PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 91–102 DOI: https://doi.org/10.2298/PIM1817091I

MAGNETIC VECTOR FIELDS: NEW EXAMPLES

Jun-ichi Inoguchi and Marian Ioan Munteanu

Abstract. In a previous paper, we introduced the notion of magnetic vector fields. More precisely, we consider a vector field ξ as a map from a Riemannian manifold into its tangent bundle endowed with the usual almost Kählerian structure and we find necessary and sufficient conditions for ξ to be a magnetic map with respect to ξ itself and the Kähler 2-form. In this paper we give new examples of magnetic vector fields.

1. Preliminaries

In [13] the authors define the notion of magnetic maps with the aim of generalizing the notion of magnetic trajectory on a Riemannian manifold. In fact, both magnetic curves and harmonic maps can be obtained as particular situations of magnetic maps.

Let f : N → M be a smooth map between two Riemannian manifolds (N, h) of dimension n and (M, g) of dimension m. Suppose that N is compact and let ξ be a global vector field on N having null divergence. Let ω be a 1-form on M. The energy of f is known as E(f) = \frac{1}{2} ∫_{N} |df|^{2}dv_{h}, where dv_{h} is the volume element on N and |df| is the Hilbert-Schmidt norm of the differential df given (in a point p ∈ N) by

|df_{p}|^{2} = ∑_{i=1}^{n} g_{f(p)} (f_{∗,p}e_{i}, f_{∗,p}e_{i}) .

Here {e_{i}; i = 1, …, n} is an arbitrary orthonormal basis for T_{p}N and f_{∗,p} : T_{p}N → T_{f(p)} M is the tangent map of f at p.

A smooth map f : (N, h) → (M, g) which is a critical point of E(f) is called a harmonic map (see e.g., [11, 21]).

Let us now define the following functional for f associated to ξ and ω:

P(f) = ∫_{N} ω(df(ξ))dv_{h}.

The Landau-Hall functional associated to ξ and ω is defined by

LH(f) = E(f) + P(f).

2010 Mathematics Subject Classification: Primary 53C43; Secondary 53C07, 53C15, 58E20.

Key words and phrases: magnetic curves; harmonic maps; magnetic maps.

The first author is supported by Kakenhi 15K04834.

Let I be an open interval containing 0. A smooth variation of f is a smooth map F : N × I → M, such that F(p, 0) = f(p). For the sake of simplicity we use the notation f_{ϵ}(p) = F(p, ϵ). The variation vector field along f is a section in the induced bundle f^{−1} T (M) defined by V (x) = \frac{∂f_{ϵ}}{∂ϵ} |_{ϵ=0} (x).

Definition 1.1. [13] The map f is called magnetic with respect to ξ and ω if it is a critical point of the Landau Hall integral LH(f).

In what follows we compute the first variation \frac{d}{dϵ}LH(f_{ϵ})|_{ϵ=0} . It is known from the theory of harmonic maps that

\frac{d}{dϵ}E(f_{ϵ})|_{ϵ=0} = − ∫_{N} g(τ(f), V) ∘ f dv_{h},

where τ(f) := trace_{h} ∇df is the tension field of f.

Let us focus on the integral P and compute \frac{d}{dϵ}P(f_{ϵ})|_{ϵ=0} . Consider local coordinates x^{1}, …, x^{n} on N and y^{1}, …, y^{m} local coordinates on M. With respect to this setting, the map f_{ϵ} may be expressed as y^{α} = f_{ϵ}^{α} (x), where f_{ϵ}^{α} are smooth functions on the domain of coordinates x taking values in ℝ. From now on the indices i, j, k range from 1 to n, while the indices α, β, γ range from 1 to m.

We have

P(f_{ϵ}) = ∫_{N} ω_{α} (f_{ϵ}(x))\frac{∂f_{ϵ}^{α}}{∂x^{i}} (x)ξ^{i}(x) dv_{h}.

Compute

(1.1) \frac{d}{dϵ}P(f_{ϵ})|_{ϵ=0}

= ∫_{N} [ \frac{∂ω_{α}}{∂y^{β}} (f(x))\frac{∂f_{ϵ}^{β}}{∂ϵ} (x)|_{ϵ=0} \frac{∂f^{α}}{∂x^{i}} (x) + ω_{α} (f(x))\frac{∂}{∂ϵ}|_{ϵ=0} \frac{∂f_{ϵ}^{α}}{∂x^{i}} (x)] ξ^{i}(x)dv_{h}.

Let us define a vector field X on N by X(x) = ξ^{i}(x)ω_{α} (f(x))V^{α} (x)\frac{∂}{∂x^{i}} and compute its divergence. We obtain

div(X) = (∇^{h} _{i} ξ^{i})ω_{α} (f(x))V^{α} (x) + ξ^{α} (x)′∇_{i}ω_{α} (f(x))V^{α} (x)
+ ξ^{i}(x)ω_{α} (f(x))′∇_{i}V^{α} (x),

where ∇^{h} is the Levi-Civita connection on N and ′∇ is the induced connection. We successively have

∇^{h} _{i} ξ^{i} = div(ξ),
′∇_{i}ω_{α} (f(x)) = (′∇_{\frac{∂}{∂x^{i}}} ω(f(x))) (\frac{∂}{∂y^{α}} ∘ f) = \frac{∂}{∂x^{i}} ω_{α} (f(x)) − ω ( ′∇_{\frac{∂}{∂x^{i}}} \frac{∂}{∂y^{α}} ∘ f)

= \frac{∂ω_{α}}{∂y^{β}} (f(x))\frac{∂f^{β}}{∂x^{i}} (x) − ω_{β}(f(x))\frac{∂f^{γ}}{∂x^{i}} (x) Γ^{g} _{γα}^{β} (f(x))

= \frac{∂f^{γ}}{∂x^{i}} (x) ∇^{g} _{γ} ω_{α} (f(x));

′∇_{i}V^{α} = \frac{∂V^{α}}{∂x^{i}} (x) + \frac{∂f^{β}}{∂x^{i}} (x) Γ^{g} _{βγ}^{α} (f(x))V^{γ}(x).

As ξ is divergence free, we get

div(X) = ξ^{i}(x) [ω_{α} (f(x))\frac{∂V^{α}}{∂x^{i}} (x) + \frac{∂f^{β}}{∂x^{i}} (x)\frac{∂ω_{α}}{∂y^{β}} (f(x))V^{α} (x)] .

Since ∫_{N} div(X)dv_{h} = 0, we obtain

(1.2) ∫_{N} ξ^{i}(x)ω_{α} (f(x))\frac{∂V^{α}}{∂x^{i}} (x)dv_{h} = − ∫_{N} ξ^{i}(x)\frac{∂ω_{α}}{∂y^{β}} (f(x))\frac{∂f^{β}}{∂x^{i}} (x)V^{α} (x)dv_{h}.

Combining (1.1) and (1.2) we find

\frac{d}{dϵ}|_{ϵ=0} P(f_{ϵ}) = ∫_{N} ξ^{i}(x)\frac{∂ω_{α}}{∂y^{β}} (f(x)) ( \frac{∂f^{α}}{∂x^{i}} (x)V^{β}(x) − \frac{∂f^{β}}{∂x^{i}} (x)V^{α} (x)) dv_{h}

= ∫_{N} ξ^{i}(x)\frac{∂f^{α}}{∂x^{i}} (x) ( \frac{∂ω_{α}}{∂y^{β}} (f(x)) − \frac{∂ω_{β}}{∂y^{α}} (f(x))) V^{β}(x)dv_{h}

= ∫_{N} ξ^{i}(x)\frac{∂f^{α}}{∂x^{i}} (x)(dω)_{αβ} V^{β}(x)dv_{h}

= ∫_{N} dω(f_{∗}ξ, V) ∘ f dv_{h}.

Define the endomorphism ϕ, called the Lorentz force associated to the potential 1-form ω, by g(ϕ(X), Y) = dω(X, Y), for all X, Y tangent to M. It follows that

\frac{d}{dϵ}|_{ϵ=0} P(f_{ϵ}) = ∫_{N} g(ϕf_{∗}ξ, V) ∘ f dv_{h}.

We finally obtain

\frac{d}{dϵ}|_{ϵ=0} LH (f_{ϵ}) = − ∫_{N} g(τ(f) − ϕf_{∗}ξ, V) ∘ f dv_{h}.

We state the following.

Theorem 1.1. [13] Let f : (N, h) → (M, g) be a smooth map. Then f is a magnetic map with respect to ξ and ω if and only if it satisfies the Lorentz equation, that is

(1.3)  τ(f) = ϕ(f_{∗}ξ).

Sometimes, equation (1.3) will be called the magnetic equation. Recall that on a Riemannian manifold (M, g) a magnetic field is defined by a closed 2-form F and the Lorentz force associated to F is a (1, 1) tensor field ϕ on M given by g(ϕX, Y) = F (X, Y). The magnetic trajectories of F are curves γ satisfying the Lorentz equation ∇_{γ′}γ′ = ϕγ′. This equation is a particular case of equation (1.3) when N is an interval of ℝ and ξ = \frac{d}{dt}, where t is the global coordinate on ℝ. Magnetic curves were intensively studied in the last years by several geometers (including the authors of this article) in different ambient spaces. See for example [8, 9, 10, 14, 15, 18].

Remark 1.1. The Lorentz equation (1.3) was obtained from a variational principle assuming that the domain is compact and the 2-form F is exact. Since it has a tensorial character, one can define a magnetic map f : (N, h) → (M, g) without the assumptions N compact and F exact (but only closed). Moreover, we will remove also the assumption for ξ to be divergence free.

Let ξ be a global vector field on N and F be a magnetic field on M with the associated Lorentz force ϕ. Similarly to magnetic curves, we may also introduce a strength (i.e., a real number) in the equation. Hence, we give the following.

Definition 1.2. We say that f is a magnetic map with strength q ∈ ℝ associated to ξ and F if the Lorentz equation

τ(f) = q ϕ(f_{∗}ξ)

is satisfied.

2. Vector fields as magnetic maps

In our previous paper [14] we ask when a vector field is a magnetic map. More precisely, we consider a Riemannian manifold (M, g) of dimension n and its tangent bundle (T (M), g_{S}) equipped with the Sasaki metric. On T (M) we also define an almost complex structure J_{S} by

J_{S}X^{H} = X^{V}, J_{S}X^{V} = −X^{H} , for all X ∈ 𝔛(M).

It is known that (T (M), g_{S}, J_{S}) is an almost Käherian manifold [6]. Hence, the Kähler 2-form Ω_{S} = g_{S}(J_{S} ·, ·) may be considered as a magnetic field on T (M).

A vector field ξ ∈ 𝔛(M) will be thought as a map from (M, g) to (T (M), g_{S}, J_{S}). In the book of Dragomir and Perrone [7], the authors write the following formula

τ(ξ) = − {(trace_{g} R(∇_{•}ξ, ξ)•)^{H} + (Δ_{g}ξ)^{V}} ∘ ξ.

Here Δ_{g} denotes the rough Laplacian on vector fields, defined by

Δ_{g}X = − ∑_{k=1}^{n} [∇_{e_{k}} ∇_{e_{k}} X − ∇_{∇_{e_{k}} e_{k}} X],

where {e_{k}}_{k=1,...,n} is an orthonormal frame on M. We also have

J_{S}(ξ_{∗}ξ) = ξ^{V} − (∇_{ξ}ξ)^{H} .

We state the following.

Theorem 2.1. [14] Let (M, g) be a Riemannian manifold and (T (M), g_{S}, J_{S}) its tangent bundle endowed with the usual almost Kählerian structure. Let ξ be a vector field on M. Then ξ is a magnetic map with strength q associated to ξ itself and the Kähler magnetic field Ω_{S} if and only if the following conditions hold:

(2.1)  trace_{g} R(∇_{•}ξ, ξ)• = q ∇_{ξ} ξ,
(2.2)  Δ_{g}ξ = −qξ.

Consider a Killing vector field ξ on the Riemannian manifold (M, g). We know that:

Lemma 2.1. A Killing vector field ξ on a Riemannian manifold (M, g) satisfies the equation ∇_{XY}^{2} ξ = −R(ξ, X)Y, for all X, Y ∈ 𝔛(M).

We ask now for ξ : (M, g) → (T (M), g_{S}, J_{S}) to be a magnetic map. Then ξ must satisfy (2.2). But Δ_{g}ξ = − trace_{g} ∇^{2}ξ. Using the previous lemma, we get

Δ_{g}ξ = trace_{g} R(ξ, •) • .

On the other hand, we have

Ric(ξ, X) = trace_{g}{Z ↦ R_{Zξ} X} = ∑_{i=1}^{n} g(e_{i}, R_{e_{i} ξ}X) = − ∑_{i=1}^{n} g(R_{e_{i} ξ}e_{i}, X)
= g(trace_{g} R(ξ, •)•, X) = −qg(ξ, X), for all X ∈ 𝔛(M).

So, if Q is the Ricci operator, that is g(QX, Y) = Ric(X, Y), for all X, Y tangent to M, then we get that Qξ = −qξ. We give the following.

Proposition 2.1. If a Killing vector field is a magnetic map with strength q, then it is an eigenvector of the Ricci operator corresponding to the eigenfunction (−q).

Remark 2.1. In the special case of Einstein manifolds, the strength q is related to the scalar curvature, namely q = −\frac{scal}{n} .

Suppose that M is a real space form M^{n}(c), case when the curvature tensor is expressed as R_{XY} Z = c(g(Y, Z)X − g(X, Z)Y ), for all X, Y, Z ∈ 𝔛(M). We can easily compute trace_{g} R(∇_{•}ξ, ξ)• = c(∇_{ξ}ξ − div(ξ)). As ξ is Killing, its divergence is zero and thus, the magnetic equation becomes

(2.3)  (c − q)∇_{ξ}ξ = 0.

We obtained the following.

Theorem 2.2. Let ξ be a Killing vector field on a real space form M^{n}(c), n ⩾ 2. If ξ is a non-harmonic magnetic map with strength q, then q = (1 − n)c and ξ is self parallel, case in which it has constant length.

Proof. Note that a real space form M^{n}(c) is Einstein and its scalar curvature is scal = cn(n−1). So, as ξ is magnetic, cf. Remark 2.1, we must have q = (1−n)c. Obviously, equation (2.3) is satisfied if q = c. In this situation we get that M is flat and q = 0, that is ξ is a harmonic vector field. If q ≠ c then ∇_{ξ}ξ = 0. As ξ is Killing, we have

g(∇_{ξ}ξ, X) + g(ξ, ∇_{X} ξ) = 0, for all X ∈ 𝔛(M).

It follows that the length of ξ is constant. □

In the end of this section we propose the study of the following problem: Study non-harmonic magnetic Killing vector fields on the unit sphere 𝕊^{n}.

3. Magnetic vector fields on almost contact metric manifolds

A (φ, ξ, η)-structure on a manifold M is defined by a field φ of endomorphisms of tangent spaces, a vector field ξ and a 1-form η satisfying

η(ξ) = 1, φ^{2} = − I+η ⊗ ξ, φξ = 0, η ∘ φ = 0.

If (M, φ, ξ, η) admits a compatible Riemannian metric g, namely

g(φX, φY) = g(X, Y) − η(X)η(Y), for all X, Y ∈ 𝔛(M),

then M is said to have an almost contact metric structure, and (M, φ, ξ, η, g) is called an almost contact metric manifold. It follows that η(X) = g(ξ, X), for any X ∈ 𝔛(M) and ξ is unitary.

The fundamental 2-form Ω is defined by Ω(X, Y) = g(φX, Y), for any vector fields X and Y. Recall that a contact metric manifold is an almost contact metric manifold such that Ω = dη. If in addition the structure is normal, that is the normality tensor field N = [φ, φ] + 2dη ⊗ ξ vanishes, then the manifold M is called a Sasakian manifold. Here [φ, φ] denotes the Nijenhuis tensor of φ. Denoting by ∇ the Levi-Civita connection associated to g, the Sasakian manifold (M, φ, ξ, η, g) is characterized by (∇_{X} φ)Y = −g(X, Y)ξ + η(Y)X, for any X, Y ∈ 𝔛(M). As a consequence, we have ∇_{X} ξ = φX, for all X ∈ 𝔛(M). A systematic study of these structures is presented in the two books of Blair [4, 5]. However, we use the sign convention given by Sasaki, see e.g., [12].

On the other hand, a Kenmotsu manifold can be defined as a normal almost contact metric manifold such that dη = 0 and dΩ = 2η ∧ Ω. These manifolds can be characterized using their Levi-Civita connection, by requiring

(∇_{X} φ)Y = g(φX, Y)ξ − η(Y)φX, for every X, Y ∈ 𝔛(M).

In our previous paper [14], we find some conditions when the Reeb vector field ξ on a Sasakian space form is magnetic, that is satisfies the condition in Theorem 2.1. We obtain that q = −2n.

Let us analyze the property of the characteristic vector field ξ on a Kenmotsu manifold to be magnetic. Recall the following two useful formulas:

∇_{X} ξ = X − η(X)ξ,
R(X, ξ)Y = g(X, Y)ξ − η(Y)X, for every X, Y ∈ 𝔛(M).

Compute trace_{g} R(∇_{•}ξ, ξ)•. To do this, consider as usual, a φ adapted orthonormal basis {e_{i}, φe_{i}, ξ}, i = 1, …, n. We have ∇_{e_{i}} ξ = e_{i}, ∇_{φe_{i}} ξ = φe_{i}, ∇_{ξ}ξ = 0. Hence

trace_{g} R(∇_{•}ξ, ξ)• = ∑_{i=1}^{n} [R(e_{i}, ξ)e_{i} + R(φe_{i}, ξ)φe_{i}]

= ∑_{i=1}^{n} [g(e_{i}, e_{i})ξ + g(φe_{i}, φe_{i})ξ] = 2nξ.

Thus, the equation (2.1) becomes 2nξ = 0, which is a contradiction.

As a matter of fact, for the second condition of Theorem 2.1, we have

Δ_{g}ξ = − ∑_{i=1}^{n} [( ∇_{e_{i}} ∇_{e_{i}} ξ − ∇_{∇_{e_{i}} e_{i}} ξ) + ( ∇_{φe_{i}} ∇_{φe_{i}} ξ − ∇_{∇_{φe_{i}} φe_{i}} ξ) ]

= − ∑_{i=1}^{n} [η(∇_{e_{i}} e_{i})ξ + η(∇_{φe_{i}} φe_{i})ξ] = 2nξ.

Therefore, ξ is an eigenvector of the rough Laplacian with corresponding eigenfunction q = −2n. We conclude with the following.

Proposition 3.1. The characteristic vector field of a Kenmotsu manifold is not magnetic.

Next we would like to make some comments on the same problem in a cosymplectic manifold. Recall that a cosymplectic manifold is an almost contact metric manifold for which the three tensor fields φ, ξ and η are parallel. Therefore, the first condition in the Theorem 2.1 is automatically satisfied. Since Δ_{g}ξ = 0, the second condition implies q = 0, that is ξ is a harmonic map. We conclude with the following.

Proposition 3.2. If the characteristic vector field of a cosymplectic manifold is magnetic, then it is harmonic.

At this point we propose another problem: Study the property of ξ of being a magnetic map on a generalized Sasakian space form. See [1].

We end this section with some comments concerning the condition div(ξ) = 0 used in finding the magnetic equation. Because some readers may think that the divergence free condition for ξ is too strong or artificial, we mention that this condition is often satisfied. For example, on almost contact metric manifolds, we know the following:

• The characteristic vector field ξ of a contact metric manifold is divergence free.

• In addition, cosymplectic manifolds have divergence free ξ.

• However, ξ is not always divergence free; e.g. on Kenmotsu manifolds, we have

div ξ = ∑_{i=1}^{n} g(∇_{e_{i}} ξ, e_{i}) + ∑_{i=1}^{n} g(∇_{φe_{i}} ξ, φe_{i})

= ∑_{i=1}^{n} g(e_{i}, e_{i}) + ∑_{i=1}^{n} g(φe_{i}, φe_{i}) = 2n ≠ 0.

4. More examples of magnetic maps

4.1. H-minimal submanifolds. Let N be an n-dimensional Lagrangian submanifold in a Kähler manifold M. Then ζ := −JH/n is a globally defined tangent vector field on M. Here H is the mean curvature vector field. In our previous paper [13], we showed that the inclusion map ι : N → M satisfies τ(ι) = Jι_{∗}ζ.

According to Oh [19], a Lagrangian submanifold N is said to be Hamiltonianminimal (in short H-minimal) if it is a critical point of the volume functional under compactly supported smooth variations arising from Hamiltonian deformations.

The Euler–Lagrange equation of this variational problem is div (JH) = 0, that is ζ is divergence free.

This implies that every H-minimal Lagrangian submanifold N is magnetic with respect to ζ = −JH/n and the Kähler form of M.

4.2. L-minimal submanifolds. In Sasakian geometry, one introduces the notion of L-minimal immersion as follows:

Definition 4.1. [16] An n-dimensional Legendrian submanifold N in a Sasakian manifold M is said to be L-minimal if it is a critical point of the volume functional under compactly supported smooth variations arising from Legendre deformations.

The Euler–Lagrange equation of this variational problem is div(φH) = 0.

One can check that every Legendrian submanifold satisfies τ(ι) = ϕι_{∗}ζ, where the vector field ζ is defined globally on N by ζ := −φH/n.

Thus every L-minimal Legendrian submanifold in a Sasakian manifold is magnetic with respect to the divergence free vector field ζ and the contact form on M.

4.3. Magnetic hypersurfaces in complex space forms. Let (M, g, J) be a Kähler manifold of complex dimension n and let f : N → (M, g, J) be an orientable real hypersurface with unit normal vector field ν. Then the Kähler structure (g, J) induces an almost contact metric structure (φ, ξ, η, h) on N as follows. First, define the vector field ξ by f_{∗}ξ = −Jν. Next (φ, η) are defined by the formula

Jf_{∗}X = f_{∗}φX + η(X)ν

for all tangent vector X on N. Finally, we set h = f^{∗}g.

Then the Levi-Civita connections ˜∇ of M and ∇ of N are related by the following Gauss formula and Weingarten formula:

˜∇_{X} f_{∗}Y = f_{∗}∇_{X} Y + g(AX, Y)ν, ]; ˜∇_{X} ν = −f_{∗}AX, X ∈ 𝔛(N).

The endomorphism field A is called the shape operator of N derived from ν. We know that

(∇_{X} φ)Y = η(Y)AX − g(AX, Y)ξ, ∇_{X} ξ = φAX.

The following result is fundamental (see [7]).

Proposition 4.1. The structure vector field ξ is divergence free.

Proof. We can compute

div ξ = ∑_{i=1}^{n−1} g(∇_{e_{i}} ξ, e_{i}) + ∑_{i=1}^{n−1} g(∇_{ϕe_{i}} ξ, ϕe_{i}) + g(∇_{ξ}ξ, ξ)

= ∑_{i=1}^{n−1} g(ϕAe_{i}, e_{i}) + ∑_{i=1}^{n−1} g(ϕAϕe_{i}, ϕe_{i}) + g(ϕAξ, ξ).

We note that g(ϕAξ, ξ) = 0. Next we have

⟨ϕAϕe_{i}, ϕe_{i}⟩ = −⟨Aϕe_{i}, ϕ^{2}e_{i}⟩ = ⟨Aϕe_{i}, e_{i}⟩ = ⟨ϕe_{i}, Ae_{i}⟩ = −⟨e_{i}, ϕAe_{i}⟩.

Thus ξ is divergence free. □

The tension field τ(f) is given by τ(f) = (2n − 1)Hν. Here H is the mean curvature function. If Ω = g(J·, ·) is considered as a magnetic field on M, then the magnetic equation for the immersion f with respect to {ξ, Ω} and strength q is computed as

(2n − 1)Hν = qJ(f_{∗}ξ) = qJ(−Jν) = qν.

Thus f is magnetic with respect to {ξ, Ω} if and only if q = (2n − 1)H.

Proposition 4.2. [13] Let f : N → (M, g, J) be an orientable real hypersurface of constant mean curvature H with induced almost contact metric structure (φ, ξ, η, h). Then f is a magnetic map with respect to the structure vector field ξ and the Kähler magnetic field Ω with strength q = (2n − 1)H.

Now, we add one more example to our previous list of magnetic real hypersurfaces in complex space forms and complex Grassmannian manifolds given in [13], namely magnetic real hypersurfaces in complex quadrics.

Example 4.1. In [3], Berndt and Suh studied real hypersurfaces in the Grassmannian manifold G̃r_{2}(ℝ^{m+2} ) of oriented 2-planes in Euclidean (m+2)-space. As is well known, the Grassmannian manifold G̃r_{2}(ℝ^{m+2} ) is identified with the complex quadric

𝒬_{m} = {[z_{1} : z_{2} : ⋯: z_{m+2} ] ∈ ℂP^{m+1} | z_{1}^{2} + z_{2}^{2} + ⋯+ z_{m+2}^{2} = 0}

in the complex projective (m + 1)-space.

When we equip the ambient projective space with the Fubini–Study metric of constant holomorphic sectional curvature 4, then 𝒬_{m} = SO(m+2)/ SO(2)×SO(m) is a Hermitian symmetric space of rank 2 and maximal sectional curvature 4 with respect to the induced metric g. The Ricci tensor is given by Ric = 2mg.

Hereafter we assume that m ⩾ 3. For m = 2k, the map

[z_{1} : z_{2} : ⋯: z_{k+1} ] ↦ [z_{1} : z_{2} : ⋯: z_{k+1} : iz_{1} : iz_{2} : ⋯: iz_{k+1} ]

defines a totally geodesic complex immersion of ℂP^{k} into 𝒬_{2k} ⊂ ℂP^{2k+1} .

For r ∈ (0, π/2), the tube around ℂP^{k} is a homogeneous real hypersurface with principal curvatures λ_{1} = 2 cot(2r), λ_{2} = 0, λ_{3} = − tanr, λ_{4} = cotr and multiplicities m_{1} = 1, m_{2} = 2, m_{3} = m_{4} = 2k − 2.

In case m = 2, i.e., k = 1, we have ℂP^{1} ⊂ 𝒬_{2} = 𝕊^{2} × 𝕊^{2}. The principal curvatures of a tube around ℂP^{1} are 0 and 2 cot(2r).

The inclusion map of a tube M_{r} of radius r around ℂP^{k} into 𝒬_{2k} is a magnetic immersion with respect to the magnetic field F = Ω with strength

q = m_{1}λ_{1} + m_{2}λ_{2} + m_{3}λ_{3} + m_{4}λ_{4} = 2(2k − 1) cot2r.

4.4. Harmonic unit vector fields as magnetic maps. A unit vector field ξ on a Riemannian manifold (M, g) is said to be a harmonic unit vector field if it is a critical point of the energy functional over the space 𝔛_{1}(M) of all smooth unit vector fields on M. The Euler-Lagrange equation of this variational problem is Δ_{g}ξ = |ξ|^{2}ξ. Moreover it is known that ξ is a harmonic map from (M, g) into the unit tangent sphere bundle U(M) with the metric induced from g_{S} if and only if ξ is a harmonic unit vector field and satisfies trace_{g} R(∇_{•}ξ, ξ)• = 0 (see [7]).

Comparing the harmonic map equation for ξ : M → U(M) and magnetic equation for ξ : M → T (M) we have

Proposition 4.3. Let ξ be a unit vector field on a Riemannian manifold (M, g). Assume that ξ satisfies

• ξ is divergence free, (optional condition)

• ∇_{ξ}ξ = 0,

• |∇ξ| is constant

• ξ : M → U(M) is a harmonic map. Then ξ is a magnetic map into T (M) with strength q = −|∇ξ|^{2}.

4.5. Magnetic vector fields on real hypersurfaces. An oriented real hypersurface N of a Kähler manifold M is said to be Hopf if the structure vector field ξ introduced in subsection 4.3 is a principal vector field. In that case, if Aξ = αξ, then α is called the Hopf principal curvature on N. It is easy to check that ξ satisfies ∇_{ξ}ξ = 0 if and only if N is Hopf.

The following results are direct consequences of [20, Theorem 3.2] due to Perrone.

Proposition 4.4. Let N ⊂ M be an oriented Hopf hypersurface of a Kähler-Einstein manifold. Then the structure vector field ξ satisfies:

(1) ξ is a harmonic unit vector field if and only if grad H = ξ(H)ξ, where H is the mean curvature function.

(2) If the principal curvature α corresponding to ξ is constant along the trajectories of ξ then ξ(H) = 0.

Corollary 4.1. Let N ⊂ M be an oriented Hopf hypersurface of a Kähler– Einstein manifold satisfying ξ(α) = 0. Then ξ is a harmonic map into U(N) if and only if the mean curvature is constant.

Complex space forms are typical examples of Kähler–Einstein manifolds.

Theorem 4.1. Let N be an oriented Hopf hypersurface with constant principal curvatures in a complex space form M. Then the characteristic vector field ξ of N is a magnetic map with strength q = −|A|^{2} + α^{2}.

Proof. Let N be an oriented Hopf hypersurface with constant principal curvatures in a complex space form M. Then ξ satisfies

Δ_{g}ξ = |∇ξ|^{2}ξ, trace_{g} R(∇_{•}ξ, ξ)• = 0, ∇_{ξ}ξ = 0.

Since all the principal curvatures are constant and ∇ξ = φA, we have |∇ξ|^{2} = |A|^{2} − α^{2}. Hence ξ is a magnetic map with strength q = −|A|^{2} + α^{2}. □

As is well known, a complete and simply connected complex space form is a complex projective space ℂP^{n}(c), a complex Euclidean space ℂ^{n} or a complex hyperbolic space ℂH^{n}(c), according as c > 0, c = 0 or c < 0. Hopf hypersurfaces in ℂP^{n}(c) and ℂH^{n}(c) are classified by Kimura [17] and Berndt [2], respectively.

Of course, one can check that characteristic vector fields of all homogeneous Hopf real hypersurfaces in ℂP^{n}(c) and ℂH^{n}(c) are magnetic maps into tangent bundles. However, we exhibit here only few examples.

Example 4.2 (Type A hypersurfaces). Let us consider

M̂_{k}(r) := 𝕊^{2k+1} (cosr) × 𝕊^{2n−1−2k} (sinr) ⊂ 𝕊^{2n+1} , 0 ⩽ k < n, 0 < r < \frac{π}{2}.

Then the Hopf projection image M_{k}(r) of M̂_{k}(r) is a Hopf hypersurface in the complex projective space ℂP^{n}(4) of constant holomorphic sectional curvature 4. These hypersurfaces M_{k}(r) are referred as to type A hypersurfaces. Note that type A hypersurfaces are quasi-Sasakian. The type A hypersurface M_{k}(r) has constant principal curvatures λ_{1} = − tanr, λ_{2} = cotr, α = 2 cot(2r), 0 < r < \frac{π}{2} with multiplicities m_{1} = 2k, m_{2} = 2(n − k − 1), m_{α} = 1. Then the characteristic vector field ξ is a magnetic map into T (M) with strength

q = −(m_{1}λ_{1}^{2} + m_{2}λ_{2}^{2}) = −2{k tan^{2} r + (n − k − 1) cot^{2} r} < 0.

Example 4.3 (Horospheres). Let M be a horosphere in the complex hyperbolic n-space ℂH^{n}(−4). It is known that the horosphere in ℂH^{n}(−4) is a Sasakian space form of constant holomorphic sectional curvature −3. The horosphere has constant principal curvatures λ = 1 with multiplicity 2n − 2 and α = 2 with multiplicity 1. Then the strength is q = −2(n − 1). This is consistent with Section 2.

Example 4.4 (Type B hypersurfaces). Let M be a tube over totally real and totally geodesic real hyperbolic space ℍ^{n} in the complex hyperbolic n-space ℂH^{n}(−4) of constant holomorphic sectional curvature −4. Then M is a Hopf hypersurface with constant principal curvatures having the form

λ_{1} = \frac{1}{r} coth u, λ_{2} = \frac{1}{r} tanh u, α = \frac{2}{r} tanh (2u)

with multiplicities m_{1} = m_{2} = n − 1, m_{α} = 1. Hence the characteristic vector field ξ is a magnetic map into T (M) with strength

q = −(m_{1}λ_{1}^{2} + m_{2}λ_{2}^{2}) = −\frac{n − 1}{r^{2}} \{coth^{2} u + tanh^{2} u\} < 0.

Acknowledgments. The second author would like to thank organizers for warm hospitality, cordially atmosphere created in Zlatibor, as well as for supporting him to participate at this Seminar.

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Institute of Mathematics University of Tsukuba 1-1-1 Tennodai Tsukuba, Ibaraki 305-8571, Japan

inoguchi@math.tsukuba.ac.jp

Al. I. Cuza University of Iasi Faculty of Mathematics Iasi, Romania

marian.ioan.munteanu@gmail.com