QR-Hypersurfaces of Quaternionic Kähler Manifolds

Vittorio Mangione

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

We prove that the basic manifold of a submersion from a QR-hypersurface of a quaternionic Kähler manifold to an almost quaternionic Hermitian manifold is quaternionic Kähler. Then we prove some results involving the sectional curvatures.

Mathematics Subject Classification: 53B35, 53C25, 53C26. Key words: Kähler manifolds, Hermitian manifolds, quaternionic manifolds, CRmanifolds.

Introduction

Real hypersurfaces of quaternionic space forms have been studied by many authors ([1], [2], [3], [4], [5], [11], [12]) under conditions concerning their shape operator. It is known that real hypersurface of quaternionic Kähler manifolds are not CRhypersurface in general ([2]). The study of CR-submanifolds of a quaternionic Kähler manifolds has been carried out in the paper [1]. S. Kobayashi considered the similarity between the total space of a Riemannian submersion and a CR-submanifold of a Kähler manifold in terms of distributions ([9]). In this paper we study Riemannian submersions from QR-hypersurface of a quaternionic Kähler manifold over an almost quaternionic Hermitian manifold (second section). In the last section we study some curvature properties induced on the basic manifold by the submersion.

1 Hypersurfaces of quaternionic Kähler manifolds

We say that a 4(m + 1) − dimensional manifold M̃ with a metric g̃ is a quaternionic Kähler manifold (m ≥ 1) if there exists a 3−dimensional vector bundle V of tensors of type (1, 1) on M̃ satisfying the following conditions:

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 63-70.

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(a) In any coordinate neighborhood Ũ on M̃ there is a local basis of almost Hermitian structures {J_{a}, g̃}, such that J_{a}^{2} = −Id, a ∈ {1, 2, 3} and J_{a} ∘ J_{b} = −J_{b} ∘ J_{a} = J_{c} for any cyclic permutation (a, b, c) of (1, 2, 3).

(b) For any local section φ of V and any tangent vector X to M̃, ˜∇_{X}φ is also a local section in V, where ˜∇ denotes the Levi-Civita connection of g̃.

Condition (b) is equivalent to the following:

(b′) There exist local 1 − forms ω_{ab}, a, b ∈ {1, 2, 3} on Ũ such that ω_{ab} + ω_{ba} = 0, and

˜∇_{x}J_{a} = ω_{ab}(x)J_{b} + ω_{ac}(x)J_{c} (1)

for any cyclic permutation (a, b, c) of (1, 2, 3).

Given two local bases {J_{a}} and {J′_{a}} of V defined on coordinate neighborhoods Ũ and Ũ′ such that Ũ ∩ Ũ′≠∅, we have on Ũ ∩ Ũ′:

J′_{a} = ∑_{b=1}^{3} C_{ab}J_{b} (2)

where [C_{ab}] is an element of the special orthogonal group SO(3) (see [8]). Let M be an orientable hypersurface of M̃ and ξ a unit normal field defined on M. On Ũ ξ_{a}= −J_{a}(ξ), a ∈ {1, 2, 3} defines a tangent vector field to M. Similarly, we define ξ′_{a} on Ũ′ and on Ũ ∩ Ũ′≠∅ we have:

ξ′_{a} = ∑_{b=1}^{3} C_{ab}ξ_{b}, b ∈ {1, 2, 3} (3)

so that one obtains a distribution V on M which is locally represented by {ξ_{a}}, 1 ≤ a ≤ 3, on Ũ. Let H be the orthogonal complementary distribution to V with respect to the Riemannian metric g induced by g̃ on M. We see that for each x ∈ M, H_{x} is J_{a} − invariant, but V_{x} is not an anti-invariant subspace of T_{x}M̃ with respect J_{a}, a = {1, 2, 3}. It is easy see that J_{a}(V_{x}) = T_{x}M^{⊥}, x ∈ M, where T_{x}M^{⊥} is the normal space at x to the hypersurface M in M̃. In general, when the previous conditions are satisfied, we say that M is a QR − hypersurface of M̃(see [3]). Now, let B be the second fundamental form of M in M̃. Then, for any E, F ∈ Γ(T M) we have the Gauss formula

˜∇_{E}F = ∇_{E}F + B(E, F), (4)

where ˜∇ and ∇ are the Levi-Civita connections on M̃ and M, respectively. If L denotes the fundamental tensor of Weingarten with respect to ξ, we have the Weingarten formula ˜∇_{E}ξ = −L(E), (5)

and for any E, F ∈ Γ(T M) the following formula

g(L(E), F) = g(B(E, F), ξ) (6) holds.

The integrability of the distributions V and H on M has been studied by A. Bejancu ([2]). We recall that the vertical distribution V is integrable if and only if

B(U, X) = 0 (7)

for any U ∈ Γ(V) and X ∈ Γ(H). If (7) is satisfied, we say that M is a mixed geodesic QR − hypersurface of M̃.

2 Riemannian submersions of QR-hypersurfaces

Let M be a mixed geodesic QR-hypersurface of a quaternionic Kähler manifold M̃. We denote by (M′, g′, J′_{a}), a ∈ {1, 2, 3}, an almost quaternionic Hermitian manifold ( i.e. satisfying the condition (a)). We say that a Riemannian submersion π: M → M′ is a QR-submersion if the following conditions are satisfied:

i) V is the kernel of π_{∗};

ii) for each x ∈ M, π_{∗} : H_{x} → T_{π(x)}M′ is an isometry with respect to each complex structure of H_{x} and T_{π(x)}M′, where T_{π(x)}M′ denotes the tangent space to M′ at π(x).

As in the paper [10], the letters U, V, W, W′ will always denote vertical vector fields and X, Y, Z, Z′ horizontal vector fields. A horizontal vector field X on M is said to be basic if it is π − related to a vector field X′ on M′. We denote by T and A O’Neill’s fundamental tensors (see [13], [11]).

Lemma 2.1 Let X and Y be basic vector fields on M. Then the following conditions hold:

a) The horizontal component h[X, Y] of [X, Y] is a basic vector field and π_{∗}h[X, Y] = [X′, Y′] ∘ π;

b) h(∇_{X}Y ) is basic vector field corresponding to ∇′_{X′}Y′ where ∇ and ∇′ are the the Levi-Civita connections on M and M′, respectively;

c) [X, U] ∈ Γ(V), for any vertical field U ∈ Γ(V);

where h denotes the horizontal component of a vector E on M.

We define a skew-symmetric tensor field C by

˜∇_{x}Y = h ˜∇_{x}Y + C(X, Y) (1)

for all X, Y ∈ Γ(H). The second fundamental form B of M in M̃ is:

B(E, F) = ˜∇_{E}F − ∇_{E}F (2)

for all E, F ∈ Γ(T M).

Theorem 2.2 Let M be a mixed geodesic QR-hypersuface of a quaternionic Kähler manifold M̃. If π : M → M′ is a QR-submersion of M on an almost quaternionic Hermitian manifold, then M′ is a quaternionic Kähler manifold.

Proof. By using Gauss formula and (1) we obtain

h∇_{X}J_{a}Y − J_{a}(h∇_{X}Y) = ω_{ab}(X)J_{b}Y + ω_{ac}(X)J_{c}Y, (3)

for any local basic vector fields X, Y on M and for any cyclic permutation (a, b, c) of (1, 2, 3). Then we can define 1 − forms ω′_{ab} on M′ by

ω′_{ab}(X′) ∘ π = ω_{ab}(X), a, b, c ∈ {1, 2, 3}, (4)

for any local vector field X′ on M′ and X a real basic vector field on M such that π_{∗}X = X′. On the other hand, by the definition of a QR-submersion we have

π_{∗} ∘ J_{a} = J′_{a} ∘ π_{∗}. (5)

Using Lemma 2.1, from (3)-(5) we obtain

h(∇′_{X}J′_{a})Y′ = ω′_{ab}(X′)J′_{b}Y′ + ω′_{ac}(X′)J′_{c}Y′,

where ∇′ is the Levi-Civita connection on M′ and X′, Y′ any local vector fields on M′. We conclude that (M′, J′_{a}, g′) is a quaternionic Kähler manifold. □

3 Totally umbilical QR-hypersurfaces

In the sequel we shall denote by ⟨ · , · ⟩ the scalar product induced on the tangent spaces of M and M̃ by the Riemannian metric g. We recall that a hypersurface M of M̃ is totally umbilical if the first and the second fundamental forms are proportional, that is B(E, F) = ⟨E, F ⟩H (1)

for any E, F ∈ Γ(T M), where H is the mean curvature vector of M, defined by the formula,

H = \frac{1}{4m + 3}T raceB. (2)

We have the Gauss equation:

R̃(E, E′, F, F′) = R(E, E′F, F′) − ⟨B(E, F), B(E′, F′)⟩ + (3)

+⟨B(E, F′), B(F, E′)⟩.

Taking account of the formula (1), the Gauss equation for a totally umbilical hypersurface M in M̃ becomes:

R̃(E, E′, F, F′) = R(E, E′, F, F′) − (⟨E, F ⟩⟨E′, F′⟩ + (4)

−⟨E, F′⟩⟨F, F′⟩)∥H∥^{2},

where ∥H∥^{2} = ⟨H, H⟩. We see that, if M is a totally umbilical QR-hypersurface of M̃, then it is a mixed geodesic QR-hypersurface, i.e. B(V, X) = 0 for any V ∈ Γ(V) and X ∈ Γ(H). Consequently, the vertical distribution V is integrable.

Moreover, it is easy to check that each leaf of V is totally geodesic in M (see, for example [3], p. 121). Then we conclude that the first fundamental tensor T of the Riemannian submersion π : M → M′ vanishes, because T_{U}V is the second fundamental form of each fibre for any U, V ∈ Γ(V) (see [7], [13]). Let us now recall the following two Gray-O’Neill curvature equations for a Riemannian submersion:

R(U, V, U′, V′) = R̂(U, V, U′, V′) + ⟨T_{U}V′, T_{V} U′⟩ + (5)

−⟨T_{V} V′, T_{U}U′⟩,

R(X, Y, X′, Y′) = R^{∗}(X, Y, X′, Y′) + 2⟨C(X, Y), C(X′, Y′)⟩+ (6)

+⟨C(Y, X′), C(X, Y′)⟩ − ⟨C(X, X′), C(Y, Y′)⟩,

for all U, V, U′, V′ ∈ Γ(V) and X, Y, X′, Y′ ∈ Γ(H), where, for any quadruplet of horizontal vector fields (X, Y, X′, Y′), R^{∗}(X, Y, X′, Y′) = R′(π_{∗}X, π_{∗}Y, π_{∗}X′, π_{∗}Y′) ∘ π, with R^{∗} Riemannian curvature on the fibres of H. Here R′ is the Riemannian curvature of the metric g′ on M′.

Lemma 3.1 Let M be a totally umbilical, not totally geodesic, QR-hypersurface of a quaternionic Kähler manifold. Then the tensor field C wich measures the integrability of the horizontal distribution H, is given by the formula

C(X, Y) = ∥H∥ ∑_{a=1}^{3}⟨X, J_{a}Y ⟩ξ_{a}. (7)

Proof. Using (1), (4), (5) and (7), we obtain

J_{a}(LX) − ∇_{X}ξ_{a} = ω_{ac}(X)ξ_{b} − ω_{ab}(X)ξ_{c}, (8)

for any X ∈ Γ(H). Now, by (6) in (8), we have

⟨∇_{X}Y, ξ_{a}⟩ = ⟨B(X, J_{a}Y), ξ⟩, (9)

for any X, Y ∈ Γ(H), and a ∈ {1, 2, 3}. Taking into account that the mean curvature vector H of M is a global vector field and it is non vanishing on M (see [3]), we take

ξ = \frac{H}{∥H∥}. Then we have

C(X, Y) = V ∇_{X}Y = ∥H∥ ∑_{a=1}^{3}⟨X, J_{a}Y ⟩ξ_{a}, (10)

from which formula (7) follows. □ Theorem 3.1 Let M be a totally umbilical, not totally geodesic, QR-hypersurface of a quaternionic Kähler manifold. Then,

a) K̃(U, V) = K(U, V)−∥H∥^{2}, where {U,V} is an orthonormal basis of the vertical 2 − plane α, α ⊂ V_{x}, x ∈ M, and K̃, K denote the sectional curvatures of α on M̃, M, respectively.

b) K(X, Y) = K′(X′, Y′) − 3∥H∥^{2} ∑_{a=1}^{3} ⟨X, J_{a}Y ⟩^{2}, where X,Y is an orthonormal

basis of a horizontal 2−plane α ⊂ H_{x}, K(X, Y) denoting the sectional curvature of α, and K′(X′, Y′) denotes the sectional curvature in M′ of the 2 − plane spanned by X′ = π_{∗}X and Y′ = π_{∗}Y.

Proof. Property a) is easily obtained from (4) and (5). From (6), as an immediate consequence of the skew-symmetry of C, we have

R(X, Y, X, Y) = R′(X′, Y′.X′, Y′) − 3∥C(X, Y)∥^{2}. (11)

Lemma 3.1 and (11) directly give b). □

We recall that a totally umbilical, not totally geodesic, hypersurface M of a Riemannian manifold M̃ is an extrinsic hypersphere if the mean curvature vector field H is parallel with respect to the linear normal connection ∇^{⊥} or, equivalently, ∥H∥ = c is a constant c ≠ 0 on M. Then we have the following

Theorem 3.2 Let M be an extrinsic hypersurface of a flat quaternionic Kähler manifold M̃ and π : M → M′ a QR-submersion of M on a quaternionic Kähler manifold M′. Then M′ is a quaternionic Kähler manifold with constant quaternionic sectional curvature c > 0.

Proof. By (4), (6) and Lemma 3.1 we have

R′(X′, Y′)Z′ = ∥H∥^{2}{g′(Y′, Z′)X′ − g′(X′, Z′)Y′ +

+ ∑_{a=1}^{3}(g′(J′_{a}Y′, Z′)J′_{a}X′ − g(J′_{a}X′, Z′)J′_{a}Y′) +

+2g′(X′, J′_{a}Y′)J′_{a}Z′).

where ∥H∥ is a constant on M′ and X′, Y′, Z′ ∈ Γ(T M′). □

Remark There exist no proper totally umbilical QR-submanifolds in positively or negatively curved quaternionic Kähler manifolds (see [3]).

Acknowledgment: The author thanks to Prof. V. Balan for his suggestions on improving the exposition and for pointing out some errors.

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Dipartimento di Matematica, Università degli Studi di Parma Via M. D’Azeglio, 85A, 43100 Parma, Italia. email: vittorio.mangione@unipr.it