Three-Dimensional Slant Submanifolds of K–Contact Manifolds

Antonio Lotta

Abstract

The intrinsic geometry of 3–dimensional non anti–invariant slant submanifolds of K–contact manifolds is studied. It is proved that these submanifolds are Riemannian manifolds of quasi constant sectional curvature. Moreover explicit formulae are given for sectional curvatures, Ricci tensor, and scalar curvature of a slant submanifold. A necessary condition is proved for the minimality of a 3–dimensional slant submanifold of a Sasakian manifold of constant φ–sectional curvature. Finally it is proved that every 3–dimensional slant submanifold of a K-contact manifold is locally homogeneous provided it has constant horizontal curvature with respect to the induced metric.

Mathematics Subject Classifications: 53C25, 53C42, 53C30 Key Words: slant submanifolds–contact manifolds

Introduction

Let M be an almost contact metric manifold with structure (φ, ξ, η, g) (our standard reference for contact geometry is [2]). By a slant submanifold of M, we mean an immersed submanifold N such that for any x ∈ N and any X ∈ T_{x}N linearly independent on ξ, the angle between φX and T_{x}N is a constant ϑ ∈ [0, \frac{π}{2} ], called the slant angle of N in M and denoted by sla(N). General properties of slant submanifolds according to this definition are studied by the author in [8].

The results given here apply to non anti–invariant three–dimensional slant submanifolds in K–contact manifolds, that is contact metric manifolds whose characteristic vector field is Killing (see [2]). In particular in section 2 we study the intrinsic geometry of a slant submanifold as a Riemannian manifold endowed with the induced metric. This is made using some fundamental properties of the so–called Riemannian manifolds of quasi–constant sectional curvatures (QC–manifolds),which have been studied in [5]. The results used here are recalled in section 1. Indeed, we prove that any 3–dimensional slant submanifold of a K–contact manifold is a QC–manifold with respect to the induced metric. As a consequence, we get explicit formulae for the sectional curvatures, Ricci tensor and scalar curvature of a slant submanifold.

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 37-51 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Moreover, we find a necessary condition for the minimality of a slant submanifold of a Sasakian manifold of constant φ–sectional curvature. This condition involves the so-called horizontal curvatures of the submanifold.

Section 3 deals with three dimensional slant submanifolds having constant horizontal curvature. We give a characterization of these slant submanifolds by means of an algebraic property of the Ricci tensor. The main result shows that slant submanifolds of constant horizontal curvature are locally homogeneous Riemannian manifolds with respect to the induced metric.

We also remark that this study is motivated by the existence of examples. In fact, it can be shown that the standard Sasakian space form R^{2n+1} admits 3–dimensional harmonic foliations whose leaves are all slant submanifolds with constant horizontal curvature (see [10]).

1 Riemannian manifolds of quasi–constant sectional curvatures

In this section we recall some fundamental facts about Riemannian manifolds of quasi–constant sectional curvatures (QC–manifolds), which can be found in [5].

We denote by (M, g, ξ) a Riemannian manifold together with a globally defined unit vector field. Let π ⊂ T_{p}M be a plane with sectional curvature K(π). If ξ_{p} is orthogonal (resp. belongs) to π, K(π) is said to be horizontal (resp. vertical) and the plane itself is called horizontal (resp. vertical).

(M, g, ξ) is called a Riemannian manifold of quasi–constant sectional curvatures (QC–manifold) if for all planes π ⊂ T_{p}M, K(π) depends only on the point p and on the angle ψ_{π} between ξ_{p} and π. In this case there exist differentiable functions a, b on M such that for all p ∈ M and π ⊂ T_{p}M:

(1.1)  K(π) = a(p) + b(p) cos^{2} ψ_{π}.

Hence in a QC–manifold, all horizontal (resp. vertical) sectional curvatures at the same point p are equal to a(p) (resp. a(p) + b(p)).

Let Γ and Ω be the tensor fields on M defined by

Γ(X, Y, Z, U) = g(X, Z)g(Y, U) − g(Y, Z)g(X, U)
Ω(X, Y, Z, U) = g(Y, U)η(X)η(Z) − g(X, U)η(Y)η(Z)
+ g(X, Z)η(Y)η(U) − g(Y, Z)η(X)η(U).

The following result gives a useful characterization of QC–manifolds by means of the curvature tensor:

Proposition 1.1 ([5]) (M, g, ξ) is a Riemannian manifold of quasi–constant sectional curvatures if and only if there exist functions a, b on M such that

(1.2)  R = aΓ + bΩ,

where R is the curvature tensor of (M, g).

The functions a, b are those in formula (1.1). In particular, if (1.1) or (1.2) holds, it follows that (M, g) has constant sectional curvature if and only if b = 0.

We say (M, g, ξ) has constant vertical curvature k ∈ R, if for all p ∈ M and π ⊂ T_{p}M vertical plane section, K(π) = k. Analogously we speak of manifolds (M, g, ξ) with constant horizontal curvature.

The following proposition easily follows from (1.1):

Proposition 1.2 Suppose that (M, g, ξ) is a QC–manifold and k ∈ R. The following properties are equivalent:

i) (M, g) has constant sectional curvature k.

ii) (M, g, ξ) has constant horizontal curvature k and constant vertical curvature k.

Finally we denote by η the canonical 1–form defined by:

η(X) = g(X, ξ).

We recall that (see [5]), if (M, g, ξ) is a QC–manifold with b ≠ 0, such that dim(M) ≥ 4, then for all X, Y ∈ D we have

dη(X, Y) = 0,

where D is the distribution orthogonal to ξ, given by

D_{p} = {X ∈ T_{p}M|η(X) = 0}.

2 Three-dimensional slant submanifolds of K–contact manifolds

We start this section recalling some definitions. Let M be a manifold of dimension 2m + 1, m ≥ 1, and λ a nonzero real number. By a λ–homotetic contact metric structure on M we mean any almost contact metric structure (φ, ξ, η, g) such that

dη = λΦ

where Φ is the associated fundamental 2–form defined by

Φ(X, Y) = g(X, φY).

A λ–homotetic Sasakian structure on M is a λ–homotetic contact metric structure which is normal in the usual sense, that is:

[φ, φ] + 2dη ⊗ ξ = 0.

Obviously if λ = 1 these notions are exactly those of contact metric structure and Sasakian structure. The following proposition holds (see the theorem at page 73 of [2]): Proposition 1.1. A λ–homotetic contact metric structure on M is λ–homotetic Sasakian if and only if

(∇_{X}φ)Y = λ{g(X, Y)ξ − η(Y)X}.

Now let N be a slant submanifold of odd dimension of an almost contact metric manifold M with structure (φ, ξ, η, g). Let ϑ be the slant angle of N in M. Supposing that N is not anti–invariant, that is ϑ ≠ \frac{π}{2} , it is known (see [8]) that ξ is tangent to N. Moreover N inherits an almost contact metric structure (φ̄, ξ, η, g) with

φ̄ = \frac{1}{cos ϑ}P,

where P is the tensor field of type (1,1) on N define by the decomposition

φX = P X + F X

with P X (resp. F X) tangent (resp. normal) component of φX with respect to N, for any vector field X tangent to N. We remark that the tensor field Q = P^{2} satisfies

(2.1)  QX = cos^{2}ϑ(−X + η(X)ξ).

The following lemma is useful for the local study of 3–dimensional slant submanifolds: Lemma 2.1. Let N be a 3–dimensional slant submanifold of an almost contact metric manifold M. Suppose that N is not anti–invariant. If p ∈ N, then in a neighborhood of p there exist vector fields e_{1}, e_{2} tangent to N, such that {ξ, e_{1}, e_{2}} is a local orthonormal frame satisfying

(2.2)  P e_{1} = cos ϑe_{2}, P e_{2} = − cos ϑe_{1}.

Proof. Since N has dimension 3, ξ is tangent to N. Hence by the Gram– Schmidt process one gets a local orthonormal frame

{ξ_{|U}, ϵ_{1}, ϵ_{2}}

defined in a neighborhood U of p. Then the vector fields

e_{1} = ϵ_{1}, e_{2} = \frac{1}{cos ϑ}P e_{1}

have the required properties.

We denote by Φ̄ the fundamental 2–form of (φ̄, ξ, η, g). If the structure of the ambient manifold M is contact metric, then (cfr. [8], theorem 3.6)

dη = cos ϑΦ̄,

hence N is a cos ϑ–homotetic contact metric manifold.

In the following we shall prove that if dim(N) = 3 and the ambient manifold is K–contact, then (φ̄, ξ, η, g) is a cos ϑ–homotetic Sasakian structure. First of all we recall that supposing M is K–contact, then for any vector field X tangent to N ([8], lemma 4.1):

(2.3)  P X = −∇_{X}ξ,

where ∇ is the Levi–Civita connection on N. Moreover the following theorem holds Theorem 2.1. ([8]) Let N be an immersed submanifold of a K-contact manifold M with structure (φ, ξ, η, g), and suppose that ξ is tangent to N. Then if ϑ ∈ [0, \frac{π}{2} ], the following properties are equivalent:

a) N is slant in M with slant angle ϑ

b) (N, g, ξ) has constant vertical curvature cos^{2}ϑ. Now we prove Theorem 2.2. Let N be a 3–dimensional slant submanifold of a K-contact manifold M with structure (φ, ξ, η, g). Suppose that ϑ = sla(N) ≠ \frac{π}{2} . Then for any X, Y vvector fields tangent to N

(2.4)  (∇_{X}P)Y = cos^{2}ϑ{g(X, Y)ξ − η(Y)X}.

It follows that the cos ϑ–homotetic contact metric structure induced on N is cos ϑ–homotetic Sasakian. Proof. Let X, Y be vector fields tangent to N. Let p ∈ N and {e_{1}, e_{2}} the orthonormal frame on N defined in a neighborhood U of p, given by lemma 3.2. We shall prove that the vector fields on the two sides of formula (2.4) coincide on U. Put ξ_{|U} = e_{o}, and let w_{i}^{j} be the structural 1–forms defined by

∇_{X}e_{i} = ∑_{j=0}^{2} w_{i}^{j}(X)e_{j}.

Obviously

w_{j}^{i} = −w_{i}^{j}, w_{i}^{i} = 0.

Notice that by virtue of (2.3)

(∇_{X}P)e_{o} = ∇_{X}P e_{o} − P (∇_{X}e_{o}) = QX.

Moreover, using (2.2) we get

(∇_{X}P)e_{1} = ∇_{X}(cos ϑe_{2}) − P (w_{1}^{0}(X)e_{o} + w_{1}^{1}(X)e_{1} + w_{1}^{2}(X)e_{2})
= cos ϑ(w_{2}^{0}(X)e_{o} + w_{2}^{1}(X)e_{1}) + cos ϑw_{1}^{2}(X)e_{1} =
= cos ϑw_{2}^{0}(X)e_{0}

and analogously

(∇_{X}P)e_{2} = − cos ϑw_{1}^{0}(X)e_{0}.

Moreover, writing

Y = η(Y)e_{o} + g(Y, e_{1})e_{1} + g(Y, e_{2})e_{2}

and using the formulas above it follows

(2.5) (∇_{X}P)Y = η(Y)QX + {cos ϑw_{2}^{0}(X)g(Y, e_{1}) − cos ϑw_{1}^{0}(X)g(Y, e_{2})}e_{o}.

Finally, notice that by virtue of (2.3), we have

w_{2}^{0}(X) = g(∇_{X}e_{2}, ξ) = −g(e_{2}, ∇_{X}ξ) = g(e_{2}, P X)

and

w_{1}^{0}(X) = g(e_{1}, P X).

By substituting in (2.5), since P is skew–symmetric we get

(∇_{X}P)Y = η(Y)QX + cos ϑ{g(P X, e_{2})g(Y, e_{1}) − g(P X, e_{1})g(Y, e_{2})}ξ
= η(Y)QX + cos^{2}ϑ{g(X, e_{1})g(Y, e_{1}) + g(X, e_{2})g(Y, e_{2})}ξ
= η(Y)QX + cos^{2}ϑ{g(X, Y) − η(X)η(Y)}ξ
= cos^{2}ϑ{−η(Y)X + η(Y)η(X)ξ + g(X, Y)ξ − η(Y)η(X)ξ

and (2.4) is proved. Finally, for the induced structure (φ̄, ξ, η, g) on N, it follows that

(∇_{X} φ̄)Y = cos ϑ{g(X, Y)ξ − η(Y)X}.

Hence by virtue of proposition 1.1 this structure is cos ϑ–homotetic Sasakian. Corollary 2.1. Every K–contact structure on a three-dimensional manifold is Sasakian. Proof. Let M be a K–contact manifold of dimension 3 with structure (φ, ξ, η, g). It is obvious that the immersed submanifold (M, i), where i is the identity, is an invariant submanifold of this K–contact manifold. On the other hand, it is clear that the induced structure (φ̄, ξ, η, g) on M coincides with (φ, ξ, η, g). By the theorem above this structure must be 1–homotetic Sasakian, which means that it is Sasakian.

Theorem 2.2 will be used in order to study the intrinsic geometry of threedimensional slant submanifolds of K–contact manifolds. First we prove Proposition 2.1. Let M be a λ–homotetic Sasakian manifold with structure (φ, ξ, η, g). Then (M, g, ξ) is a QC–manifold if and only if either dim(M) = 3 or (M, g) has constant sectional curvature. Proof. Suppose that dim(M) = 3. For any point p of M, let D_{p} be the orthogonal complement of ξ_{p} with respect to g. We can define a function a : M ⟶ R as follows

∀p ∈ M a(p) = K(D_{p}).

We shall prove that for any plane π ⊂ T_{p}M the corresponding sectional curvature is given by

(2.6)  K(π) = a(p) + (λ^{2} − a(p)) cos^{2} ψ_{π},

where ψ_{π} is the angle between ξ_{p} and π. This implies that (M, g, ξ) is a QC– manifold. Put

(2.7)  η^{⋆} = λη, ξ^{⋆} = \frac{1}{λ}ξ, g^{⋆} = λ^{2}g.

Since the Levi–Civita connections of g and g^{⋆} coincide, by using proposition 1.1 it is easy to verify that (φ, ξ^{⋆}, η^{⋆}, g^{⋆}) is a Sasakian structure on M. Let π ⊂ T_{p}M be a plane; then we have

(2.8)  K(π) = λ^{2}K^{⋆}(π),

where we denote by K^{⋆}(π) the sectional curvature of π with respect to the metric g^{⋆}. Now fix a basis {X^{⋆}, Y^{⋆}} of π orthonormal with respect to g^{⋆}. Notice that if the vectors X^{⋆}, Y^{⋆}, ξ^{⋆} are linearly dependent then we have K^{⋆}(π) = 1 hence K(π) = λ^{2} and (2.6) is satisfied since obviously ψ_{π} = 0. Thus we can suppose that X^{⋆}, Y^{⋆}, ξ are linearly independent so that we can write

X^{⋆} = η^{⋆}(X^{⋆})ξ^{⋆} + αZ, Y^{⋆} = η^{⋆}(Y^{⋆})ξ^{⋆} + βW

with α, β ∈ R and W, Z unit vectors orthogonal to ξ^{⋆} with respect to g^{⋆}. Observe that being dim(M) = 3, Span{Z, W } coincides with the orthogonal complement of ξ^{⋆} in T_{p}M with respect to g^{⋆}. On the other hand (2.7) obviously implies that this orthogonal complement is D_{p}. Moreover, the following formula holds (see [2] page 96)

(2.9) K^{⋆}(π) = η^{⋆}(X^{⋆})^{2} + η^{⋆}(Y^{⋆})^{2} + {1 − η^{⋆}(X^{⋆})^{2} − η^{⋆}(Y^{⋆})^{2}}K^{⋆}(Z, W).

Finally the vectors

X = λX^{⋆}, Y = λY^{⋆}

make up an orthonormal basis of π with respect to g. Hence by using (2.7), (2.8) formula (2.9) can be rewritten as

(2.10)  \frac{1}{λ^{2}} K(π) = η(X)^{2} + η(Y)^{2} + \{1 − η(X)^{2} − η(Y)^{2}\}\frac{1}{λ^{2}} K(D_{p}).

Since

cos^{2} ψ_{π} = η(X)^{2} + η(Y)^{2}

substituting in (2.10) we get

K(π) = λ^{2} cos^{2} ψ_{π} + (1 − cos^{2} ψ_{π})a(p),

which is exactly (2.6).

To prove the converse, suppose that (M, g, ξ) is a QC–manifold and that (M, g) has not constant sectional curvature. We prove by contradiction that it must be dim(M) = 3. Assume that dim(M) ≥ 4. Then by virtue of the remarks at the end of section §1, for all X, Y vector fields orthogonal to ξ we would have

dη(X, Y) = 0

which is impossible, since λ ≠ 0. Theorem 2.3. Let N be a non anti–invariant slant submanifold of a K-contact manifold M with structure (φ, ξ, η, g). Let ϑ be the slant angle of N in M. Then (N, g, ξ) is a Riemannian manifold of quasi–constant sectional curvatures if and only if either dim(N) = 3 or (N, g) has constant sectional curvature cos^{2}ϑ. Moreover if dim(N) = 3, the following properties hold:

I. For any p ∈ N denote by ξ_{p}^{⊥} the orthogonal complement of ξ_{p} in T_{p}N and let a : N ⟶ R the function given by

(2.11)  a(p) = K(ξ_{p}^{⊥}).

Then for any plane π ⊂ T_{p}N:

(2.12)  K(π) = a(p) + (cos^{2}ϑ − a(p)) cos^{2} ψ_{π},

where ψ_{π} is the angle between ξ_{p} and π.

II. The Ricci tensor of N has the following expression

(2.13)  S(X, Y) = (a + cos^{2}ϑ)g(X, Y) + (cos^{2}ϑ − a)η(X)η(Y)

III. The scalar curvature of N is given by

(2.14)  r = 2a + 4cos^{2}ϑ.

Proof. Supposed that N has dimension 3, by virtue of theorem 2.2 the induced structure (φ̄, ξ, η, g) on N is cos ϑ–homotetic Sasakian and this implies by virtue of proposition 2.1 that (N, g, ξ) is QC. To prove the converse, suppose that (N, g, ξ) is a QC–manifold. First notice that if (N, g) has constant sectional curvature, then the constant sectional curvature must be cos^{2}ϑ by virtue of theorem 2.1. If (N, g) has not constant sectional curvature, we prove by contradiction that dim (N) = 3. Assume that dim(N) ≥ 4. As in the proof of proposition 2.1, it would follow that

dη(X, Y) = 0

for all vector fields tangent to N and orthogonal to ξ. Hence, since the induced structure (φ̄, ξ, η, g) is cos ϑ–homotetic

cos ϑΦ̄(X, Y) = 0

which is absurd because N is not anti–invariant. We conclude that dim(N) = 3.

Now assume dim(N) = 3. Formula (2.12) follows from (2.6) putting λ = cos ϑ. On the other hand since N is a QC–manifold, (2.12) can also be obtained directly by the general formula (1.1). In fact, notice that in the present case we have b = cos^{2}ϑ − a by virtue of theorem 3.3. Moreover using this expression for b, formula (2.1) gives for the curvature tensor R of N:

R(X, Y, Z, U) = a{g(X, Z)g(Y, U) − g(Y, Z)g(X, U)} +

+(cos^{2}ϑ − a){g(Y, U)η(X)η(Z) − g(X, U)η(Y)η(Z)

+g(X, Z)η(Y)η(U) − g(Y, Z)η(X)η(U)}.

In order to prove (2.13), let p ∈ N and fix an orthonormal basis {ξ_{p}, e_{1}, e_{2}} of T_{p}N. Let e_{o} = ξ_{p}. Then for all X, Z ∈ T_{p}N we get

S_{p}(X, Z) = ∑_{i=0}^{2} R(X, e_{i}, Z, e_{i})

= a ∑_{i=0}^{2}{g(X, Z)g(e_{i}, e_{i}) − g(e_{i}, Z)g(X, e_{i})} +

+(cos^{2}ϑ − a) ∑_{i=0}^{2}{g(e_{i}, e_{i})η(X)η(Z) − g(X, e_{i})η(e_{i})η(Z)

+g(X, Z)η(e_{i})^{2} − g(e_{i}, Z)η(X)η(e_{i})}

= a{3g(X, Z) − g(X, Z)} +

+(cos^{2}ϑ − a){3η(X)η(Z) − η(X)η(Z) + g(X, Z) − η(Z)η(X)}

= (a + cos^{2}ϑ)g(X, Z) + (cos^{2}ϑ − a)η(X)η(Z)

and (2.13) is proved. (2.14) soon follows by the definition of scalar curvature:

r(p) = ∑_{i=0}^{2} S_{p}(e_{i}, e_{i})

= (cos^{2}ϑ + a) ∑_{i=0}^{2} g(e_{i}, e_{i}) + (cos^{2}ϑ − a) ∑_{i=0}^{2} η(e_{i})^{2} = 2a + 4cos^{2}ϑ.

Corollary 2.2. Let (N, g) a Riemannian manifold of dimension 3 and let ϑ ∈ [0, \frac{π}{2} ]. Suppose that N admits two ϑ–slant isometric immersions into two K-contact manifolds M_{1}, M_{2}, whose characteristic vector fields ξ_{1}, ξ_{2} satisfy

g(ξ_{1}, ξ_{2}) = 0.

Then N has constant sectional curvature cos^{2}ϑ.

Proof. By the theorem just proved, (N, g, ξ_{1}) and (N, g, ξ_{2}) are both QC– manifolds. In particular theorem 2.1 implies that they have both constant vertical curvature cos^{2}ϑ. Now, if p ∈ N, by the hypothesis, the plane orthogonal to ξ_{1}_{p} in T_{p}N is a vertical plane of (N, g, ξ_{2}), hence its sectional curvature is cos^{2}ϑ. This shows that (N, g, ξ_{1}) has also constant horizontal curvature cos^{2}ϑ and the assertion follows from proposition 1.2.

Remark. In all that follows if N is a slant submanifold of dimension 3 of a K–contact manifold M, we denote by ξ_{p}^{⊥} the orthogonal complement of ξ_{p} in T_{p}N, while a denotes the function a : N ⟶ R defined by (2.11). Moreover the expressions ‘slant submanifold’ and ‘slant immersion’ will always mean ‘non anti–invariant slant submanifold ’and ‘ slant immersion with slant angle different from \frac{π}{2} ’ respectively. Lemma 2.2. Let N be a slant submanifold of dimension 3 of a Sasakian manifold M with constant φ–sectional curvature k. For any p ∈ N, the sectional curvature in M of the plane ξ_{p}^{⊥} equals

\frac{1}{4}\{(k + 3) + 3cos^{2}ϑ(k − 1)\}.

Proof. It is known that in a Sasakian manifold the following formula holds (cfr [2] page 95):

K(X, Y) = \frac{1}{8}\{3(1 + cos ζ)^{2}H(X + φY) + 3(1 − cos ζ)^{2}H(X − φY)
−H(X + Y) − H(X − Y) − H(X) − H(Y) + 6 sin^{2} ζ}

where {X, Y } are orthonormal tangent vectors in a point p ∈ M, which are both orthogonal to ξ_{p}, ζ ∈ [0, π] is the angle between X e φY, and H(X) = K(X, φX). Now let p a point of the slant submanifold N and let X be a unit vector in ξ_{p}^{⊥}. Since

g(P X, P X) = −g(X, QX) = cos^{2}ϑ

it follows that, putting Y = \frac{1}{cos ϑ}P X, we get an orthonormal basis \{X, Y \} of

ξ_{p}^{⊥}. Moreover notice that

g(X, φY) = g(X, P Y) = g(X, \frac{1}{cos ϑ}QX) = − cos ϑ.

By the above remark it follows that the sectional curvature of the plane ξ_{p}^{⊥} in the ambient manifold M equals

\frac{1}{8}\{3(1 − cos ϑ)^{2}k + 3(1 + cos ϑ)^{2}k − 4k + 6(1 − cos^{2}ϑ)\}

that is

\frac{1}{4}\{3k(1 + cos^{2}ϑ) − 2k + 3 − 3cos^{2}ϑ)\}.

Theorem 2.4. Let M be a Sasakian manifold with constant φ–sectional curvature k. A necessary condition for a slant 3–dimensional submanifold N of M to be minimal is

a ≤ \frac{1}{4}\{(k + 3) + 3cos^{2}ϑ(k − 1)\}.

Proof. Suppose N is minimal and let p ∈ N. Fix an orthonormal basis {ξ_{p}, e_{1}, e_{2}} of T_{p}N. By the above lemma and Gauss’ equation we have

(2.15) a(p) = \frac{1}{4}\{(k+3)+3cos^{2}ϑ(k−1)\}+g(α(e_{1}, e_{1}), α(e_{2}, e_{2}))−∥α(e_{1}, e_{2})∥^{2}.

On the other hand the minimality of N implies

α(ξ_{p}, ξ_{p}) + α(e_{1}, e_{1}) + α(e_{2}, e_{2}) = 0.

But since ξ is Killing, α(ξ_{p}, ξ_{p}) = 0, hence we get

α(e_{1}, e_{1}) = −α(e_{2}, e_{2}).

Now formula (2.15) can be rewritten as

a(p) = \frac{1}{4}\{(k + 3) + 3cos^{2}ϑ(k − 1)\} − ∥α(e_{1}, e_{1})∥^{2} − ∥α(e_{1}, e_{2})∥^{2}

and the assertion follows.

Remark. If M has constant φ–sectional curvature k = −3, the necessary condition for the minimality of a 3–dimensional slant submanifold becomes:

a ≤ −3cos^{2}ϑ.

There exist examples of three-dimensional minimal slant submanifold of the Sasakian space form R^{2n+1}, for which a = −3cos^{2}ϑ identically (see [10]). On the other hand, when k = 1 the above condition becomes

a ≤ 1.

The standard 3–sphere S^{3} is an example of totally geodesic invariant submanifold of the Sasakian space form S^{2n+1}, satisfying a = 1.

3 Local homogeneity of slant submanifolds with constant horizontal curvature

Let N be a non anti–invariant, 3–dimensional slant submanifold of a K–contact manifold M with structure (φ, ξ, η, g). According to the definition given in section 2, to say that (N, g, ξ) has constant horizontal curvature c ∈ R is equivalent to saying that the function a : N ⟶ R in theorem 2.3 is constant and equals c. We shall prove that in this case N is a locally homogeneous Riemannian manifold with respect to the induced metric. We first prove some results about the Ricci tensor of N.

Proposition 3.1 Let N be a 3–dimensional slant submanifold of a K–contact manifold M with structure (φ, ξ, η, g). The covariant derivative of the Ricci tensor S of N is given by:

(∇_{X}S)(Y, Z) = X(a)(g(Y, Z) − η(Y)η(Z)) +
+ cos ϑ(cos^{2}ϑ − a){Φ̄(X, Y)η(Z) + Φ̄(X, Z)η(Y)},

where Φ̄ denotes the fundamental 2–form of the induced structure (φ̄, ξ, η, g) on N. In particular, if N has constant horizontal curvature c, we have

(∇_{X}S)(Y, Z) = cos ϑ(cos^{2}ϑ − c){Φ̄(X, Y)η(Z) + Φ̄(X, Z)η(Y)}.

Proof. By using (2.13) we get

(∇_{X}S)(Y, Z) = ∇_{X}S(Y, Z) − S(∇_{X}Y, Z) − S(Y, ∇_{X}Z)
= ∇_{X}((a + cos^{2}ϑ)g(Y, Z)) + ∇_{X}((cos^{2}ϑ − a)η(Y)η(Z))
− (a + cos^{2}ϑ)g(∇_{X}Y, Z) − (cos^{2}ϑ − a)η(∇_{X}Y)η(Z)
− (a + cos^{2}ϑ)g(Y, ∇_{X}Z) − (cos^{2}ϑ − a)η(∇_{X}Z)η(Y)
= X(a)(g(Y, Z) − η(Y)η(Z)) + (cos^{2}ϑ − a){∇_{X}(η(Y)η(Z))
− η(∇_{X}Y)η(Z) − η(∇_{X}Z)η(Y)

On the other hand

∇_{X}(η(Y)η(Z)) = (g(∇_{X}Y, ξ)+g(Y, ∇_{X}ξ))η(Z)+(g(∇_{X}Z, ξ)+g(Z, ∇_{X}ξ))η(Y)

hence

(∗) \begin{aligned}(∇_{X}S)(Y, Z) & = & X(a)(g(Y, Z) − η(Y)η(Z))+ \\ & & +(cos^{2}ϑ − a)\{g(Y, ∇_{X}ξ)η(Z) + g(Z, ∇_{X}ξ)η(Y)\}. \end{aligned}

Now recall that ∇_{X}ξ = ˜∇_{X}ξ − α(X, ξ) and φX = − ˜∇_{X}ξ, so we write

g(Y, ∇_{X}ξ) = g(X, φY) = g(X, P Y) = cos ϑg(X, φ̄Y) = cos ϑΦ̄(X, Y).

Substituting this formula in (*) we finally obtain

(∇_{X}S)(Y, Z) = X(a)(g(Y, Z) − η(Y)η(Z)) +
+ cos ϑ(cos^{2}ϑ − a){Φ̄(X, Y)η(Z) + Φ̄(X, Z)η(Y)}

and this proves the assertion.

The following theorem characterizes slant submanifolds with constant horizontal curvature by means of a remarkable property of S:

Theorem 3.1. Let N be a 3–dimensional slant submanifold of a K–contact manifold M with structure (φ, ξ, η, g). The following properties are equivalent:

a) (N, g, ξ) has constant horizontal curvature

b) For all vector fields X tangent to N (∇_{X}S)(X, X) = 0, where S is the Ricci tensor of N.

Proof. If N has constant horizontal curvature then b) follows immediately by proposition 3.1. Vice-versa, suppose b) holds. A general result (see [1],page 432) insures that any Riemannian manifold whose Ricci tensor satisfies b) must have constant scalar curvature. In the present case, by virtue of (2.14) it follows that a is constant.

Theorem 3.2. Any connected, 3–dimensional, non anti–invariant slant submanifold N of a K–contact manifold M, with constant horizontal curvature, is a locally homogeneous Riemannian manifold with respect to the induced metric.

Proof. The proof is based on the theory of homogeneous structures on Riemannian manifold developed by Tricerri and Vanhecke in [11]. In particular we shall prove that N admits a homogeneous structure of type T_{3}. This suffices to prove our assertion by virtue of theorem 1.10 in [11]. On the other hand, since (N, g) is 3–dimensional, connected and orientable, the problem of finding such a structure is equivalent to finding a tensor field T in N of type (1,2) satisfying (see theorem 6.3 and the proof of theorem 6.4 in [11]):

(3.2)  (∇_{X}S)(Y, Z) = −S(T_{X}Y, Z) − S(Y, T_{X}Z)

(3.3)  T = λdV, λ ∈ R.

In formula (3.2) T is thought as a tensor of type (0,3) in the usual way:

T (X, Y, Z) = g(T_{X}Y, Z)

while dV is the volume form with respect to a suitable orientation of N. Now one easily sees that the orientation of N can be chosen in such a way that

(3.1)  dV (X, Y, ξ) = −Φ (X, Y).

Hence we define a tensor T putting

T = cos ϑdV.

By the above remarks, to prove the theorem we just need to show that this tensor satisfies (3.2). In fact, by virtue of (2.13) we have

S(T_{X}Y, Z) = (c + cos^{2}ϑ)g(T_{X}Y, Z) + (cos^{2}ϑ − c)η(T_{X}Y)η(Z) =
= cos ϑ{(c + cos^{2}ϑ)dV (X, Y, Z) + (cos^{2}ϑ − c)dV (X, Y, ξ)η(Z)}

S(Y, T_{X}Z) = cos ϑ{(c + cos^{2}ϑ)dV (X, Z, Y) + (cos^{2}ϑ − c)dV (X, Z, ξ)η(Y)}

hence

S(T_{X}Y, Z)+S(Y, T_{X}Z) = cos ϑ(cos^{2}ϑ−c){dV (X, Y, ξ)η(Z)+dV (X, Z, ξ)η(Y)}

that is

S(T_{X}Y, Z) + S(Y, T_{X}Z) = − cos ϑ(cos^{2}ϑ − c){Φ̄(X, Y)η(Z) + Φ̄(X, Z)η(Y)}

and the assertion follows by virtue of proposition 3.1.

References

[1] A.Besse, Einstein manifolds, Springer Verlag.

[2] D.E.Blair, Contact manifolds in Riemannian Geometry. Lect. Notes In Mat. 509, Springer Verlag 1976.

[3] B.Y.Chen, Slant immersions. Bull.Austral.Math.Soc. vol. 41 (1990) [135– 147].

[4] B.Y.Chen, Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, 1990.

[5] G. Ganchev–V.Mihova, Riemannian manifolds of quasi–constant sectional curvatures Preprint.

[6] S.Kobayashi-N.Nomizu, Foundations of differential geometry vol.II. Interscience Publishers, 1969.

[7] M.Kon, Invariant submanifolds in Sasakian manifolds. Math. Ann. 219 (1976) [277–290].

[8] A.Lotta, Slant submanifolds in contact geometry (to appear in Bull. Math. Soc. Sc. Math. Roumanie)

[9] A.Lotta, Immersioni inclinate in geometria di contatto. Rapporti interni dell’Università di Bari–Dipartimento di Matematica 2/96 (1996).

[10] A.Lotta, Foliations of the Sasakian space R^{2n+1} by minimal slant submanifolds, Rapporti interni dell’Università di Bari–Dipartimento di Matematica 49/96 (1996).

[11] F.Tricerri–L.Vanhecke, Homogeneous Structures on Riemannian Manifolds, London Mathematical Society Lect. Notes Series 83, Cambridge University Press, 1983.

[12] K.Yano–M.Kon, Anti-invariant submanifolds. Lecture Notes In Pure And Applied Mathematics, vol 21, Marcel Dekker, 1976.

[13] C.Udrişte, Almost coquaternion metric strcutures on 3–dimensional manifolds, Kodai Math. Sem. Rep. 26 (1995), 318-326.

[14] Gh.Mocică and C.Udrişte, Asupra 3–structurilor Sasaki pe varietăţi cu trei dimensiuni, Bul. IPB, 39 (1977), 3–8.

Author address: ANTONIO LOTTA Dipartimento di Matematica Via Orabona, 4 70125 BARI, ITALY E-mail: lotta@pascal.dm.uniba.it

Ordinary address: ANTONIO LOTTA Dipartimento di Matematica Dottorato XII Ciclo Via Buonarroti 2 56100 PISA, ITALY