Reflector Spaces Over the 4-Dimensional Kaneyuki-Kozai’s Para-Hermitian Symmetric Spaces

P.M. Gadea and L. Hernández Encinas

Abstract

The reflector spaces over the 4-dimensional para-Hermitian symmetric spaces in Kaneyuki-Kozai’s classification, are determined.

Mathematics Subject Classification: 53C35, 53C50. Key words: para-Hermitian symmetric spaces, para-Kähler manifolds.

1 Introduction

As is well known, the non-simplicity of SO(4) enriches 4-dimensional Riemannian manifolds with some special features. Similarly, the non-simplicity of SO(2, 2) endows 4-dimensional neutral manifolds with special characteristics. For instance, in the almost para-Hermitian case (see [2, 3]), Jensen and Rigoli [9] have defined analogs to Riemannian twistors, called reflectors. In the case of a Riemannian 4-manifold (see [13, p. 97]), the twistor space can be defined as the set of oriented almost complex structures, and we thus have a bundle with fibre SO(4)/U(2) ≈ S^{2}. The reflector spaces are defined as the sets of either positively or negatively oriented almost para-Hermitian structures on an oriented neutral 4-manifold. So, analogously to the complex case, we have almost paracomplex spaces of real dimension 6 as total spaces of bundles over such 4-manifolds, with fiber also an almost paracomplex space, specifically one of the homogeneous spaces SO(2, 2)/B_{±}(2) ≈ S_{1}^{2}, which are the factors of the decomposition of the Grassmannian SO(2, 2)/SO(1, 1) × SO(1, 1) (see §2).

In the present paper we determine the reflector spaces over the 4-dimensional symmetric spaces corresponding to the symmetric pairs in Kaneyuki-Kozai’s table in [10]. (We shall not consider here the 4-dimensional Kaneyuki-Kozai’s symmetric spaces obtained as combinations—direct products or not—of two 2-dimensional spaces.) We give as a corollary the homotopy classification of oriented almost para-Hermitian structures on such spaces.

Balkan Journal of Geometry and Its Applications, Vol.2, No.1, 1997, pp. 41-49 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 Preliminaries

Definitions 1. An almost paracomplex manifold (M, J) is an almost product manifold (i.e. J is an (1, 1) tensor field on M with J^{2} = 1) such that the two eigenbundles T^{+}M and T^{−}M associated to the two eigenvalues +1 and −1 of J, respectively, have the same rank. An almost para-Hermitian manifold (M, g, J) is a differentiable manifold M equipped with an almost paracomplex structure J and a neutral metric g, compatible in the sense that

g(JX, Y) + g(X, JY) = 0, X, Y ∈ X (M) .

For more details on (almost) paracomplex and (almost) para-Hermitian manifolds see [3] and references therein.

On the other hand, let B_{+}(2) denote the usual almost para-Hermitian group in the 4-dimensional case

B_{+}(2) = \{( \begin{matrix}A \\ & ^{t}A^{−1} \end{matrix} ) : A ∈ GL(2, ℝ)\} ⊂ SL(4, ℝ) ;

and B_{−}(2) denote its conjugate subgroup [9, p. 430], defined by

B_{−}(2) = { A ∈ SO(2, 2) : AI_{−} = I_{−}A, I_{−} = diag(−1, 1, 1, −1) } .

The groups SO(2, 2)/B_{±}(2) are ([9, p. 430]) diffeomorphic to the Lorentz space form S_{1}^{2} ≈ S^{1} × ℝ.

The nonsimplicity of SO(2, 2) induces the decomposition of the Grassmannian G_{1,1}(2, 2) of oriented neutral planes in ℝ^{2,2} as

G_{1,1}(2, 2) ≈ SO(2, 2)/SO(1, 1) × SO(1, 1) ≈
≈ (SO(2, 2)/B_{+}(2)) × (SO(2, 2)/B_{−}(2)) ≈ S_{1}^{2} × S_{1}^{2}.

This fact determined Jensen and Rigoli to define the reflector spaces on an oriented neutral manifold: Definitions 2 ([9]). The reflector spaces r_{±}: Z_{±}(M) → M on an oriented neutral manifold M are the two bundles of positively (resp. negatively) oriented almost para-Hermitian structures on M, with respective total spaces

Z_{±}(M) = {(p, J) : J is a para-Hermitian tensor on (T_{p}M, g|_{p}) of ± orientation}

≈ SO(M) ×_{SO(2,2)} (SO(2, 2)/B_{±}(2)) ≈ SO(M)/B_{±}(2) ,

where SO(M) stands for the total space of the bundle of oriented null frames over M; and the projections are given by r_{±}(p, J) = p.

The fact that the groups SO(2, 2)/B_{±}(2) are diffeomorphic to the Lorentz space form S_{1}^{2} ≈ S^{1} × ℝ, can be easily visualized at the Lie algebra level: in fact, since we have ([7, p. 520]) the isomorphism 𝔰𝔬(2, 2) ≈ 𝔰𝔩(2, ℝ) × 𝔰𝔩(2, ℝ), from the Iwasawa decomposition of 𝔰𝔩(2, ℝ) one has

(1)  𝔰𝔬(2, 2) ≈ 𝔰𝔬(2) + ℝ + ℝ + 𝔰𝔬(2) + ℝ + ℝ ,

in such a way that any element of 𝔰𝔬(2, 2) can be written as

( \begin{matrix}& −s \\ s \end{matrix} ) + ( \begin{matrix}a \\ & −a \end{matrix} ) +^{( c )} + ( \begin{matrix}& −t \\ t \end{matrix} ) + ( \begin{matrix}b \\ & −b \end{matrix} ) +^{( d )}

so that we can write 𝔰𝔬(2, 2) ≈ 𝔤𝔩(2, ℝ) + 𝔰𝔬(2) + ℝ.

3 The reflector spaces

Proposition. The total spaces and the fibres of the reflector bundles

H/K ↪ Z_{±}(M) ≈ G/K
↓
M ≈ G/H ≈ T^{∗}N

of either positively or negatively oriented almost para-Hermitian structures over the 4-dimensional Kaneyuki and Kozai’s symmetric spaces M = G/H (which are diffeomorphic to the cotangent bundle T^{∗}N of a Riemannian symmetric space N) are diffeomorphic to the spaces G/K and H/K, respectively, given in the two tables:

H/K  G/K  M = G/H  N
1a  \frac{SL(2,ℝ)}{S(GL_{0}(1,ℝ)^{2})}  \frac{SL(3,ℝ)}{S(GL_{0}(1,ℝ)^{3})}  \frac{SL(3,ℝ)}{S(GL_{0}(2,ℝ)×GL_{0}(1,ℝ))}  S^{2}
1b \frac{SL(2,ℝ)}{S(GL(1,ℝ)^{2})} × ℤ_{2} \frac{SL(3,ℝ)}{S((GL(1,ℝ)^{2})_{0}×GL_{0}(1,ℝ))}  \frac{SL(3,ℝ)}{S(GL(2,ℝ)×GL(1,ℝ))} P_{2}(ℝ)
2  \frac{GL(1,ℂ)^{2}}{△GL(1,ℂ)}  \frac{GL(2,ℂ)}{△GL(1,ℂ)}  \frac{GL(2,ℂ)}{GL(1,ℂ)^{2}}  P_{1}(ℂ)
3  \frac{GL(2,ℂ)}{ρ(SO(3,ℂ))}  \frac{SO(4,ℂ)}{SO(3,ℂ)}  \frac{SO(4,ℂ)}{GL(2,ℂ)}  \frac{SO(4)}{U(2)}
4a  SO(2) · ℝ^{+}  SO_{0}(3, 1)  \frac{SO_{0}(3,1)}{SO(2)·ℝ^{+}}  Q_{3,1}(ℝ)
4b  \frac{SO_{0}(1,1)·ℝ^{+}}{ℤ}  \frac{SO_{0}(2,2)}{ℤ}  \frac{SO_{0}(2,2)}{SO_{0}(1,1)·ℝ^{+}}  Q_{2,2}(ℝ)
5  GL(1, ℂ)  SO(3, ℂ)  \frac{SO(3,ℂ)}{GL(1,ℂ)}  Q_{1}(ℂ)
6  GL(1, ℂ)  Sp(1, ℂ)  \frac{Sp(1,ℂ)}{GL(1,ℂ)}  \frac{Sp(1)}{U(1)}

Proof. The infinitesimal classification of para-Hermitian symmetric spaces with semisimple group was obtained in [10], from which we have the following table of para-Hermitian symmetric (semi)-simple Lie algebras:

(𝔤, 𝔥)  M_{0}^{∗} (= N)
1  (𝔰𝔩(m + n, ℝ), 𝔰𝔩(m, ℝ) + 𝔰𝔩(n, ℝ) + ℝ)  G_{m,n}(ℝ)
2  (𝔰𝔩(m + n, ℂ), 𝔰𝔩(m, ℂ) + 𝔰𝔩(n, ℂ) + ℂ)  G_{m,n}(ℂ)
3  (𝔰𝔲^{∗}(2m + 2n), 𝔰𝔲^{∗}(2m) + 𝔰𝔲^{∗}(2n) + ℝ)  G_{m,n}(ℍ)
4  (𝔰𝔲(n, n), 𝔰𝔩(n, ℂ) + ℝ)  U(n)
5  (𝔰𝔬(n, n), 𝔰𝔩(n, ℝ) + ℝ)  SO(n)
6  (𝔰𝔬^{∗}(4n), 𝔰𝔲^{∗}(2n) + ℝ)  U(2n)/Sp(n)
7  (𝔰𝔬(2n, ℂ), 𝔰𝔩(n, ℂ) + ℂ)  SO(2n)/U(n)
8  (𝔰𝔬(m + 1, n + 1), 𝔰𝔬(m, n) + ℝ)  Q_{m+1,n+1}(ℝ)
9  (𝔰𝔬(n + 2, ℂ), 𝔰𝔬(n, ℂ) + ℂ)  Q_{n}(ℂ)
10  (𝔰𝔭(n, ℝ), 𝔰𝔩(n, ℝ) + ℝ)  U(n)/O(n)
11  (𝔰𝔭(n, n), 𝔰𝔲^{∗}(2n) + ℝ)  Sp(n)
12  (𝔰𝔭(n, ℂ), 𝔰𝔩(n, ℂ) + ℂ)  Sp(n)/U(n)
13  (E_{6}^{1}, 𝔰𝔬(5, 5) + ℝ)  G_{2,2}(ℍ)/ℤ_{2}
14  (E_{6}^{4}, 𝔰𝔬(1, 9) + ℝ)  P_{2}(𝕆)
15  (E_{6}^{ℂ}, 𝔰𝔬(10, ℂ) + ℂ)  E_{6}/Spin(10) × T^{1}
16  (E_{7}^{1}, E_{6}^{1} + ℝ)  SU(8)/Sp(4) × ℤ_{2}
17  (E_{7}^{3}, E_{6}^{4} + ℝ)  T^{1} × E_{6}/F_{4}
18  (E_{7}^{ℂ}, E_{6}^{ℂ} + ℂ)  E_{7}/E_{6} × T^{1}

In the above list, G_{m,n}(𝔽) denotes the Grassmann manifold of m-planes in 𝔽^{m+n}, where F = ℝ, ℂ or ℍ. Q_{m,n}(ℝ) denotes the real quadric in P_{m+n−1}(ℝ) defined by the quadratic form of signature (m, n). Q_{n}(ℂ) denotes the complex quadric in P_{n+1}(ℂ). P_{2}(𝕆) denotes the octonion projective plane. The list on the right of the table contains those Riemannian symmetric spaces M_{0}^{∗} (which we have denoted in the present paper by N) with the property that if M = G/H is a para-Hermitian symmetric space corresponding to the symmetric pair (𝔤,𝔥) associated to the specific M_{0}^{∗}, then M is diffeomorphic to the cotangent bundle T^{∗}M_{0} of a covering manifold M_{0} of M_{0}^{∗}.

It is immediate that the respective (real) dimensions of the spaces G/H are:

\begin{matrix}1) 2mn & 2) 4mn & 3) 8mn & 4) 2n^{2} & 5) n(n − 1) & 6) 2n(2n − 1) \\ 7) 2n(n − 1) & 8) 2(m + n) & 9) 4n & 10) n(n + 1) & 11) 2n(2n + 1) & 12) 2n(n + 1) \\ 13) 32 & 14) 32 & 15) 64 & 16) 54 & 17) 52 & 18) 108. \end{matrix}

Thus, as it is easily seen, the only 4-dimensional Kaneyuki-Kozai’s symmetric spaces, corresponding to symmetric pairs in the table, are the eight spaces M = G/H appearing in the fourth column of the table of the assertion in the Proposition, which we have numbered 1a to 6. Each such M = G/H is diffeomorphic to the cotangent bundle of the corresponding 2-dimensional Riemannian symmetric space N appearing in the fifth column of that table. Since they are almost para-Hermitian manifolds, all of them are orientable and we can define their reflector spaces, which:

(i) are in fact bundles because of their structure H/K ↪ G/K → G/H; (ii) have as total space a space of real dimension 6; (iii) have fibre diffeomorphic to S_{1}^{2} (hence diffeomorphic to SO(2, 2)/B_{+}(2) and

SO(2, 2)/B_{−}(2)), so the fibre is almost paracomplex.

Now, we consider the six cases in the Proposition. The first case has two subcases, with base spaces G/H ≈ GL_{0}(3, ℝ)/GL_{0}(2, ℝ) × GL_{0}(1, ℝ) (the 0 denoting the identity component) and G/H ≈ SL(3, ℝ)/S(GL(2, ℝ) × GL(1, ℝ)), which are respectively diffeomorphic to the paracomplex projective model P_{2}(𝔹) ≈ T^{∗}S^{2} and to the reduced paracomplex projective model P_{2}(𝔹)/ℤ_{2} ≈ T^{∗}P_{2}(ℝ) (see for instance [5, 3]). The respective reflector spaces have the flag spaces in the table as total spaces.

As for the fibre in 1a, from the Iwasawa decomposition of SL(2, ℝ) = KAN we have:

\frac{SL(2, ℝ)}{S(GL_{0}(1, ℝ)^{2})} ≈ (KAN)/A ≈ KN ≈ SO(2) × ℝ ≈ S_{1}^{2} ;

and for 1b:

\frac{GL(2, ℝ) × GL(1, ℝ)}{(GL(1, ℝ)^{2})_{0} × GL_{0}(1, ℝ)} ≈ \frac{SL(2, ℝ)}{S((GL(1, ℝ)^{2})_{0})} × ℤ_{2} ≈ SO(2) × ℝ .

Note that we have bundles with fibre a Lorentzian space form and with base space a para-Kähler space form [4].

In the case 2, the base space is GL(2, ℂ)/GL(1, ℂ) × GL(1, ℂ) ≈ T^{∗}P_{1}(ℂ). Considering the usual diffeomorphism in terms of the diagonal △GL(1, ℂ), i.e. GL(1, ℂ) ≈ (GL(1, ℂ) × GL(1, ℂ))/△GL(1, ℂ), we have the reflector space as in the table, with fibre (GL(1, ℂ) × GL(1, ℂ))/△GL(1, ℂ) ≈ GL(1, ℂ) ≈ U(1) × ℝ ≈ S_{1}^{2}.

In the third case, the base space is SO(4, ℂ)/GL(2, ℂ) ≈ T^{∗}(SO(4)/U(2)) ≈ T^{∗}S^{2} (the last diffeomorphism by [14, p. 215]). We first consider [11, p. 893]:

gl(2, C) = \{( \begin{matrix}O & S \\ S & O \end{matrix} ) : O ∈ so (∈, C), S ∈ sym (∈, C)\} =

= \{( \begin{matrix}& a + bi & p + qi & r + si \\ −a − bi & & r + si & x + yi \\ p + qi & r + si & & a + bi \\ r + si & x + yi & −a − bi \end{matrix} )\} .

Now, it is immediate that the subset 𝔥 of elements obtained deleting the last row and the last column in the above expression is a Lie subalgebra of 𝔤𝔩(2, ℂ). We have

h ≈ \{( \begin{matrix}& a & b \\ −a & & r \\ p & r \end{matrix} ) + i ( \begin{matrix}& b & q \\ −b & & s \\ q & s \end{matrix} )\} ,

that is, h ≈ 𝔰𝔬(2, 1) + i 𝔰𝔬(2, 1), but we know (see for instance [1, p. 32]) that 𝔰𝔬(2, 1) is a real form of 𝔰𝔬(3, ℂ), so we have 𝔥 ≈ ρ_{∗}(𝔰𝔬(3, ℂ)), where ρ_{∗} is ([12, p. 53]) the differential of an analytic representation ρ of SO(3, ℂ) in GL(2, ℂ), and the reflector space is thus the one given in the table, where ρ denotes the given representation of SO(3, ℂ). This reflector has fibre

GL(2, ℂ)/ρ(SO(3, ℂ)) ≈ {x + yi ∈ ℂ^{∗}} ≈ GL(1, ℂ) ≈ U(1) × ℝ ≈ S_{1}^{2} ,

as it must be.

The fourth case also has two subcases. In the case 4a, the base space is SO_{0}(3, 1)/SO(2)· ℝ^{+} ≈ T^{∗}Q_{3,1}(ℝ), the cotangent bundle of the subspace of P_{3}(ℝ) obtained under the identification (x, y) ∼ (−x, −y) in S^{2} × S^{0}, that is, S^{2}. In fact, the space Q_{3,1}(ℝ) in Kaneyuki-Kozai’s notation is the space Q_{n,ν}(ℝ) = Q_{4,1}(ℝ) in Takeuchi’s notation ([15, pp. 144–8]), which is the quadric x_{0}^{2} − x_{1}^{2} − x_{2}^{2} − x_{3}^{2} = 0 in P_{3}(ℝ). As for the sense of the dot in SO(2) · ℝ^{+} see [11, p. 893]. The total space of its reflector bundle is the identity component SO_{0}(3, 1) of the homogeneous Lorentz group, and the fibre H/K is thus SO(2) · ℝ^{+} ≈ S_{1}^{2}.

In the case 4b, the base space is SO_{0}(2, 2)/SO_{0}(1, 1) · ℝ^{+} ≈ T^{∗}Q_{2,2}(ℝ), the cotangent bundle of the sphere bundle in P_{3}(ℝ) obtained under the identification (x, y) ∼ (−x, −y) in S^{1} × S^{1}, which is a new S^{1} × S^{1}. In fact, the space Q_{2,2} in Kaneyuki-Kozai’s notation is the space Q_{n,ν}(ℝ) = Q_{4,2}(ℝ) in Takeuchi’s notation ([15, pp. 144–8]), which is the quadric x_{0}^{2} + x_{1}^{2} − x_{2}^{2} − x_{3}^{2} = 0 in P_{3}(ℝ).

In order to obtain its reflector spaces, we consider the Lie algebra 𝔰𝔬(2, 2)′ isomorphic to 𝔰𝔬(2, 2), given by

𝔰𝔬(2, 2)′ = 𝔰𝔬(2) + ℝ + ℝ + 𝔰𝔬(2) + ℝ + iℝ ,

with elements

( \begin{matrix}& −s \\ s \end{matrix} ) + ( \begin{matrix}a \\ & −a \end{matrix} ) +^{( c )} + ( \begin{matrix}& −t \\ t \end{matrix} ) + ( \begin{matrix}b \\ & −b \end{matrix} ) + ( \begin{matrix}& −d \\ d \end{matrix} )

(see the decomposition 1), and its corresponding Lie group, which we shall denote by SO_{0}(2, 2)/ℤ, and write in terms of the Iwasawa decomposition as

SO_{0}(2, 2)/ℤ ≈ SL(2, ℝ) × (SL(2, ℝ)/ℤ) ≈ KAN × KAK .

The space SO_{0}(2, 2)/ℤ so defined is diffeomorphic to both total spaces of the reflector bundles over SO_{0}(2, 2)/SO_{0}(1, 1) · ℝ^{+}, since it has real dimension 6, and the fibre is (SO_{0}(1, 1) · ℝ^{+})/ℤ ≈ S_{1}^{2}.

Finally, in the fifth and sixth cases, with SO(3, ℂ)/GL(1, ℂ) ≈ T^{∗}Q_{1}(ℂ) and Sp(1, ℂ)/GL(1, ℂ) ≈ T^{∗}(Sp(1)/U(1)) as respective base spaces, we have the reflector spaces appearing in the table, both with fibre GL(1, ℂ) ≈ S_{1}^{2}.

Notice that in the 2nd, 3rd, 5th and 6th cases we are considering the cotangent bundles of the Hermitian symmetric spaces P_{1}(ℂ), SO(4)/U(2), Q_{1}(ℂ) and Sp(1)/U(1), respectively. Corollary 2. The 4-dimensional Kaneyuki-Kozai’s para-Hermitian symmetric spaces have only a homotopy class of either positively or negatively oriented almost para-Hermitian structures. Proof. The set of positively oriented (resp. negatively oriented) almost para-Hermiti-an structures on each space M = G/H in the table in the Proposition, coincides with the set of differentiable sections of the corresponding reflector space. The homotopy classes of such sections are a subset of the whole set of homotopy classes [G/H, G/K]. To compute the last sets of classes we shall determine the homotopy type (h.t.) of each of the total spaces G/K and base spaces G/H. We have the table

topology of G/K h.t. of G/H  topology of G/K h.t. of G/H
1a  SO(3) × ℝ^{3}  S^{2}  4a (SU(2)/ℤ_{2}) × ℝ^{3}  S^{2}
1b (SO(3) · N)/ℤ_{2}  P_{2}(ℝ)  4b S^{1} × S^{1} × S^{1} × ℝ^{3}  S^{1} × S^{1}
2  SU(2) × ℝ^{3}  S^{2}  5  SU(2) × ℝ^{3}  S^{2}
3  SU(2) × ℝ^{3}  S^{2}  6  SU(2) × ℝ^{3}  S^{2}

In fact, the column at the right is obtained from the fact M ≈ T^{∗}N as in the table in the Proposition, and the homeomorphisms SO(4)/U(2) ≈ Sp(1)/U(1) ≈ S^{2}. As for the column at the left, we have:

1a and 1b) Let SL(3, ℝ) = SO(3) · AN denote the Iwasawa decomposition of SL(3, ℝ). It is immediate that

SL(3, ℝ)/S(GL_{0}(1, ℝ)^{3}) ≈ (SO(3) · AN)/A ≈ SO(3) · N ≈ SO(3) · ℝ^{3} .

Similarly

\frac{SL(3, ℝ)}{S((GL(1, ℝ)^{2})_{0} × GL_{0}(1, ℝ))} ≈ (SO(3) · N)/ℤ_{2} ,

SO(3) · N as in 1a and ℤ_{2} being the subgroup {diag(1, 1, 1) ∪ diag(−1, −1, 1)} of SL(3, ℝ).

2) We have G/K ≈ (U(2) · ℝ^{4})/(U(1) · ℝ) ≈ SU(2) · ℝ^{3}.

3, 5 and 6) They follow from the isomorphisms SO(4, ℂ) ≈ SL(2, ℂ) × SL(2, ℂ) and SO(3, ℂ) ≈ Sp(1, ℂ) ≈ SL(2, ℂ) ≈ SU(2) · ℝ^{3}.

4a) It follows from SO_{0}(3, 1) ≈ SL(2, ℂ)/ℤ_{2} ≈ (SU(2)/ℤ_{2})·ℝ^{3}, where ℤ_{2} denotes the center of SU(2) (see [1, pp. 108, 513–515]).

Hence, the homotopy type of G/K is that of SO(3) for 1a, that of (SO(3) · N)/ℤ_{2} for 1b, that of SU(2) for 2, 3, 5 and 6, that of SU(2)/ℤ_{2} for 4a and that of S^{1}×S^{1}×S^{1} for 4b. Consequently, we have:

For 1a: [G/H, G/K] ≈ π_{2}(SO(3)) = 0;

For 1b: [G/H, G/K] ≈ [P_{2}(ℝ), (SO(3) · N)/ℤ_{2}] ⊂ π_{2}(SO(3)) = 0;

For 2, 3, 5 and 6: [G/H, G/K] ≈ π_{2}(SU(2)) = 0;

For 4a: [G/H, G/K] ≈ [S^{2}, SU(2)/ℤ_{2}] = π_{2}(P_{3}(ℝ)) = 0 (see [8, p. 105]).

For 4b one has [G/H, G/K] ≈ [S^{1} × S^{1}, S^{1} × S^{1} × S^{1}]. Now, according to [8, p. 4–5], given the topological spaces X, Y and Z, the C^{0} map

θ: Map (Z × X, Y) → Map (Z, Map (X, Y)) ,

which assigns to f(z, x) the C^{0} map Z → Map (X, Y), where the image of z ∈ Z is the map x ↦ f(z, x), is a homeomorphism onto its image set if the spaces are Hausdorff, and it is a bijective map if X is locally compact. Accordingly, denoting by ΩX the space of loops of a given space X, we have, applying also [8, Th. 3.6, p. 6]: Map (S^{1} × S^{1}, S^{1} × S^{1} × S^{1}) = (Map (S^{1} × S^{1}, S^{1}))^{3} ≈ (Map (S^{1}, Map (S^{1}, S^{1})))^{3} = (Map (S^{1}, Ω(S^{1}))^{3} ≈ (Map (S^{2}, S^{1}))^{3} so that

[S^{1} × S^{1}, S^{1} × S^{1} × S^{1}] ≈ (π_{2}(S^{1}))^{3} = 0 .

Finally, given two differentiable manifolds M and N, any continuous map from M to N is homotopic to a differentiable map ([6, p. 64, Prop. 4.6]; and, moreover, two differentiable maps which are homotopic by a continuous homotopy are in fact homotopic by a differentiable homotopy ([6, p. 67, Prop. 4.11]. So, we have the above results in this proof for differentiable maps, thus concluding.

Acknowledgements. Partially supported by DGICYT, Spain, under Project n. PB95–0124. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

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P. M. Gadea Institute of Mathematics and Fundamental Physics, CSIC, Serrano 123, 28006-Madrid, Spain. e-mail: pmgadea@gugu.usal.es

L. Hernández Encinas Faculty of Education, University of Salamanca, Paseo de Canalejas 169, 37008-Salamanca, Spain. e-mail: encinas@gugu.usal.es