Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 1, 267-274.

Generalized Hopf Constructions and Eigenmaps between Spheres

Ye-Lin Ou*

Department of Mathematics, Guangxi University for Nationalities 530006 Nanning, P. R. China e-mail: ylou@gxu.edu.cn

Abstract. In this paper, we give two kinds of generalizations of Hopf constructions, which can be used to produce many quadratic eigenmaps between spheres (Theorem 2.3) including some well-known examples (e.g. Veronese map) and some new ones. We also obtain some non-existence results (Theorems 3.2, 3.3) for eigenmaps into low dimensional spheres. Finally, we give an existence result (Theorem 3.6) for quadratic eigenmaps S^{n−2} × S^{1} → S^{n−1}. MSC 1991: 58E20 Keywords: harmonic maps, Hopf constructions, spheres, eigenmaps

1. Introduction

A map between spheres φ: S^{m} → S^{n} is called an eigenmap with eigenvalue λ if it is a harmonic map satisfying |dφ|^{2} ≡ λ. It is well-known that if ϕ: R^{m+1} → R^{n+1} is a map whose components are given by harmonic polynomials of homogeneous degree k and ϕ restricts to a map of spheres φ: S^{m} → S^{n} the φ is an eigenmap with eigenvalue λ_{k} = k(k + m − 1). In fact, the description of all eigenfunctions of the Laplacian on S^{m} implies that any eigenmap φ: S^{m} → S^{n} is of this form [4]. We say φ is a λ_{k}-eigenmap induced by ϕ. A map φ: S^{m} → S^{n} is said to be full if the image of φ is not contained in any proper great sphere of S^{n}. Clearly, if φ is induced by ϕ: R^{m+1} → R^{n+1}, the φ is full if and only if the n + 1 component functions of ϕ are linearly independent.

The classification of eigenmaps between spheres is an interesting problem in the study of harmonic maps, see R. T. Smith [16], Eells and Lemaire [8,9,10], and the recent works [12], [19]. The problem is still far from being solved except for the following few cases:

(i) For φ: S^{2} → S^{n}, we have Calabi’s rigidity theorem [5] which asserts that any full λ_{k}-eigenmap from S^{2} must be the standard minimal immersion f_{λ_{k}}: S^{2} → S^{2k}.

* Supported by the NSF of Guangxi (9611004) P.R.China (ii) For φ: S^{3} → S^{n}, Toth [17] obtains a complete classification of full λ_{2}-eigenmaps. (iii) For φ: S^{m} → S^{2}, Gauchman and Toth [12] give a rigidity theorem saying that any full

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1999 Heldermann Verlag

λ_{2}-eigenmap is isometric to the classical Hopf fibration S^{3} → S^{2}. Concerning the construction of λ_{2}-eigenmaps, the following methods are known:

a) Gradients of cubic eiconals (see e.g. [11, VIII (1.23)]): A k-homogeneous polynomial F: R^{m+1} → R is an eiconal if its gradient satisfies |∇F (x)|^{2} = |x|^{2k−2} for all x ∈ R^{m+1}. Clearly φ = ∇F is a (k − 1)-homogeneous polynomial map S^{m} → S^{m}. Furthermore, if F is also harmonic, then φ is an eigenmap. All cubic harmonic eiconals were classified by Cartan [6], and their gradients produce λ_{2}-eigenmaps S^{m} → S^{m} for m = 4, 7, 13, 25.

b) The classical Hopf constructions: Let μ: R^{p} × R^{p} → R^{n} be an orthogonal multiplication (see [11] and [18] for existence of such orthogonal multiplications), i.e. an R-bilinear map satisfying |μ(x, y)|^{2} = |x|^{2}|y|^{2}, ∀(x, y) ∈ R^{p} × R^{p}. Then the classical Hopf construction on μ is the map defined by H_{μ}: R^{p} × R^{p} → R^{n+1}, H_{μ}(x, y) = (|x|^{2} − |y|^{2}, 2μ(x, y)); this induces an eigenmap S^{2p−1} → S^{n}. The standard multiplications of the real algebras of real, complex, quaternionic and Cayley numbers are examples of orthogonal multiplications; their Hopf constructions induce, respectively, the well-known Hopf fibrations S^{3} → S^{2}, S^{7} → S^{4}, S^{15} → S^{8}, which are eigenmaps with eigenvalues λ_{2} = 8, 16, 32. The concept of orthogonal multiplications has been generalized, by Baird and the author [3], to multilinear norm-preserving mappings; these can be used to construct some interesting harmonic maps and morphisms. On the other hand, Ding [7] uses the so-called α-Hopf constructions to give many harmonic (not necessarily eigen-) maps between spheres.

c) In [12], a construction of raising the source dimension is given so that for any λ_{2}- eigenmap φ: S^{m} → S^{n} one can construct another \overline{φ}: S^{m+1} → S^{m+n+2}; [12] also provides the so-called harmonic product construction.

d) Let φ_{1}: S^{m−1} → S^{p−1} and φ_{2}: S^{n−1} → S^{q−1} be two λ_{2}-eigenmaps, and let μ: R^{m} × R^{n} → R^{r} be an orthogonal multiplication. Then it is easily checked that f(x, y) := (φ_{1}(x), φ_{2}(y), \sqrt{2}μ(x, y)) induces a λ_{2}-eigenmap S^{m+n−1} → S^{p+q+r−1}. By using the existence of orthogonal multiplication of the form μ: R^{2} × R^{n} → R^{r}, Ueno [19] is able to construct some new examples of λ_{2}-eigenmaps between spheres.

2. Generalized Hopf constructions

Let μ: R^{p} × R^{p} → R^{n} be an orthogonal multiplication, and k ∈ N. We introduce the following two kinds of generalized Hopf constructions:

Definition 2.1. i) The kp-Hopf construction on μ is a map F_{1}: R^{kp} ≅ \underbrace{R^{p} × ⋯ × R^{p}}_{k} → R^{\frac{1}{2} k(k−1)(n+1)}, (k ≥ 2), defined by

(1)  F_{1}(x_{1}, …, x_{k}) = \frac{1}{\sqrt{k − 1}} (|x_{i}|^{2} − |x_{j}|^{2}, \sqrt{2k}μ(x_{i}, x_{j}))_{1≤i<j≤k} .

ii) The 2kp-Hopf construction on μ is a map F_{2}: R^{2kp} ≅ \underbrace{R^{p} × ⋯ × R^{p}}_{2k} → R^{k(2k−1)n+k},

(k ≥ 1), defined by

(2) F_{2}(x_{1}, …, x_{2k}) = (|x_{2i−1}|^{2} − |x_{2i}|^{2}, 2μ(x_{2i−1}, x_{2i}), \sqrt{2}μ(x_{t}, x_{s}))_{{1≤i≤k, 1≤t<s≤2k: \atop (t,s)≠(2i−1,2i)}} .

Remark 2.2. Clearly, the kp-Hopf construction for k = 2 and the 2kp-Hopf construction for k = 1 are both identical to the classical Hopf construction. When p = n = 2, μ: C × C → C with μ(z, w) = z\overline{w}, and k = m + 1 (m as in [12]), then F_{1} is the generalized Hopf map mentioned in [12], whilst F_{2} is different from the generalized complex Veronese map given in [12].

Theorem 2.3. Let μ: R^{p}×R^{p} → R^{n} be an orthogonal multiplication. Then the generalized Hopf constructions F_{1} and F_{2} induce, respectively, the following full λ_{2}-eigenmaps

(3)  f_{1}: S^{kp−1} → S^{\frac{k}{2} (k−1)n+k−2} ,
(4)  f_{2}: S^{2kp−1} → S^{k(2k−1)n+k−1} .

Proof. A direct computation shows that |F_{1}(x_{1}, …, x_{k})|^{2} = (|x_{1}|^{2} + ⋯ + |x_{k}|^{2})^{2}, |F_{2}(x_{1}, …, x_{2k})|^{2} = (|x_{1}|^{2} + ⋯ + |x_{2k}|^{2})^{2}. Thus, F_{1} and F_{2} indeed induce λ_{2}-eigenmaps between spheres. We need only to check the fullness and the dimensions of the range spheres. Note that the components of orthogonal multiplication μ(x_{i}, x_{j}) are all bilinear functions in x_{t} and x_{s}, and it is not difficult to see that those components of F_{1}(x_{1}, …, x_{k}) given by (\sqrt{2k}μ(x_{i}, x_{j}))_{1≤i<j≤k} form a linearly independent set consisting of \frac{1}{2}k(k−1)n el-

ements. On the other hand, the set of components, given by (|x_{i}|^{2} − |x_{j}|^{2})_{1≤i<j≤k} contains only k − 1 linearly independent elements. It is easily checked that no linear combination of these k − 1 elements can be linearly represented by the above-mentioned linearly independent set. It follows that the maximum linearly independent set of all components of F_{1} contains \frac{1}{2}k(k − 1)n + k − 1 elements. Therefore F_{1} induces a λ_{2}-eigenmap f_{1}. The proof of the statement on F_{2} is similar and is omitted. □

Remark 2.4. The classical Hopf constructions can only produce full λ_{2}-eigenmaps from odd-dimensional spheres whilst the generalized Hopf constructions can produce full λ_{2}- eigenmaps from both even- and odd-dimensional spheres which include some well-known examples as well as some new ones as the following examples show.

Example 2.5. Let μ: R × R → R, μ(x, y) = xy be the multiplication of real numbers. Then the kp-Hopf construction for k = 3 is given by

F_{1}(x, y, z) = \frac{1}{\sqrt{2}} (x^{2} − y^{2}, x^{2} − z^{2}, y^{2} − z^{2}, \sqrt{6}xy, \sqrt{6}xz, \sqrt{6}yz) ,

which induces a λ_{2}-eigenmap f_{1}: S^{2} → S^{5}. However, f_{1} is not full as one can check that its image is contained in the proper great sphere of S^{5} cut out by the hyperplane orthogonal to the vector ν = (1, −1, 1, 0, 0, 0) ∈ R^{6}. In fact, one can further check that after the orthogonal transformation of coordinates

(ε_{1}, …, ε_{6}) = (e_{1}, …, e_{6}) ( \begin{matrix}\begin{matrix}\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ −\frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & −\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \end{matrix} & 0 \\ 0 & 1 0 0 0 1 0 0 0 1 \end{matrix} ) ,

F_{1} can be written as F_{1}(x, y, z) = (0, \frac{1}{2}(x^{2} + z^{2} − 2y^{2}), \frac{1}{2} \sqrt{3}(x^{2} − z^{2}), \sqrt{3}xy, \sqrt{3}xz, \sqrt{3}yz) ∈ S^{4}. This is the well-known Veronese map.

Example 2.6. For k = 2, the 2kp-Hopf construction on the real multiplication is given by F_{2}: R^{4} → R^{8}, with F_{2}(x_{1}, x_{2}, x_{3}, x_{4}) = (x_{1}^{2} −x_{2}^{2}, x_{3}^{2} −x_{4}^{2}, 2x_{1}x_{2}, 2x_{3}x_{4}, \sqrt{2}x_{1}x_{3}, \sqrt{2}x_{1}x_{4}, \sqrt{2}x_{2}x_{3}, \sqrt{2}x_{2}x_{4}), which induces a full λ_{2}-eigenmap S^{3} → S^{7}.

Example 2.7. For k = 4, the kp-Hopf construction on the real multiplication is defined by F_{1}: R^{4} → R^{12} with

\begin{matrix}F_{1}(x_{1}, …, x_{4}) = \frac{1}{\sqrt{3}}( & x_{1}^{2} − x_{2}^{2}, x_{1}^{2} − x_{3}^{2}, x_{1}^{2} − x_{4}^{2}, x_{2}^{2} − x_{3}^{2}, x_{2}^{2} − x_{4}^{2}, x_{3}^{2} − x_{4}^{2}, \\ \sqrt{8}x_{1}x_{2}, \sqrt{8}x_{1}x_{3}, \sqrt{8}x_{1}x_{4}, \sqrt{8}x_{2}x_{3}, \sqrt{8}x_{2}x_{4}, \sqrt{8}x_{3}x_{4}) \end{matrix}

which induces a λ_{2}-eigenmap S^{3} → S^{11}. Similarly to Example 2.4, one can check that the induced map is not full, and after an orthogonal change of coordinates, F_{1} can be written as

\begin{aligned}F_{1}(x_{1}, …, x_{4}) = ( & \frac{1}{3} (3x_{2}^{2} − x_{1}^{2} − x_{3}^{2} − x_{4}^{2}) , \frac{\sqrt{2}}{3} (2x_{3}^{2} − x_{1}^{2} − x_{4}^{2}) , \frac{\sqrt{2}}{3} (x_{1}^{2} − x_{4}^{2}) , \\ & \begin{matrix}√ & \overline{8} \\ √ & \overline{3} \end{matrix}x_{1}x_{2}, \begin{matrix}√ & \overline{8} \\ √ & \overline{3} \end{matrix}x_{1}x_{3}, \begin{matrix}√ & \overline{8} \\ √ & \overline{3} \end{matrix}x_{1}x_{4}, \begin{matrix}√ & \overline{8} \\ √ & \overline{3} \end{matrix}x_{2}x_{3}, \begin{matrix}√ & \overline{8} \\ √ & \overline{3} \end{matrix}x_{2}x_{4}, \begin{matrix}√ & \overline{8} \\ √ & \overline{3} \end{matrix}x_{3}x_{4}, 0, 0, 0) ∈ S^{8}. \end{aligned}

Thus F_{1}, in fact, induces a full λ_{2}-eigenmap S^{3} → S^{8}, which is isometric to the standard minimal immersion as in the classification of Toth [17].

Corollary 2.8. For natural numbers m and k ≥ 2, and the pairs (p, n) = (1,1), (2,2), (3,4), (3,7), (3,8), (3,9), (4,4), (5,8), (6,8), (7,8), (8,8), (9,16), (10,16), (11,20), (11,26), (12,20), (12,26), (13,24), (13,28), (14,24), (14,32), (15,24), (15,32), (16,24), (16,32), (17,32), etc., there exist full λ_{2}-eigenmaps

\begin{matrix}f_{1}: S^{kp−1} → S^{\frac{k}{2} (k−1)n+k−2}, & f_{2}: S^{2mp−1} → S^{m(2m−1)n+m−1}, \\ f_{3}: S^{kp} → S^{kp+ \frac{1}{2} k(k−1)n+k−1}, & f_{4}: S^{2mp} → S^{m[2p+(2m−1)n+1]}, \end{matrix}

and λ_{3}-eigenmaps

\begin{matrix}S^{kp−1} → S^{\frac{1}{2} k(k−1)(nk+2)p−1}, & S^{kp} → S^{k(kp+1)[p+1+ \frac{1}{2} (k−1)n]−1}, \\ S^{2mp−1} → S^{2m^{2}p[(2m−1)n+1]−1}, & S^{2mp} → S^{(2mp+1)\{2mp+1+m[(2m−1)n+1]\}−1}. \end{matrix}

Proof. f_{1} and f_{2} are obtained by using Theorem 2.3 and the fact that for the above pairs of (p, n) values, there exist (see [1], [14]) full orthogonal multiplications of the form R^{p} × R^{p} → R^{n}, whilst f_{3} and f_{4} are given by raising the source dimensions (see [12]) of f_{1} and f_{2}. Finally the degree raising method [12] applies to give λ_{3}-eigenmaps. □

Remark 2.9. 1) Let p = n = 2. The generalized Hopf constructions provide full λ_{2}- eigenmaps S^{7} → S^{14}, S^{7} → S^{13}, which cannot be obtained by the methods of [12] or [19].

2) In [17], Toth shows that quadratic harmonic maps S^{3} → S^{n} exist if and only if 2 ≤ n ≤ 8 and n ≠ 3, and that for such n, there is just one class of quadratic harmonic maps up to suitable equivalence. By appropriate choice of p, n and k, the generalized Hopf constructions produce representatives for four of the six classes, namely n = 2, 4, 7, 8.

By using Corollary 2 in [15] we immediately obtain the following

Corollary 2.10. For any pairs of eigenmaps of the same degree constructed by Corollary 2.8 there exists an equivariant harmonic join, e.g. f_{1} ∗ f_{2}: S^{3kp−1} → S^{\frac{1}{2} k(5k−3)n+2(k−1)}.

3. Some existence and non-existence results

In this section, the following lemma will be used for several times.

Lemma 3.1. If m ≥ 2n, then every polynomial map S^{m} → S^{n} is a constant map.

Proof. See Eells and Ratto [11, Corollary VIII: 3.16]. □

Theorem 3.2. For m ≥ 8, n ≤ 7, there exists no full eigenmap S^{m} → S^{n}.

Proof. First note that a constant map is never full and that the composition of two full maps is again full. Thus, it follows from Lemma 3.1 that there exists no full polynomial map S^{m} → S^{n} (m ≥ 8, n = 1, 2, 3, 4). The proof of the theorem is completed by the following

Claim. There exists no full polynomial map S^{m} → S^{n} (m ≥ 8, n = 7, 6, 5).

Proof of Claim. By using the multiplication of quaternions and its restrictions we know that there exist full orthogonal multiplications R^{4} × R^{4} → R^{4}, R^{3} × R^{4} → R^{4}, R^{3} × R^{3} → R^{4}, on which the Hopf constructions induce full quadratic (not necessarily harmonic) maps: S^{7} → S^{4}, S^{6} → S^{4}, S^{5} → S^{4} respectively. If there were a full polynomial map S^{m} → S^{n} (m ≥ 8, n = 7, 6, 5), then, by composing it with one of the above three full quadratic maps, there would be a full (and hence non-constant) polynomial map S^{m} → S^{4} (m ≥ 8), which is impossible by Lemma 3.1. This contradiction gives the proof of the claim. □

In contrast to the rigidity theorem [12] for full λ_{2}-eigenmaps S^{m} → S^{2}, we have

Theorem 3.3. There exists no full λ_{2}-eigenmap S^{m} → S^{3} for any m ∈ N.

Proof. It follows from Lemma 3.1 that the possible domain dimensions of a full λ_{2}-eigenmap φ: S^{m} → S^{3} are m = 5, 4, 3, 2, 1. Note that the Hopf fibration S^{3} → S^{2} is full. Thus, by using the composition and Lemma 3.1 again we can exclude the cases for m = 5, 4. On the other hand, it is known [17] that there exists no λ_{2}-eigenmap S^{3} → S^{3}. The case for m = 1 can be excluded by the fact (see e.g. [12]) that the maximum range dimension for a full λ_{2}-eigenmap S^{m} → S^{n} is \frac{m(m+3)}{2} − 1. Now we are left with the only possible case m = 2, which, however, can be excluded by applying Calabi’s rigidity theorem [5].

Remark 3.4. In their attempt to list all full λ_{2}-eigenmaps S^{5} → S^{n}, the authors of [12] and [19] raise the existence question for full λ_{2}-eigenmaps S^{5} → S^{3}. Theorem 3.3 gives a negative answer to that question.

For higher degree eigenmaps, Eells raised the following questions during the first International Conference on “Harmonic morphisms and the related topics” held in July 1997 in Brest (France):

Problem 1. Is there any λ_{k}-eigenmap S^{m} → S^{n} for k ≥ 3 and m ≥ n?

Problem 2. If S^{m} → S^{n} is a λ_{k}-eigenmap for k ≥ 3, and m ≥ n, then it is a locally trivial fibration. Is it a Hopf fibration?

Now we give an example which gives a positive answer to Problem 1 and a negative answer to Problem 2.

Example 3.5. Let ϕ: S^{4} → S^{4} be the gradient map of the first cubic harmonic eiconal F: R^{5} → R with

F (x) = \frac{1}{3}x_{1}^{3} − x_{1}x_{2}^{2} + \frac{1}{2}x_{1}(x_{3}^{2} + x_{4}^{2} − 2x_{5}^{2}) + \frac{\sqrt{3}}{2}x_{2}(x_{3}^{2} − x_{4}^{2}) + \sqrt{3}x_{3}x_{4}x_{5},

and φ: S^{7} → S^{4} be the Hopf fibration. It is known that ϕ is a λ_{2}-eigenmap and φ is a harmonic morphism (for definition of this interesting special harmonic maps see e.g. [10, (2.30)]) which is also a λ_{2}-eigenmap. By using the composition law [10, (2.31)] we see that ϕ ∘ φ: S^{7} → S^{4} is an eigenmap of degree 4, which is obviously not a Hopf fibration.

Theorem 3.6. For n = 2s, (s ≥ 2), there exist full quadratic eigenmaps S^{n−2} × S^{1} → S^{n−1}.

Proof. It is well-known (see e.g. [11]) that any orthogonal multiplication R^{p} × R^{q} → R^{n} restricts to a bi-eigenmap [11, VIII (2.11)] S^{p−1} × S^{q−1} → S^{n−1}. On the other hand, it follows from [19] that there exists an orthogonal multiplication R^{p} ×R^{2} → R^{n} for any even n and any p. Now let n = 2s and p = n − 1 we obtain the required results. □

Example 3.7. It can be checked that μ: R^{7} × R^{2} → R^{8} with

\begin{aligned}μ(x_{3}, …, x_{9}; x_{1}, x_{2}) = ( & − x_{1}x_{5} + x_{2}x_{7}, x_{1}x_{6} − x_{2}x_{8}, x_{1}x_{3} + x_{2}x_{9}, −x_{1}x_{4}, \\ & − x_{1}x_{9} − x_{2}x_{3}, −x_{2}x_{4}, x_{1}x_{7} + x_{2}x_{5}, −x_{1}x_{8} − x_{2}x_{6}) \end{aligned}

is a full orthogonal multiplication. Thus, it restricts to a full quadratic eigenmap S^{6}×S^{1} → S^{7}.

Remark 3.8. Other related constructions of harmonic maps S^{n−2} × S^{1} → S^{n−1} have been found by Hsiang and Yu [13] and by Baird and Eells [2].

Acknowledgments. I would like to thank J. C. Wood for sending me reference [19]. I am also grateful to the referee for some useful comments and suggestions which help to clarify and refine the original manuscript.

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Received December 1, 1997, revised version March 20, 1998