DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 1–5

SO(n)–INVARIANT EINSTEIN

METRICS ON STIEFEL MANIFOLDS

Andreas Arvanitoyeorgos

Abstract. A Stiefel manifold V_{k}R^{n} is the set of orthonormal k-frames in R^{n}, and it is diffeomorphic to the homogeneous space SO(n)/SO(n − k). We study SO(n)-invariant Einstein metrics on this space by using the variational approach. We determine when the standard metric on SO(n)/SO(n − k) is Einstein, and we give an explicit solution for the space V_{2}R^{n}.

1. Introduction

An Einstein metric is a Riemannian metric g with the property Ric(g) = cg. In this paper we are interested in SO(n)-invariant Einstein metrics on Stiefel manifolds V_{k}R^{n} of orthonormal k-frames in R^{n}. These are diffeomorphic to the homogeneous space SO(n)/SO(n − k). The existence of SO(n)-invariant Einstein metrics on Stiefel manifolds is due to A. Sagle [Sa] and G. Jensen [Jen]. The first author reduced the problem to an algebraic system of equations, and the second used the method of Riemannian submersions. A special case is the unit tangent bundle T_{1}S^{n} = V_{2}R^{n} of S^{n}, which follows from [Ko] (see also [Be, 9.77]). Here we are interested in explicit solutions for the Einstein equation on the Stiefel manifolds V_{2}R^{n}.

First we examine when the standard metric on V_{k}R^{n} is Einstein. We use representation theory and a result from [Wa-Zi1] to show that this happens if and only if V_{k}R^{n} = S^{n} or SO(n). Next we study a certain family of SO(n)-invariant metrics on V_{k}R^{n} and obtain a unique solution for the Einstein equation on V_{2}R^{n} for that particular family. An alternative approach to these results can be found in [A].

2. Preliminaries

Let M = G/H be a compact, simply connected homogeneous space, where G is a semisimple, compact and connected Lie group, and H is a closed subgroup of G. Let g and h denote the Lie algebras of G and H respectively. We denote by Ad_{G} : G → Aut(g) (respectively ad_{g} : g → End(g)) the adjoint representation of G (respectively g).

1991 Mathematics Subject Classification. 53C25, 53C30.

Key words and phrases. Homogeneous spaces, Einstein metrics, Stiefel manifolds.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Since G is semisimple and compact the Killing form B(X, Y) = tr ad(X) ad(Y) of g is nondegenerate and negative definite on g, thus giving rise to an orthogonal decomposition of g as the direct sum g = h ⊕ m. This sum is Ad(H)-invariant, i.e. [m, h] ⊂ m. The tangent space T_{o}M of M at the identity coset o = eH can be identified with m by the map

X ↦ X^{∗}(o) = \frac{d}{d t}(exp tX · o)|_{t=0} , X ∈ m.

The restriction of −B onto the complement m induces a G-invariant Riemannian metric g_{B} on M which is called the standard metric on M.

We recall the isotropy representation χ of H in T_{o}M. For any γ ∈ G, let L_{γ} denote the diffeomorphism of G/H given by left translation. The isotropy representation χ of H on T_{o}M is defined by χ(h) = (d L_{h})_{0}. Using the identification of m with T_{o}M, this representation is identified with the adjoint representation of H on m.

3. The standard metric on Stiefel manifolds

In this section we find the Stiefel manifolds V_{k}R^{n} = SO(n)/SO(n − k), k ≥ 1 for which the standard homogeneous metric g_{B} is Einstein. For a homogeneous space G/H let χ = χ_{o} ⊕ χ_{1} ⊕ ⋯ ⊕ χ_{s} be the decomposition of the isotropy representation of H in m as a direct sum of a trivial representation χ_{o} and Ad(H)- irreducible representations χ_{i} (i = 1, …, s). Let m = m_{o} ⊕ m_{1} ⊕ ⋯ ⊕ m_{s} be the corresponding decomposition of m. According to [Wa-Zi1, Corollary 1.3] if the metric g_{B} is Einstein, then either H is trivial or m_{o} = 0.

Theorem 1. The standard metric on V_{k}R^{n} (k ≥ 1) is Einstein if and only if V_{k}R^{n} = S^{n} or SO(n).

Proof. We compute the isotropy representation for the Stiefel manifolds V_{k}R^{n} = SO(n)/SO(n − k), k ≥ 1. Let λ_{n} denote the standard representation of SO(n) (given by the natural action of SO(n) on R^{n}). If ∧^{2}λ_{n} denote the second exterior power of λ_{n}, then Ad_{SO(n)} = ∧^{2}λ_{n}. The isotropy representation χ of SO(n − k) is characterized by the property Ad_{SO(n)}|_{SO(n−k)} = Ad_{SO(n−k)} ⊕χ. We compute

Ad_{SO(n)}|_{SO(n−k)} = ∧^{2}λ_{n}|_{SO(n−k)} = ∧^{2}(λ_{n−k} ⊕ k) = ∧^{2}λ_{n−k} ⊕ ∧^{2}k ⊕ (λ_{n−k} ⊗ k),

where k denotes the trivial k-dimensional representation. The subrepresentation ∧^{2}λ_{n−k} is Ad_{SO(n−k)}, thus χ = ∧^{2}k ⊕ λ_{n−k} ⊕ ⋯ ⊕ λ_{n−k} (k summands). If we further decompose ∧^{2}k into a sum of \binom{k}{2} 1-dimensional subrepresentations, we obtain

(1)  χ = 1 ⊕ ⋯ ⊕ 1 ⊕ λ_{n−k} ⊕ ⋯ ⊕ λ_{n−k}.

Hence, in our case, if g_{B} is Einstein, then either H is trivial (i.e. V_{k}R^{n} = SO(n)) or m_{o} = 0 in which case k = 1 hence χ = λ_{n−1} (i.e. V_{k}R^{n} = SO(n)/SO(n−1) = S^{n}).

SO(n)–INVARIANT EINSTEIN METRICS ON STIEFEL MANIFOLDS 3

4. Homogeneous Einstein metrics on V_{2}R^{n}

In this section we are interested in SO(n)-invariant Einstein metrics on the Stiefel manifolds V_{2}R^{n} = SO(n)/SO(n − 2). We will use the variational approach to Einstein metrics.

Let M = G/H be a homogeneous space and

(2)  m = m_{1} ⊕ ⋯ ⊕ m_{s}

the decomposition of m into Ad(H)-invariant subspaces. We consider the family of Ad(H)-invariant diagonal metrics

(3)  ⟨ , ⟩ = x_{1} Q|_{m_{1}} ⊕ ⋯ ⊕ x_{s} Q|_{m_{s}} ,  x_{i} > 0

where Q is some multiple of the Killing form of G. If the m_{i}’s are pairwise inequivalent representations, then any G-invariant metric on M is determined by ⟨ , ⟩. Otherwise, by fixing the decomposition (2) of m there may exist G-invariant metrics on M not of this form. We will see later on that we face this problem with the Stiefel manifolds. For the following we use the notation from [Wa-Zi2].

Let S(g) denote the scalar curvature of a Riemannian metric g on M. According to [Ber], [Hil] the Einstein metrics on M are the critical points of the total scalar curvature functional T (g) = ∫_{M} S(g)d vol_{g} on the space of Riemannian metrics of

volume one. Now, let M_{G} denote the set of all G-invariant metrics of volume one on M, and note that on M_{G}, T (g) = S(g). Then the critical points of T |_{M_{G}} are precisely the G-invariant Einstein metrics of volume one [Be, Theorem 4.23].

Let {e_{α}} be a Q-orthonormal basis adapted to the decomposition (2) of m, i.e. e_{α} ∈ m_{i} for some i and α < β if i < j. We set

(4)  C_{ij}^{k} = ∑_{\begin{matrix}e_{α}∈m_{i} \\ e_{β}∈m_{j} \\ e_{γ}∈m_{k} \end{matrix}} Q([e_{α}, e_{β}], e_{γ})^{2}.

Notice that C_{ij}^{k} is independent of the Q-orthonormal bases chosen for m_{i}, m_{j} and m_{k}, but it depends on the choice of the decomposition of m. Also, C_{ij}^{k} is symmetric in all three indices. Let ⟨ , ⟩ be the Ad(H)-invariant metric having the form (3) with respect to a fixed decomposition of m. Then \{e_{α}/\sqrt{x_{i}} : e_{α} ∈ m_{i}\} is a ⟨ , ⟩- orthonormal basis of m. According to [Wa-Zi2, equation (1.3)] the scalar curvature of ⟨ , ⟩ is given by

(5)  S = \frac{1}{2} ∑_{i=1}^{s} \frac{d_{i}b_{i}}{x_{i}} − \frac{1}{4} ∑_{i,j,k} C_{ij}^{k} \frac{x_{k}}{x_{i}x_{j}}

where d_{i} = dim m_{i} and B|_{m_{i}} = −b_{i} Q|_{m_{i}}.

We now come to the Stiefel manifolds. As shown in section 3, the isotropy representation of V_{k}R^{n} is decomposed as χ = 1 ⊕ ⋯ ⊕ 1 ⊕ λ_{n−k} ⊕ ⋯ ⊕ λ_{n−k}. We observe that the first \frac{k(k−1)}{2} summands are equivalent to each other and the same is true for the remaining k summands.

We restrict ourselves to V_{2}R^{n} and we look for SO(n)-invariant Einstein metrics of the form (3). The isotropy representation is decomposed as χ = 1⊕λ_{n−2}⊕λ_{n−2}, hence these metrics depend on three parameters x_{1}, x_{2}, x_{3}, x_{i} > 0. Let Q(X, Y) = −\frac{B(X,Y)}{2(n−2)} = −\frac{tr XY}{2} and let E_{ij} denote the n × n matrix with 1 at the (ij)-entry and 0 elsewhere. Then the set B = {e_{ij} = E_{ij} − E_{ji} : 1 ≤ i ≤ k, 1 ≤ i < j ≤ n\} constitutes a Q-orthonormal basis of m (containing \frac{k(2n−k−1)}{2} = dim V_{k}R^{n} elements).

We adapt the basis B to the decomposition m = m_{1} ⊕ m_{2} ⊕ m_{3}, so that m_{1} = span{e_{12}}, m_{2} = span{e_{1j} : 3 ≤ j ≤ n}, m_{3} = span{e_{2j} : 3 ≤ j ≤ n}. The scalar curvature is given by (5) and in our case b_{1} = b_{2} = b_{3} = 2(n − 2) and d_{1} = 1, d_{2} = d_{3} = n − 2. For the computation of the numbers in (4) we need the following lemma.

Lemma. The multiplication table of the elements in B is given as follows:

[e_{ij}, e_{kl}] = \{ \begin{matrix}0 & i ≠ k, l j ≠ k, l \\ −e_{ik} & i ≠ k, l j = l \\ e_{il} & i ≠ k, l j = k \\ −e_{jl} & i = k, j ≠ k, l \\ e_{jk} & i = l, j ≠ k, l. \end{matrix}

Proof. The computation is straightforward.

By using this lemma we find that C_{12}^{3} = n − 2 and its rearrangements are the only non-zero triples. The scalar curvature is given by

S = (n − 2) [\frac{1}{x_{1}} + \frac{n − 2}{x_{2}} + \frac{n − 2}{x_{3}} − \frac{1}{2}(\frac{x_{1}}{x_{2}x_{3}} + \frac{x_{2}}{x_{1}x_{3}} + \frac{x_{3}}{x_{1}x_{2}} )]

and the volume condition is x_{1}x_{2}^{n−2}x_{3}^{n−2} = 1. Let S̃ = S − λ(x_{1}x_{2}^{n−2}x_{3}^{n−2}). Then

\frac{∂S̃}{∂x_{1}} = \frac{2(n − 2)}{x_{1}^{2}} + (n − 2)(\frac{x_{1}}{x_{2}x_{3}} − \frac{x_{2}}{x_{1}^{2}x_{3}} − \frac{x_{3}}{x_{1}^{2}x_{2}} ) − μx_{2}^{n−2}x_{3}^{n−2}

\frac{∂S̃}{∂x_{2}} = \frac{2(n − 2)}{x_{2}^{2}} − \frac{x_{1}}{x_{2}^{2}x_{3}} + \frac{x_{2}}{x_{1}x_{3}} − \frac{x_{3}}{x_{1}x_{2}^{2}} − μx_{1}x_{2}^{n−3}x_{3}^{n−2}

\frac{∂S̃}{∂x_{3}} = \frac{2(n − 2)}{x_{3}^{2}} − \frac{x_{1}}{x_{2}x_{3}^{2}} − \frac{x_{2}}{x_{1}x_{3}^{2}} + \frac{x_{3}}{x_{1}x_{2}} − μx_{1}x_{2}^{n−2}x_{3}^{n−3}.

Setting the above equations equal to zero and by using the volume condition we obtain the following system of equations:

2(n − 2)x_{2}x_{3} + (n − 2)x_{1}^{2} − (n − 2)x_{2}^{2} − (n − 2)x_{3}^{2} = 2(n − 2)x_{1}x_{3} − x_{1}^{2} + x_{2}^{2} − x_{3}^{2}
= 2(n − 2)x_{1}x_{2} − x_{1}^{2} − x_{2}^{2} + x_{3}^{2}.

Thus we have obtained:

Proposition. The Einstein equation for V_{2}R^{n} reduces to the following system of algebraic equations:

2(n − 2)x_{3}(x_{1} − x_{2}) = (n − 1)x_{1}^{2} − (n − 1)x_{2}^{2} − (n − 3)x_{3}^{2}
(n − 2)x_{1}(x_{2} − x_{3}) = x_{2}^{2} − x_{3}^{2}

This system of equations has only one positive solution, namely x_{2} = x_{3} = \frac{n−1}{2(n−2)}x_{1}. One can also check that as long as the Q-orthonormal basis B of m is adapted to the decomposition m = m_{1} ⊕ m_{2} ⊕ m_{3} so that m_{1} = span{e_{12}}, m_{2} = span{e_{ij} : 1 ≤ i, j ≤ n, j ≠ 2}, and m_{3} = span{e_{kl} : e_{kl} ≠ e_{ij}}, then the Einstein equation for V_{2}R^{n} reduces to the same algebraic system given in the previous proposition. Thus we have obtained

Theorem. The Stiefel manifolds V_{2}R^{n} = SO(n)/SO(n − 2) admit (up to scale) at least one diagonal SO(n)-invariant Einstein metric, explicitly given as x_{2} = x_{3} = \frac{n−1}{2(n−2)}x_{1}. If the Q-othonormal basis B is adapted to the decomposition (2) the way described above, then this metric is unique.

References

[A] A. Arvanitoyeorgos: Homogeneous Einstein metrics on Stiefel manifolds, to appear in Comment. Math. Univ. Carolinae (1996).

[Be] A. L. Besse: Einstein Manifolds, Springer–Verlang, Berlin 1987.

[Ber] M. Berger: Quelques formules de variation pour une structure riemannienne, Ann. Sci. Éc. Norm. Sup. 3 (1970), 285–294.

[Hil] D. Hilbert: Die Grundlagen der Physik, Nachr. Akad. Wiss. Gött. (1915), 395–407.

[Ja] I. M. James: The Topology of Stiefel Manifolds, Cambridge University Press, Lecture Note Series 24, Great Britain 1976.

[Jen] G. R. Jensen: Einstein metrics on principal fibre bundles, J. Diff. Geom. 8 (1973), 599–614.

[Ko] S. Kobayashi: Topology of positively pinched Kähler manifolds, Tôhoku Math. J. 15 (1963), 121–139.

[Sa] A. A. Sagle: Some homogeneous Einstein manifolds, Nagoya Math. J. 39 (1970), 81–106.

[Wa-Zi1] M. Wang–W. Ziller: On normal homogeneous Einstein metrics, Ann. Sci. Éc. Norm. Sup. 18 (1985), 563–633.

[Wa-Zi2] M. Wang–W. Ziller: Existence and non-existence of homogeneous Einstein metrics, Invent. math. 84 (1986), 177–194.

UNIVERSITY OF CRETE, DEPARTMENT OF MATHEMATICS GR-714 09 HERAKLION, GREECE E-mail: arvanga@talos.cc.uch.gr

Current address: THE BRITISH COUNCIL-GREECE AND UNIVERSITY OF ESSEX-ENGLAND 17 KOLONAKI SQUARE, GR-102 10 ATHENS, GREECE E-mail: taarv@prometheus.hol.gr