On Hamiltonian minimality of isotropic non-homogeneous tori in $\mathbb{H}^n$ and $\mathbb{C} \mathrm{P}^{2n+1}$

TL;DR

This paper constructs a family of flat isotropic non-homogeneous tori in hyperbolic and complex projective spaces and determines the precise conditions under which they are Hamiltonian minimal.

Contribution

It introduces a new family of tori in hyperbolic and complex projective spaces and characterizes their Hamiltonian minimality conditions.

Findings

Explicit construction of flat isotropic non-homogeneous tori.

Necessary and sufficient conditions for Hamiltonian minimality.

Extension of minimality theory to non-homogeneous tori.

Abstract

We construct a family of flat isotropic non-homogeneous tori in $\mathbb{H}^n$ and $\mathbb{C} \mathrm{P}^{2n+1}$ and find necessary and sufficient conditions for their Hamiltonian minimality.

Equations

$${\omega }_I=(I(\cdot ),\cdot ),{\omega }_J=(J(\cdot ),\cdot ),{\omega }_K=(K(\cdot ),\cdot )$$

$$\psi :{\mathbb{R}}^l\times {\mathbb{R}}^m\to {\mathbb{H}}^n,\psi (x,y)=(r_1{e}^{j(e_1,x)}{e}^{k(f_1,y)},\dots ,r_n{e}^{j(e_n,x)}{e}^{k(f_n,y)}),$$

$$x\in {\mathbb{R}}^l,y\in {\mathbb{R}}^m,r_d>0,e_d\in {\mathbb{R}}^l,f_d\in {\mathbb{R}}^m,d=1,\dots ,n.$$

$$e^{j(e_d,x)}=\cos (e_d,x)+\sin (e_d,x)j,e^{k(f_d,y)}=\cos (f_d,y)+\sin (f_d,y)k.$$

$$G_1=({\psi }_{x_p},{\psi }_{x_q}{)}_{pq},G_2=({\psi }_{y_r},{\psi }_{y_s}{)}_{rs},$$

$$p,q=1,\dots ,l;r,s=1,\dots ,m.$$

$$e_d^{\prime }=(e_d;\Vert e_d{\Vert }_{G_1^{-1}}^2+\Vert f_d{\Vert }_{G_2^{-1}}^2)\in {\mathbb{R}}^{l+1},\Vert e_d{\Vert }_{G_1^{-1}}^2=(G_1^{-1}{e}_d,e_d),$$

$$f_d^{\prime }=(f_d;\Vert e_d{\Vert }_{G_1^{-1}}^2+\Vert f_d{\Vert }_{G_2^{-1}}^2)\in {\mathbb{R}}^{m+1},\Vert f_d{\Vert }_{G_2^{-1}}^2=(G_2^{-1}{f}_d,f_d).$$

$${\pi }_v:S^{4n+3}(1)\to \mathbb{C}{\mathrm{P}}^{2n+1}$$

$$e_d^{\prime \prime }=(e_d;\Vert e_d{\Vert }_{G_1^{-1}}^2+\Vert f_d{\Vert }_{G_2^{-1}}^2-(l+m))\in {\mathbb{R}}^{l+1},$$

$$f_d^{\prime \prime }=(f_d;\Vert e_d{\Vert }_{G_1^{-1}}^2+\Vert f_d{\Vert }_{G_2^{-1}}^2-(l+m))\in {\mathbb{R}}^{m+1}.$$

$${\mathrm{rk}}_{\mathbb{Z}}⟨e_1^{\prime \prime },\dots ,e_n^{\prime \prime }⟩=l.$$

$${\mathrm{rk}}_{\mathbb{Z}}⟨f_1^{\prime \prime },\dots ,f_n^{\prime \prime }⟩=m.$$

$$⟨x,y⟩=\sum x_i{\bar{y}}_i,x,y\in {\mathbb{H}}^n,$$

$$⟨x,y⟩=(x,y)-{\omega }_I(x,y)i-{\omega }_J(x,y)j-{\omega }_K(x,y)k.$$

$${\omega }_I({\psi }_{x_p},{\psi }_{y_q})=e_{1p}{f}_{1q}{r}_1^2+\cdots +e_{np}{f}_{nq}{r}_n^2.$$

$$H=-\frac{1}{l+m}({\alpha }_1{r}_1{e}^{j(e_1,x)}{e}^{k(f_1,y)},\dots ,{\alpha }_n{r}_n{e}^{j(e_n,x)}{e}^{k(f_n,y)}).$$

$$H=\frac{1}{l+m}\Delta \psi ,$$

$$\Delta =\sum _{p,q=1}^l{G}_1^{pq}{\partial }_{x_p}{\partial }_{x_q}+\sum _{r,s=1}^m{G}_2^{rs}{\partial }_{y_r}{\partial }_{y_s},$$

$$\sum _{p,q=1}^l{G}_1^{pq}{\partial }_{x_p}{\partial }_{x_q}{e}^{j(e_d,x)}=-(G_1^{-1}{e}_d,e_d)e^{j(e_d,x)},$$

$$\sum _{r,s=1}^m{G}_2^{rs}{\partial }_{y_r}{\partial }_{y_s}{e}^{k(f_d,y)}=-(G_2^{-1}{f}_d,f_d)e^{k(f_d,y)}$$

$${\mathrm{rk}}_{\mathbb{Z}}⟨e_1^{\prime },\dots ,e_n^{\prime }⟩=l.$$

$$H=-\frac{1}{l+m}({\alpha }_1{r}_1{e}^{j(e_1,x)}{e}^{k(f_1,y)},\dots ,{\alpha }_n{r}_n{e}^{j(e_n,x)}{e}^{k(f_n,y)})$$

$$=j({\lambda }_1{\partial }_{x_1}\psi +\cdots +{\lambda }_l{\partial }_{x_l}\psi +{\eta }_1{\partial }_{y_1}\psi +\cdots +{\eta }_m{\partial }_{y_m}\psi ).$$

$$\frac{{\alpha }_d}{l+m}=(\lambda ,e_d)-i(\eta ,f_d),d=1,\dots ,n,$$

$$\frac{{\alpha }_d}{l+m}=(\lambda ,e_d),d=1,\dots ,n.$$

$${\mathrm{rk}}_{\mathbb{R}}⟨e_1^{\prime },\dots ,e_n^{\prime }⟩={\mathrm{rk}}_{\mathbb{R}}⟨e_1,\dots ,e_n⟩=l.$$

$${\mathrm{rk}}_{\mathbb{R}}⟨e_1,\dots ,e_n⟩={\mathrm{rk}}_{\mathbb{Z}}⟨e_1,\dots ,e_n⟩.$$

$${\mathrm{rk}}_{\mathbb{R}}⟨e_1^{\prime },\dots ,e_n^{\prime }⟩={\mathrm{rk}}_{\mathbb{Z}}⟨e_1^{\prime },\dots ,e_n^{\prime }⟩=l.$$

$$({\partial }_{z_p}\psi ,{\partial }_{z_q}\psi )={\delta }_{pq},({\partial }_{w_r}\psi ,{\partial }_{w_s}\psi )={\delta }_{rs},$$

$$\delta {\omega }_J(H,\cdot )=\mathrm{div}JH=\sum _{p=1}^l({\nabla }_{{\psi }_{z_p}}JH,{\psi }_{z_p})+\sum _{q=1}^m({\nabla }_{{\psi }_{w_q}}JH,{\psi }_{w_q}),$$

$${\nabla }_{{\psi }_{z_p}}JH=\mathrm{Pr}({\partial }_{z_p}JH)$$

$$=-\frac{1}{l+m}\mathrm{Pr}({\partial }_{z_p}j({\alpha }_1{r}_1{e}^{j(e_1,z)}{e}^{k(f_1,w)},\dots ,{\alpha }_n{r}_n{e}^{j(e_n,z)}{e}^{k(f_n,w)}))$$

$$=\frac{1}{l+m}\mathrm{Pr}(e_{1p}{\alpha }_1{r}_1{e}^{j(e_1,z)}{e}^{k(f_1,w)},\dots ,e_{np}{\alpha }_n{r}_n{e}^{j(e_n,z)}{e}^{k(f_n,w)}),$$

Full text

The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh 5913.2018.1) and the Russian Foundation for Basic Research (Grant 18-01-00411)

1. Introduction

A submanifold of real dimension $k$ in a Kähler manifold of complex dimension $n\geq k$ is called isotropic if the Kähler form of the manifold vanishes on it. For $k=n$ the submanifold is called Lagrangian. An isotropic submanifold in a Kähler manifold is called Hamiltonian-minimal (shortly, H-minimal) if the variations of its volume along the Hamiltonian fields with compact support are zero. In particular, any minimal isotropic submanifold is H-minimal.

The notion of H-minimality was introducted in the paper [Oh93] in connection with the Arnold conjecture on the number of fixed points of a Hamiltonian symplectomorphism. The examples of H-minimal Lagrangian surfaces in $\mathbb{C}^{2}$ and $\mathbb{C}\mathrm{P}^{2}$ were found in [CU98, HR00, HR02] and [Mir03, Ma05]. A family of H-minimal Lagrangian submanifolds in $\mathbb{C}^{n}$ and $\mathbb{C}\mathrm{P}^{n}$ arising from intersections of real quadrics was considered in [Mir04, MP13]. In [Sha91] and [Yer18] minimal Lagrangian and isotropic tori in $\mathbb{C}\mathrm{P}^{2}$ and $\mathbb{C}\mathrm{P}^{3}$ were studied in connection with the soliton equations: as a smooth periodic solutions of the Tzizeica equation and smooth periodic solutions of the sinh-Gordon equation accordingly. Necessary and sufficient conditions for H-minimality of isotropic homogeneous tori in $\mathbb{C}^{n}$ and $\mathbb{C}\mathrm{P}^{n}$ were found in [Ovc18].

Let us recall that a Riemannian manifold $M$ of real dimension $4n$ endowed with three complex structures $I,J,K$ is called hyperkähler if the following holds:

• •

$M$ is Kähler with respect to these structures;

• •

$I,J,K$ satisfy the relation $I\circ J=-J\circ I=K$.

Let $(\cdot,\cdot)$ be the Riemannian metric on $M$. We will denote by

[TABLE]

the corresponding Kähler forms on $M$.

Definition 1.1**.**

We will call a submanifold of $N$ in $M$ $\omega_{I}$-isotropic if $\omega_{I|N}=0$. One can define $\omega_{J}$- and $\omega_{K}$-isotropic submanifolds in $M$ in the same way.

Let us formulate the main results.

Consider a $(l+m)$-dimensional torus in $\mathbb{H}^{n}$ ($l,m\leq n$) defined by the mapping

[TABLE]

There $(\cdot,\cdot)$ is the standard dot product, the vectors $e_{1},\dots,e_{n}$ and $f_{1},\dots,f_{n}$ must generate lattices of maximal rank in $\mathbb{R}^{l}$ and $\mathbb{R}^{m}$ accordingly. Let us recall that by Euler’s identity for quaternions we have

[TABLE]

We will denote this torus by $T^{l,m}$.

Notice that the matrix of the metric tensor of the torus $T^{l,m}$ in the coordinates $(x,y)$ is constant and of block form. Actually, we have $(\psi_{x_{p}},\psi_{y_{q}})=0$ for all $p=1,\dots,l$ and $q=1,\dots,m$.

The following statement holds (see the proof in Section 2).

Proposition 1.2**.**

The torus $T^{l,m}\subset\mathbb{H}^{n}$ is $\omega_{J}$- and $\omega_{K}$-isotropic, but not $\omega_{I}$-isotropic.

We will denote the blocks of the matrix of the metric tensor of $T^{l,m}$ by

[TABLE]

Let us also introduce the vectors

[TABLE]

We have

Theorem 1.3**.**

The torus $T^{l,m}\subset\mathbb{H}^{n}$ is H-minimal

• •

with respect to $\omega_{J}$ — if and only if $\operatorname{rk}_{\mathbb{Z}}\langle e^{\prime}_{1},\dots,e^{\prime}_{n}\rangle=l$;

• •

with respect to $\omega_{K}$ — if and only if $\operatorname{rk}_{\mathbb{Z}}\langle f^{\prime}_{1},\dots,f^{\prime}_{n}\rangle=m$.

Corollary 1.4**.**

The torus $T^{n,n}\subset\mathbb{H}^{n}$ is H-minimal with respect to $\omega_{J}$ and $\omega_{K}$ simultaneously.

Let us assume that the torus $T^{l,m}\subset\mathbb{H}^{n+1}$ is contained in the unit sphere $S^{4n+3}(1)\subset\mathbb{H}^{n+1}$, i.e., $\sum r_{d}^{2}=1$. For any $v=I,J,K$ we will consider the Hopf mapping

[TABLE]

induced by the left action of the group $\{e^{vt},t\in\mathbb{R}\}$. Notice that for any $v$ we have the Fubini — Study form $\Omega_{v}$ on $\mathbb{C}\mathrm{P}^{2n+1}$ induced by the form $\omega_{v}$ on $\mathbb{H}^{n+1}$.

Definition 1.5**.**

We will call a submanifold $N\subset S^{4n+3}(1)$ $\pi_{I}$-horizontal if $N$ is orthogonal to the fibers of the fibration $\pi_{I}$. One can define $\pi_{J}$- and $\pi_{K}$-horizontal submanifolds in $S^{4n+3}(1)$ in the same way.

Let us introduce the vectors

[TABLE]

We have

Theorem 1.6**.**

Let the torus $T^{l,m}\subset S^{4n+3}(1)$ be $\pi_{J}$-horizontal. The torus $\pi_{J}(T^{l,m})$ is H-minimal if and only if

[TABLE]

Similarly, let the torus $T^{l,m}$ be $\pi_{K}$-horizontal. The torus $\pi_{K}(T^{l,m})$ is H-minimal if and only if

[TABLE]

Corollary 1.7**.**

Let the torus $T^{n,n}\subset S^{4n+3}(1)$ be $\pi_{J}$-horizontal. Then the torus $\pi_{J}(T^{n,n})$ is H-minimal.

Similarly, let the torus $T^{n,n}$ be $\pi_{K}$-horizontal. Then the torus $\pi_{K}(T^{n,n})$ is H-minimal.

Let us notice that the tori $\pi_{J}(T^{1,1})$ and $\pi_{K}(T^{1,1})$ are minimal submanifolds in $\mathbb{C}\mathrm{P}^{3}$ (see [Yer19]).

The proof of Theorem 1.3 and 1.6 is given in Section 2 and 3 accordingly. There we also give explicit examples of tori under consideration.

The author is grateful to A. Mironov for stating the problem and useful discussions.

2. Proof of Theorem 1.3

We will denote by

[TABLE]

the standard Hermitian inner product in $\mathbb{H}^{n}$. By definition we have

[TABLE]

Proof of Proposition 1.2.

One can easily check that for any given tangent vectors $\xi$, $\eta$ on $T^{l,m}$ we have $\langle\xi,\eta\rangle\in\mathbb{C}$, i.e., $\omega_{J}(\xi,\eta)=\omega_{K}(\xi,\eta)=0$.

Nonetheless, the torus $T^{l,m}$ is not $\omega_{I}$-isotropic. Actually,

[TABLE]

The matrix of the metric tensor of $T^{l,m}$ in the coordinates $(x,y)$ is constant and of block form, so after proper linear change of coordinates on $\mathbb{R}^{l}$ and $\mathbb{R}^{m}$ we can choose $e_{1}$ and $f_{1}$ to be the first standard basis vector in $\mathbb{R}^{l}$ and $\mathbb{R}^{m}$ accordingly. Then we have $\omega_{I}(\psi_{x_{1}},\psi_{y_{1}})=r_{1}^{2}\neq 0$. ∎

To prove Theorem 1.3 we are going to use the following Chen and Morvan criterion of H-minimality for compact isotropic submanifolds.

Proposition 2.1** (see [CM94]*Proposition 3.5).**

Let $N$ be a compact isotropic submanifold of a Kähler manifold $(M,I)$ with the Kähler form $\omega$, $H$ be the mean curvature field of $N$. The submanifold $N$ is H-minimal if and only if $IH$ is a tangent vector field to $N$ and $\delta(i_{H}\omega)=0$, where $i_{H}\omega=\omega(H,\cdot)$ is a 1-form on $N$, $\delta$ is the codifferential. ∎

Consequently, a compact $\omega_{J}$-isotropic submanifold $N$ of a hyperkähler manifold $M$ is H-minimal with respect to $\omega_{J}$ if and only if $JH$ is a tangent vector field to $N$ and $\delta\omega_{J}(H,\cdot)=0$ (the statement for $\omega_{K}$ is analogous).

Lemma 2.2**.**

The mean curvature field of the torus $T^{l,m}$ in $\mathbb{H}^{n}$ has the following form:

[TABLE]

There we put $\alpha_{d}=\|e_{d}\|^{2}_{G_{1}^{-1}}+\|f_{d}\|^{2}_{G_{2}^{-1}}=(G_{1}^{-1}e_{d},e_{d})+(G_{2}^{-1}f_{d},f_{d})$.

Proof.

The mean curvature field of $T^{l,m}$ equals (see, for example, [Law80, Proposition 8])

[TABLE]

where $\Delta$ is the Laplace — Beltrami operator of the torus $T^{l,m}$ in the coordinates $(x,y)$. Since the matrices $G_{1}$ and $G_{2}$ are constant, we have

[TABLE]

where $G_{1}^{pq}$, $G_{2}^{rs}$ are the coefficients of the matrices $G_{1}^{-1}$ and $G_{2}^{-1}$ accordingly.

Consequently, we obtain

[TABLE]

for all $d=1,\dots,n$. ∎

Lemma 2.3**.**

Let $H$ be the mean curvature field of the torus $T^{l,m}$. The vector field $JH$ is tangent to $T^{l,m}$ if and only if

[TABLE]

Proof.

The vector field $JH$ is tangent to $T^{l,m}$ if and only if there exist real functions $\lambda_{1}(x,y),\dots,\lambda_{l}(x,y)$; $\eta_{1}(x,y),\dots,\eta_{m}(x,y)$ such that

[TABLE]

This is equivalent to the identities

[TABLE]

where we put $\lambda=(\lambda_{1},\dots,\lambda_{l})$, $\eta=(\eta_{1},\dots,\eta_{m})$. It is clear that $\eta$ should be zero. We have

[TABLE]

This overdetermined system of linear equations has a solution $\lambda$ (where $\lambda$ does not depend on $x$ and $y$) if and only if

[TABLE]

Recall that the vectors $e_{1},\dots,e_{n}$ define a lattice of rank $l$ in $\mathbb{R}^{l}$, hence

[TABLE]

On the other hand, we have $e_{d}^{\prime}=(e_{d};\|e_{d}\|^{2}_{G_{1}^{-1}}+\|f_{d}\|^{2}_{G_{2}^{-1}})$, so

[TABLE]

∎

Lemma 2.4**.**

Suppose that the vector field $JH$ is tangent to the torus $T^{l,m}$. Then $\delta\omega_{J}(H,\cdot)=0$.

Proof.

Notice that there is linear change of coordinates $z=Ax$ and $w=By$ on $\mathbb{R}^{l}$ and $\mathbb{R}^{m}$ accordingly such that

[TABLE]

which follows from the Gram — Schmidt process. For simplifying the proof, it is convinient to pass to the coordinates $(z,w)$.

Recall that $\omega_{J}(H,\cdot)=(JH,\cdot)$. We have (see, for example, [Bes08])

[TABLE]

where $\nabla$ is a connection on the torus $T^{l,m}$ compatible with the induced metric. Let us show that $\nabla_{\psi_{z_{p}}}JH=\nabla_{\psi_{w_{q}}}JH=0$ for any $p=1,\dots,l$ and $q=1,\dots,m$.

Actually,

[TABLE]

where $\operatorname{Pr}$ is the orthogonal projection of the vector field to the torus. Furthermore,

[TABLE]

Consequently,

[TABLE]

then

[TABLE]

for any $p,r=1,\dots,l$ and $s=1,\dots,m$.

One can prove in the same way that $\nabla_{\psi_{w_{q}}}JH=0$ for any $q=1,\dots,m$.

We obtain $\delta\omega_{J}(H,\cdot)=0$. ∎

The counterpart of Lemmas 2.3 and 2.4 for $\omega_{K}$ can be proved in the same way.

Theorem 1.3 is proved. Let us consider some examples.

Example 2.5**.**

Let us consider the torus $T^{n,m}$ in $\mathbb{H}^{n}$. Since $l=n$, the first condition of Theorem 1.3

[TABLE]

holds trivially, and the torus $T^{n,m}$ is H-minimal with respect to $\omega_{J}$.

Let $f_{1},\dots,f_{m}$ be an arbitrary basis $\mathbb{R}^{m}$, and

[TABLE]

where $f_{p}=f_{q}$ if and only if $r_{p}=r_{q}$ for any $p=1,\dots,m$; $q=m+1,\dots,n$. One can directly check that $G_{1}=\text{diag}(r_{1}^{2},\dots,r_{n}^{2})$ and $\|e_{d}\|_{G_{1}^{-1}}^{2}=1/r_{d}^{2}$. Then $f^{\prime}_{m+1},\dots,f^{\prime}_{n}\in\{f^{\prime}_{1},\dots,f^{\prime}_{m}\}$ and

[TABLE]

Consequently, $T^{n,m}$ is also H-minimal with respect to $\omega_{K}$ by Theorem 1.3.

Let us notice that the mapping

[TABLE]

defines two projections $\psi(x,0)$ and $\psi(0,y)$ on the homogeneous tori $T^{l,0}$ and $T^{0,m}$ in $\mathbb{C}^{n}$ accordingly. Necessary and sufficient conditions for their H-minimality are given by [Ovc18, Theorem 1].

Example 2.6**.**

The torus $T^{l,l}\subset\mathbb{H}^{n}$ defined by the mapping

[TABLE]

is H-minimal with respect to $\omega_{J}$ and $\omega_{K}$ simultaneously if and only if the homogeneous torus $T^{l,0}$ is H-minimal in $\mathbb{C}^{n}$.

Example 2.7**.**

The torus $T^{2,2}\subset\mathbb{H}^{4}$ defined by the mapping

[TABLE]

is not H-minimal with respect to neither $\omega_{J}$ nor $\omega_{K}$. Still, the homogeneous tori $T^{2,0}$ and $T^{0,2}$ are H-minimal in $\mathbb{C}^{4}$.

Example 2.8**.**

The torus $T^{3,3}\subset\mathbb{H}^{4}$ defined by the mapping

[TABLE]

is H-minimal with respect to $\omega_{J}$ and $\omega_{K}$ simultaneously. Still, the homogeneous tori $T^{3,0}$ and $T^{0,3}$ are not H-minimal in $\mathbb{C}^{4}$.

3. Proof of Theorem 1.6

Let us consider the torus $T^{l,m}\subset S^{4n+3}(1)$ in the unit sphere, i.e., $\sum r_{d}^{2}=1$. One can easily check that conditions of $\pi_{J}$- and $\pi_{K}$-horizontality of the torus $T^{l,m}$ are equivalent to the identities $\sum r_{d}^{2}e_{d}=0$ and $\sum r_{d}^{2}f_{d}=0$ accordingly.

To prove Theorem 1.6 we are going to use the following statement, which follows from Chen and Morvan criterion of H-minimality (see Proposition 2.1).

Lemma 3.1** (see [Ovc18]*Lemma 6).**

Let $I\subset S^{2n+1}(1)$ be a horizontal lifting of an isotropic submanifold in $\mathbb{C}\mathrm{P}^{n}$. Let us denote by $\hat{H}$ the mean curvature field of $I$ in $S^{2n+1}(1)$. Then $\pi(I)$ is H-minimal in $\mathbb{C}\mathrm{P}^{n}$ if and only if $J\hat{H}$ is a tangent vector field to $I$ and $\delta\alpha_{\hat{H}}=0$. ∎

Let $N\subset S^{4n+3}(1)$ be a compact $\pi_{J}$-horizontal $\omega_{J}$-isotropic submanifold in $\mathbb{H}^{n+1}$, $\hat{H}$ be the mean curvature field of $N$ in $S^{4n+3}(1)$. One can derive from Lemma that the submanifold $\pi_{J}(N)$ is H-minimal in $\mathbb{C}\mathrm{P}^{2n+1}$ if and only if $J\hat{H}$ is a tangent vector field to $N$ and $\delta\omega_{J}(\hat{H},\cdot)=0$ (the statement for $\omega_{K}$ is analogous).

Now we are going to prove Theorem 1.6. First of all, we need to find the mean curvature field of the torus $T^{l,m}$ in the sphere $S^{4n-1}(r)$.

Lemma 3.2**.**

The mean curvature field of the torus $T^{l,m}$ in $S^{4n-1}(r)$ has the following form:

[TABLE]

There we put $\alpha_{d}=\|e_{d}\|^{2}_{G_{1}^{-1}}+\|f_{d}\|^{2}_{G_{2}^{-1}}=(G_{1}^{-1}e_{d},e_{d})+(G_{2}^{-1}f_{d},f_{d})$.

Proof.

We have the embeddings $T^{l,m}\subset S^{4n-1}(r)\subset\mathbb{H}^{n}$. Then

[TABLE]

where P is the orthogonal projection of the vector field $H$ to $S^{4n-1}(r)$ (see [Law80]). Consequently,

[TABLE]

Let us show that $(H,\psi)=-1$. To this end, it is convinient to pass to the coordinates $(z,w)$ on $T^{l,m}$ in which

[TABLE]

(see the proof of Lemma 2.4). We have

[TABLE]

Then the statement follows from Lemma 2.2. ∎

Corollary 3.3**.**

The torus $T^{l,m}$ is minimal in $S^{4n-1}(r)$ if and only if

[TABLE]

Proposition 3.4**.**

Let $T^{l,m}$ be the torus defined by the mapping

[TABLE]

and let $T^{l,0}$ and $T^{0,m}$ be the homogeneous tori in $\mathbb{C}^{n}$ defined by projections $\psi(x,0)$ and $\psi(0,y)$ accordingly.

If the tori $T^{l,0}$ and $T^{0,m}$ are minimal in $S^{2n-1}(r)$, then the torus $T^{l,m}$ is minimal in $S^{4n-1}(r)$.

Proof.

By [Ovc18, Corollary 1] the torus $T^{l,0}$ is minimal in $S^{2n-1}(r)$ if and only if

[TABLE]

where $G_{1}$ is the matrix of the metric tensor of the torus $T^{l,0}$.

Similarly, the torus $T^{0,m}$ is minimal in $S^{2n-1}(r)$ if and only if

[TABLE]

where $G_{2}$ is the matrix of the metric tensor of the torus $T^{0,m}$.

At last, notice that $G_{1}$ and $G_{2}$ are precisely the blocks of the matrix of the metric tensor of $T^{l,m}$. ∎

Example 3.5**.**

The torus $T^{3,3}$ in $\mathbb{H}^{4}$ defined by the mapping

[TABLE]

is minimal in $S^{17}(\sqrt{5})\subset\mathbb{H}^{4}$. Still, the homogeneous tori $T^{3,0}$ and $T^{0,3}$ are not minimal in $S^{7}(\sqrt{5})\subset\mathbb{C}^{4}$ (see [Ovc18, Corollary 1]).

Lemma 3.6**.**

Let $\hat{H}$ be the mean curvature field of the torus $T^{l,m}$ in $S^{4n-1}(1)$. The vector field $J\hat{H}$ is tangent to $T^{l,m}$ if and only if

[TABLE]

Proof.

The proof is similar to the proof of Lemma 2.3. ∎

Lemma 3.7**.**

Suppose that the vector field $J\hat{H}$ is tangent to the torus $T^{l,m}$. Then $\delta\omega_{J}(\hat{H},\cdot)=0$.

Proof.

Let us notice that there is linear change of coordinages $z=Ax$ and $w=By$ on $\mathbb{R}^{l}$ and $\mathbb{R}^{m}$ accordingly such that

[TABLE]

which follows from the Gram — Schmidt process. For simplifying the proof, it is convinient to pass to the coordinates $(z,w)$.

Recall that $\omega_{J}(\hat{H},\cdot)=(J\hat{H},\cdot)$. We have (see, for example, [Bes08])

[TABLE]

where $\nabla$ is a connection on the torus $T^{l,m}$ compatible with the induced metric. Let us show that $\nabla_{\psi_{z_{p}}}J\hat{H}=\nabla_{\psi_{w_{q}}}J\hat{H}=0$ for any $p=1,\dots,l$ and $q=1,\dots,m$.

Actually,

[TABLE]

where $\operatorname{Pr}$ is the orthogonal projection of the vector field to the torus. We also put $\beta_{d}=1/r^{2}-\alpha_{d}/(l+m)$. Furthermore,

[TABLE]

Consequently,

[TABLE]

then

[TABLE]

for any $p,r=1,\dots,l$ and $s=1,\dots,m$.

One can prove in the same way that $\nabla_{\psi_{w_{q}}}J\hat{H}=0$ for any $q=1,\dots,m$.

We obtain $\delta\omega_{J}(\hat{H},\cdot)=0$. ∎

The counterpart of Lemmas 3.6 and 3.7 for $\omega_{K}$ can be proved in the same way.

Theorem 1.6 is proved. Let us consider some examples.

Recall that the mapping

[TABLE]

defines two projections $\psi(x,0)$ and $\psi(0,y)$ on the homogeneous tori $T^{l,0}$ and $T^{0,m}$ in $\mathbb{C}^{n}$ accordingly. One can easily check that conditions of $\pi_{J}$- and $\pi_{K}$-horizontality of the torus $T^{l,m}$

[TABLE]

are precisely the conditions of horizontality of the tori $T^{l,0}$ and $T^{0,m}$ accordingly.

Let us denote by

[TABLE]

the Hopf fibration. Necessary and sufficient conditions for H-minimality of the tori $\pi(T^{l,0})$ and $\pi(T^{0,m})$ in $\mathbb{C}\mathrm{P}^{n-1}$ are given by [Ovc18, Theorem 2].

Example 3.8**.**

Let $T^{l,l}\subset S^{4n+3}(1)$ be the $\pi_{J}$- and $\pi_{K}$-horizontal torus in $\mathbb{H}^{n+1}$ defined by the mapping

[TABLE]

The tori $\pi_{J}(T^{l,l})$ and $\pi_{K}(T^{l,l})$ are H-minimal in $\mathbb{C}\mathrm{P}^{2n+1}$ if and only if $\pi(T^{l,0})$ is H-minimal in $\mathbb{C}\mathrm{P}^{n}$.

Example 3.9**.**

Let $T^{2,2}$ be the torus in $\mathbb{H}^{4}$ defined by the mapping

[TABLE]

The tori $\pi_{J}(T^{2,2})$ and $\pi_{K}(T^{2,2})$ are H-minimal, but not minimal in $\mathbb{C}\mathrm{P}^{7}$. Still, the torus $\pi(T^{2,0})$ is minimal in $\mathbb{C}\mathrm{P}^{3}$, and the torus $\pi(T^{0,2})$ is H-minimal, but not minimal in $\mathbb{C}\mathrm{P}^{3}$.

Example 3.10**.**

Let $T^{2,3}$ be the torus in $\mathbb{H}^{4}$ defined by the mapping

[TABLE]

The tori $\pi(T^{2,0})$ and $\pi(T^{0,3})$ are minimal in $\mathbb{C}\mathrm{P}^{3}$ by [Ovc18, Corollary 1]. Then the minimality of $\pi_{J}(T^{2,3})$ and $\pi_{K}(T^{2,3})$ in $\mathbb{C}\mathrm{P}^{7}$ follows from Proposition 3.4.