). Kähler structure of the ambient space. Then, we obtain a ruled real hypersurface M = s∈I Ms in M We call this M a ruled real hypersurface associated with γ. Since it may in general have singularities, we must omit such points. Every ruled real hypersurface M must not be Hopf, that is, the characteristic vector field of M is not principal on its open dense subset (see Section 2). Moreover, this class of real hypersurfaces in a complex hyperbolic space CH n (c) contains nice three examples, which are the minimal homogeneous one, the complete non-homogeneous minimal one and the non-complete non-homogeneous minimal one (for details, see [2]). Needless to say that the sectional curvature is one of the most important geometric invariants in Riemannian geometry. It is well-known that every ruled surface in 3-dimensional Euclidean space R3 has nonpositive Gaussian curvature. So, we have naturally an interest in the sectional curvature of every ruled real hypern (c). In [9], the authors showed that the sectional curvature K surface in a nonflat complex space form M n of every ruled real hypersurface in CP (c) satisfies a sharp inequality −∞ < K ≤ c. In this paper, we pay attention to ruled real hypersurfaces in CH n (c) associated with smooth regular curves and investigate the sectional curvatures K of them. We shall show that such real hypersurfaces M are classified into two types with regard to the range of K, that is, M satisfies sharp inequalities either −∞ < K ≤ c/4 or c ≤ K ≤ c/4 (Theorem in Section 4). 2. Preliminaries First of all, we set up some notations. Let M be a real hypersurface of an n (≥ 2)-dimensional nonflat n (c) endowed with Riemannian metric g and the canonical Kähler structure J through complex space form M an isometric immersion. We denote by N a unit normal local vector field on M and by A the shape operator n (c). Then the Riemannian connections ∇ of M n (c) and ∇ of M are related by Gauss and of M in M Weingarten formulas X Y = ∇X Y + g(AX, Y )N , ∇ (2.1) X N = −AX ∇ for vector fields X and Y tangent to M . On the real hypersurface M , an almost contact metric structure (φ, ξ, η, g) associated with N is naturally induced as ξ = −JN ,
η(X) = g(X, ξ) and φX = JX − η(X)N
for each tangent vector X ∈ TM . We call ξ, φ and η the characteristic vector field, the characteristic tensor field and the contact form on M , respectively. It is well-known that they satisfy
η(ξ) = 1,
φξ = 0,
φ = −id + η ⊗ ξ, 2
η ◦ φ = 0, g(φX, φY ) = g(X, Y ) − η(X)η(Y )
(2.2)
and ∇X ξ = φAX, for all vectors X, Y ∈ TM .
(2.3)
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Next, we characterize ruled real hypersurfaces by the property of their shape operators. For a real hypersurface M , we define two functions μ, ν : M → R by μ := g(Aξ, ξ) and ν := Aξ − μξ .
(2.4)
Then we have n (c) the following three conditions are mutually equivalent. Lemma 1 ([8]). For a real hypersurface M in M (1) M is ruled. (2) The shape operator A of M satisfies g(AX, Y ) = 0
for ∀X, Y ∈ T 0 M = {X ∈ TM | η(X) = 0}.
(2.5)
(3) The set M1 = {p ∈ M | ν(p) = 0} is an open dense subset of M . Moreover, there exists a unit vector field U on M1 which is orthogonal to ξ and satisfies Aξ = μξ + νU,
AU = νξ
and
AX = 0
(2.6)
for any tangent vector X orthogonal to both ξ and U . The statement (3) of Lemma 1 means that the characteristic vector fields ξ of ruled real hypersurfaces are not principal, that is, they are not eigenvectors of their shape operators, at each point of M1. So, one finds that ruled real hypersurfaces are examples far from Hopf hypersurfaces. Thus, the functions μ and ν are important quantities which measure how far the characteristic vector field ξ is from being a principal curvature vector. 3. Ruled real hypersurfaces associated with smooth curves We begin this section with recalling some terminology on smooth curves in a Riemannian manifold M with Riemannian connection ∇. For a smooth curve γ = γ(s) (s ∈ I) parametrized by its arclength s, we γ˙ γ(s) call the function κ(s) := ∇ ˙ the first curvature of γ. A point γ(s0 ) is said to be an inflection point if the first curvature κ vanishes at s0 . When the curve γ does not have inflection points, we set a unit vector γ˙ γ. field Yγ along γ which is orthogonal to γ˙ by Yγ := (1/κ)∇ ˙ Then, it satisfies
γ˙ γ˙ = κYγ , ∇ γ˙ Yγ = −κγ˙ + Zγ ∇
with some vector field Zγ along γ which is orthogonal to both γ˙ and Yγ . We say γ is of proper order 2 at γ(s0 ) if Zγ(s0 ) vanishes. If all points of γ are of proper order 2, that is Zγ ≡ 0, this smooth curve γ is said to be a Frenet curve of proper order 2 with curvature κ. A Frenet curve of proper order 2 with constant γ˙ γ˙ = kYγ , ∇ γ˙ Yγ = −k γ, curvature k, that is a curve which satisfies ∇ ˙ is called a circle of curvature k(> 0). n (c). On non-inflection points of γ, We now consider a smooth curve γ in a nonflat complex space form M we define its first complex torsion by τ (s) := g(γ(s), ˙ JYγ (s)) (cf. [5]). A circle of positive curvature k in Mn (c) is said to be totally real if its first complex torsion τ vanishes. A totally real circle lies on a totally real totally geodesic surface M 2 (c) (= RP 2 (c/4) or RH 2 (c/4)). n (c). We shall investigate the Let M be a ruled real hypersurface associated with a smooth curve γ in M values of the functions μ, ν along γ.
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γ˙ γ ) n (c) (n ≥ 2) and M Lemma 2. Let γ = γ(s) (s ∈ I) be a smooth curve with first curvature κ(:= ∇ ˙ in M be a ruled real hypersurface associated with the curve γ. Then the functions μ = g(Aξ, ξ) and ν = Aξ − μξ satisfy the following. (1) If κ(s) = 0 for each s, for the first complex torsion τ of γ we have μ(γ(s)) = −κ(s)τ (s),
(3.1)
ν(γ(s)) = κ(s) (1 − τ (s) ). 2
2
2
(3.2)
(2) If κ(s0 ) = 0 for some s0 , we have μ(γ(s0 )) = ν(γ(s0 )) = 0. Proof. We should first mention that the curve γ is an integral curve of the characteristic vector field ξ of M . Since γ(s) ˙ = ξγ(s) , it follows from Gauss formula (2.1), the definition (2.4) of μ and (2.3) that γ˙ γ(s) ∇ ˙ = ∇γ˙ γ(s) ˙ + g(Aγ(s), ˙ γ(s))N ˙ γ(s)
(3.3)
= φAξγ(s) + μ(γ(s))Nγ(s) . To prove Lemma 2 we divide our arguments into four separate cases. (i) The case of ν(γ(s)) = 0. In this case, by Lemma 1(3) we can define a unit vector field U with property (2.6) in a neighborhood of the point γ(s). Then, it follows from (3.3), (2.2) and (2.3) that γ˙ γ(s) ˙ = φ μ(γ(s))ξγ(s) + ν(γ(s))Uγ(s) + μ(γ(s))Nγ(s) ∇
(3.4)
= ν(γ(s))φUγ(s) + μ(γ(s))Nγ(s) , so that the first curvature κ(s) of γ satisfies κ(s)2 = ν(γ(s))2 + μ(γ(s))2 > 0.
(3.5)
As γ(s) is not an inflection point, we have γ˙ γ(s)) τ (s) = g(γ(s), ˙ JYγ (s)) = (1/κ(s))g(ξγ(s) , J ∇ ˙ = −(1/κ(s))g(Nγ (s), ν(γ(s))φUγ(s) + μ(γ(s))Nγ(s) ) = −μ(γ(s))/κ(s). This gives (3.1) which, together with (3.5), implies (3.2). Thus, in this case we have κ(s) = 0 and the equalities (3.1) and (3.2) hold. (ii) The case that ν(γ(s0 )) = 0 and μ(γ(s0 )) > 0 for some s0 ∈ I. In this case, from the definition (2.4) of ν we see Aξγ(s0 ) = μ(γ(s0 ))ξγ(s0 ) , hence (3.3) reduces to γ˙ γ)(s (∇ ˙ 0 ) = μ(γ(s0 ))Nγ(s0 ) by use of (2.2). This means κ(s0 ) = μ(γ(s0 )) > 0 and Yγ (s0 ) = Nγ(s0 ) . Therefore τ (s0 ) can be defined and τ (s0 ) = g(γ(s ˙ 0 ), JYγ (s0 )) = g(ξγ(s0 ) , JNγ(s0 ) ) = −1. So, in this case we see κ(s0 ) = 0 and the equalities (3.1) and (3.2) hold.
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(iii) The case that ν(γ(s0 )) = 0 and μ(γ(s0 )) < 0 for some s0 ∈ I. Just as the above case (ii), one can see that κ(s0 ) = −μ(γ(s0 )) > 0, Yγ (s0 ) = −Nγ(s0 ) and τ (s0 ) = 1. In this case we also see κ(s0 ) = 0 and the equalities (3.1) and (3.2) hold. (iv) The case that ν(γ(s0 )) = 0 and μ(γ(s0 )) = 0 for some s0 ∈ I. In this case we have Aξγ(s0 ) = γ˙ γ(s μ(γ(s0 ))ξγ(s0 ) = 0, so that ∇ ˙ 0 ) = φAξγ(s0 ) + μ(γ(s0 ))Nγ(s0 ) = 0. Hence, κ(s0 ) = 0. Conversely, by our argument we can see that μ(γ(s0 )) = ν(γ(s0 )) = 0 if κ(s0 ) = 0. Altogether, we conclude the results of Lemma 2. 2 Remark 1. As a matter of course, when a curve γ has an inflection point γ(s0 ), we cannot define the complex torsion τ at s = s0 . But, in view of Lemma 2(2), we may say that both of the equalities (3.1) and (3.2) are valid at such a point in a trivial sense. By the proof of Lemma 2, it is easy to see the following: n (c) (n ≥ 2) and suppose that M has a point x0 Proposition 1. Let M be a ruled real hypersurface in M where the function ν vanishes. Let γ = γ(s) (s ∈ I) be the integral curve of the characteristic vector field ξ with γ(s0 ) = x0 . Then, the curve γ satisfies one of the following: γ˙ γ)(s (a) The point γ(s0 ) = x0 is an inflection point of the curve γ, i.e., (∇ ˙ 0 ) = 0. Moreover, in this case we have Aξx0 = 0; γ˙ γ˙ = ±κJ γ˙ holds at s = s0 , that is, orthonormal vectors γ(s (b) The equation ∇ ˙ 0 ), Yγ (s0 ) span a holomorphic plane at x0 . 4. Main result We investigate sectional curvatures of the ruled real hypersurfaces associated with smooth curves in a complex hyperbolic space. We have Theorem. Let M be a ruled real hypersurface in CH n (c) (n ≥ 2) associated with a smooth curve γ defined on an open interval I(⊂ R). Then the following two statements hold: (1) The sectional curvature K of M satisfies sharp inequalities either −∞ < K ≤ c/4 or c ≤ K ≤ c/4 on M . (2) The sectional curvature K of M satisfies c ≤ K ≤ c/4 if and only if the first curvature κ and the first complex torsion τ of γ satisfy κ(s)2 (1 − τ (s)2 ) ≤ |c|/4
(4.1)
for any s ∈ I. Here, if the curve γ has an inflection point γ(s0 ), we consider that the left hand side of (4.1) vanishes on such a point. Remark 2. In [7], the first and second authors and Kim studied ruled real hypersurfaces having the same n (c). The statement (2) of Theorem was proved in their sectional curvature as that of an ambient space M paper in case that the curve γ is a Frenet curve. That is, a Frenet curve γ of proper order d is a smooth curve that has no inflection points in a wide sense, i.e., all of its curvatures κ1 (s)(= κ(s)), · · · , κd−1 (s) are positive smooth functions along the curve γ. Our Theorem statement (2) is an improvement of theirs.
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Proof of Theorem. Let K(X, Y ) denote the sectional curvature of the plane section spanned by orthonormal tangent vectors X, Y on M . Then, the well-known equation of Gauss gives the following expression of the sectional curvature in terms of g, φ and A (for details, see [9]): K(X, Y ) = (c/4) 1 + 3g(φX, Y )2 + g(AX, X)g(AY, Y ) − g(AX, Y )2 .
(4.2)
Let p ∈ M and take an arbitrary 2-plane σ in the tangent space Tp M at the point p. Then there exist orthonormal vectors X and Y tangent to M at p, which are orthogonal to ξp , such that vectors X(t) := cos t · ξp + sin t · X and Y form the orthonormal basis of the plane σ for some real number t. First, we examine the case where ν(p) = 0. In this case, using the representation (2.6) of the shape operator A with (2.2), (2.5) and (4.2), one can find that the sectional curvature K of the plane section σ can be written as K(X(t), Y ) = (c/4) 1 + 3 sin2 t g(φX, Y )2 − ν(p)2 cos2 t g(Y, Up )2 . As c < 0, it is obvious that the maximum value of K is c/4. Moreover, we see K(X(t), Y ) ≥ (c/4)(1 + 3 sin2 t) − ν(p)2 cos2 t = (c/4) − ν(p)2 + (3c/4) + ν(p)2 sin2 t. If (3c/4) +ν(p)2 ≥ 0, the sectional curvature K takes its minimum value (c/4) −ν(p)2 at, for example, t = 0, Y = Up , X = φUp . If (3c/4) + ν(p)2 < 0, it takes the minimum value (c/4) − ν(p)2 + (3c/4) + ν(p)2 = c at, for example, t = π/2, Y = Up , X = φUp . Next, we study the case where ν(p) = 0. Since Aξp = μ(p)ξp by the definition (2.4) of ν, we see from (2.5) that K(X(t), Y ) = (c/4) 1 + 3 sin2 t g(φX, Y )2 . This implies that max Kp = c/4 and min Kp = c. Summarizing the above two cases, we have max K = c/4
(4.3)
and min K =
(c/4) − ν(p)2 c
at p ∈ M with ν(p) ≥
3|c|/2,
at p ∈ M with 0 ≤ ν(p) < 3|c|/2.
(4.4)
n−1 (c) → In [3], Lohnherr–Reckziegel studied ruled real hypersurfaces by examining the map f : I × M Mn (c) which moves a subspace Mn−1 (c) along a regular curve γ : I → Mn (c) defined on an open interval I(⊂ R). The following lemma is due to them (Corollary 3 in [3]). Lemma 3. Let M be a ruled real hypersurface M = γ : I → CH n (c), where Ms ∼ = CH n−1 (c). Then:
s∈I
Ms in CH n (c) (n ≥ 2) associated with the curve
(1) For each leaf Ms , the function ν satisfies either ν(Ms ) = 0, |c|/2 , ν(Ms ) = |c|/2 or ν(Ms ) =
|c|/2, ∞ . (2) The function ν becomes ∞ in the singularities. (3) ν has at most one zero on every leaf Ms .
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Since the initial value ν(γ(s)) of ν on each leaf Ms is given by Lemma 2, in view of (4.3), (4.4) and Lemma 3(1) we get the assertions of our theorem. 2 Remark 3. We describe the geometric meaning of the inequality (4.1). When a curve γ in CH n (c) has no γ˙ γ(s))), γ˙ γ(s)) inflection points, we see from (3.1) that μ(γ(s)) = g(J γ(s), ˙ ˙ = κ(s) cos(θ(J γ(s), ˙ ∇ ˙ where ∇ γ˙ γ(s)) γ˙ γ(s). θ(J γ(s), ˙ ∇ ˙ denotes the angle between J γ(s) ˙ and ∇ ˙ Thus, μ(γ(s)) is the curvature of γ in γ˙ γ(s))). the totally real direction of γ. ˙ Then, we can rewrite (3.2) as ν(γ(s)) = κ(s) sin(θ(J γ(s), ˙ ∇ ˙ Roughly speaking, when we construct a ruled real hypersurface M = s∈I Ms by attaching a complex hyperplane Ms γ˙ γ(s))) at each point of the curve γ as described in Introduction, if some kind of curvature κ(s) sin(θ(J γ(s), ˙ ∇ ˙ of γ is enough large, then attached hyperplanes have intersections, so that hypersurface M has singularities and the sectional curvature of M becomes −∞ (see Lemma 3(2)). Now, we give some examples. Examples 1 and 2 are related to Proposition 1. Example 1. Let γ = γ(s) (s ∈ I) be an arbitrary geodesic in CH n (c) (n ≥ 2) and consider the ruled real hypersurface M associated with γ. Then, since the first curvature κ of γ satisfies κ ≡ 0, M has the sectional curvature K with c ≤ K ≤ c/4. Moreover, Proposition 1(a) and Lemma 3(3) tell us that Aξγ(s) = 0 for any s ∈ I and that ξ is not principal at the point which does not lie on the curve γ. n (c) which satisfies ∇ γ˙ γ(s) ˙ = ±κ(s)J γ(s) ˙ for each s ∈ I is called a Kähler Example 2. A curve γ : I → M Frenet curve [6]. It is needless to say that every geodesic is a Kähler Frenet curve in a trivial sense. A Kähler 1 (c). Let M be a ruled real hypersurface in Frenet curve is lying on some totally geodesic complex line M n CH (c) associated with a Kähler Frenet curve γ. Then, we see |τ | = 1 on the curve γ and hence by Theorem hypersurface M has the sectional curvature K with c ≤ K ≤ c/4. Remark that the characteristic vector ξ is principal along the above curve γ. But, the vector ξ is not principal at the other points of M . We next give examples of minimal ruled real hypersurfaces. Example 3. Let M be a minimal ruled real hypersurface M in a complex hyperbolic space CH n (c) (n ≥ 2). Lemma 1(3) and Lemma 2 show that every integral curve γ of the characteristic vector fields ξ satisfies μ(γ(s)) = 0 (∀s ∈ I) and the first complex torsion τ of γ vanishes. In fact, it is known that every minimal ruled real hypersurface M is associated with a totally real circle γ and vice versa [3,4]. In [2], Adachi, Bao and the first author classified ruled real hypersurfaces in CH n (c) into three classes with respect to the action of its isometry group I(CH n (c)). Totally real circles in CH n (c) are classified into three classes according to their curvatures with respect to I(CH n (c)) (see [1]).
If the curvature k of a totally real circle γ is greater than |c|/2, γ is closed and bounded. If k = |c|/2, it is horocyclic, so that it is unbounded
and has a single limit point at infinity γ(∞) = γ(−∞) in the ideal boundary of CH n (c). If 0 ≤ k < |c|/2, it is unbounded and has two distinct limit points at infinity γ(∞), γ(−∞) in the ideal boundary. Corresponding to this classification of totally real circles, we call ruled real hypersurfaces associated with γ of elliptic type, parabolic type and axial type, respectively. Every minimal ruled real hypersurface is locally congruent to one of them [2]. Minimal ruled real hypersurfaces of parabolic type and axial type are complete, but those of elliptic type are not complete. Furthermore, the minimal ruled real hypersurface of parabolic type is homogeneous, i.e. this minimal ruled real hypersurface is an orbit of some subgroup of the full isometry group of the ambient space CH n (c). Now, by the criterion (4.1) one can find that minimal ruled real hypersurfaces of types axial and parabolic in CH n (c) have the sectional curvature K with c ≤ K ≤ c/4, whereas the minimal ruled real hypersurface of elliptic type satisfies −∞ < K ≤ c/4. We note that the minimal ruled real hypersurface of axial type coincides with the hypersurface in Example 1.
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Remark 4. From the aspect of Example 3, we can say that the inequality (4.1) is a generalization of a fact that a minimal ruled real hypersurface M has the sectional curvature K with c ≤ K ≤ c/4 if and only if the curvature κ(s)(≥ 0) of a generating curve γ associated to M satisfies 0 ≤ κ(s) ≤ |c| /2 for every s. n (c) we Remark 5. By the construction of ruled real hypersurfaces M in a nonflat complex space form M can see that the sectional curvature K of M satisfies a sharp inequality |c|/4 ≤ |K(X, Y )| ≤ |c| in the case of n ≥ 3 and K(X, Y ) = c in the case of n = 2 for each pair of orthonormal vectors X and Y that are orthogonal to ξp at each point p ∈ M . References [1] T. Adachi, S. Maeda, Global behaviours of circles in a complex hyperbolic space, Tsukuba J. Math. 21 (1997) 29–42. [2] T. Adachi, T. Bao, S. Maeda, Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form, Hokkaido Math. J. 43 (2014) 1–14. [3] M. Lohnherr, H. Reckziegel, On ruled real hypersurfaces in complex space forms, Geom. Dedic. 79 (1999) 267–286. [4] S. Maeda, T. Adachi, Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms, Geom. Dedic. 123 (2006) 65–72. [5] S. Maeda, Y. Ohnita, Helical geodesic immersions into complex space forms, Geom. Dedic. 30 (1989) 93–114. [6] S. Maeda, H. Tanabe, Totally geodesic immersion of Kähler manifolds and Kähler Frenet curves, Math. Z. 252 (2006) 787–795. [7] S. Maeda, H. Tanabe, Y.H. Kim, Ruled real hypersurfaces having the same sectional curvature as that of an ambient nonflat complex space form, Kodai Math. J. 39 (2016) 119–128. [8] R. Niebergall, P.J. Ryan, Real hypersurfaces in complex space forms, in: T.E. Cecil, S.S. Chern (Eds.), Tight and Taut Submanifolds, Cambridge Univ. Press, 1998, pp. 233–305. [9] H. Tanabe, S. Maeda, Sectional curvatures of ruled real hypersurfaces in a nonflat complex space form, in: Proceedings of the Workshop on Differential Geometry of Submanifolds and Its Related Topics, Saga, August 4–6, 2012, World Scientific, 2013, pp. 113–118.