Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 365–367

DECOMPOSABLE ELEMENTS IN EIGENSPACES

OF THE CURVATURE OPERATOR

Mariusz Frydrych

Abstract. We find an algebraic obstruction preventing the flatness of the normal bundle of any isometric immersion of the Riemannian manifold M into any space of constant sectional curvature κ ∈ ℝ of an arbitrary dimension N > dim M. The obstruction is expressed in terms of the curvature operator of the Riemannian manifold M. Keywords. Riemannian geometry, curvature operator. MS classification. 53 C 40, 58 C 25.

1. Result

Fix a Riemannian manifold (M, g) of dimension n ≥ 4 and a point x ∈ M.

Theorem. If, for the curvature operator

R_{x} : Λ^{2} T_{x} M → Λ^{2} T_{x} M,

there exists a one-dimensional eigenspace E (E = ker(R_{x}−λ I) for some λ ∈ R) spanned by an indecomposable element ω_{0} ∈ Λ^{2} T_{x} M, then there exists an open neighbourhood W of x such that, for each open neighbourhood U ⊂ W of x, any N > n, κ ∈ ℝ and for any isometric immersion

f : (U, g|U) → K^{N}(κ)

of the Riemannian manifold U into the space K^{N}(κ) of constant sectional curvature of dimension N and curvature κ, the normal bundle νf → U is non-flat everywhere (i.e. at any point y ∈ U).

Proof. The eigenvalue λ_{0} corresponding to the eigenvector ω_{0} is a single root, so from the implicit function theorem we get a smooth funct ion λ : W_{0} → ℝ such that, for any point y ∈ W_{0}, λ(y) is an eigenvalue of multiplicity one of the operator R_{y}, and λ(x) = λ_{0} , where W_{0} is some open neighbourhood of x. It is well known that the set of decomposable elements in Λ^{2} T M is closed. The implicit function theorem brings us again to the conclusion that there exist an open neighbourhood W ⊂ W_{0} of x and a smooth section ω : W → Λ^{2} T M such that, for an arbitrary y ∈ W, ω(y) is the indecomposable eigenvector of the operator R_{y}, corresponding to the single eigenvalue λ(y), and ω(x) = ω_{0}.

Typeset by AMS-TEX

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

Suppose that the normal bundle ν f is flat at some point y ∈ U and choose an orthonormal frame ξ_{1}, …, ξ_{N−n} of ν_{y} f.

The flatness of the normal bundle at the point y via the Ricci equation implies that all the shape operators commute with each other i.e. [A_{ξ}, A_{η}] = 0 where ξ, η ∈ ν_{y} f. The set {A_{ξ_{1}}, …, A_{ξ_{N−n}}}, as the intertwining family of self-adjoint operators, is simultaneously diagonalizable by an orthonormal basis, say X_{1}, …, X_{n} ∈ T_{y} M. So, we infer at once that our shape operators fulfil the following conditions:

A_{ξ_{s}} X_{j} = r_{s j} · X_{j} for j = 1, …, n; s = 1, …, N − n,

where r_{s j} ∈ ℝ are associated eigenvalues.

Now, the Gauss equation yields the formula

⟨R(X_{i}, X_{j})X_{k}, X_{l}⟩

= (δ_{i k}δ_{j l} − δ_{i l}δ_{j k}) · κ + ∑_{s=1}^{N−n}(⟨A_{ξ_{s}}X_{i}, X_{k}⟩ · ⟨A_{ξ_{s}}X_{j}, X_{l}⟩ − ⟨A_{ξ_{s}}X_{i}, X_{l}⟩ · ⟨A_{ξ_{s}}X_{j}, X_{k}⟩)

= (δ_{i k}δ_{j l} − δ_{i l}δ_{j k}) · (κ + ∑_{s=1}^{N−n} r_{s i} · r_{s j}).

The last equation shows that the curvature operator R_{y} can be diagonalized by the decomposable orthonormal basis X_{i} ∧ X_{j} ∈ Λ^{2} T_{y} M, i < j.

This contradicts to the fact that ω(y) ∈ Λ^{2} T_{y} M is the indecomposable eigenvector generating the one-dimensional eigenspace of R_{y}. □

2. Application

In order to show that the assumptions of the above theorem are non-void, we shall give an example:

Example. Let (M, g) be the Kaehler manifold with constant holomorphic sectional curvature c ≠ 0 of complex dimension m ≥ 2 (for instance, the complex projective space ℂ P^{m} with the Fubini-Study metric [G], [KN]). The spectral decomposition of the curvature operator is the following ([B], [W], [G]). There are three eigenvalues λ_{0} = 0, λ_{1} = c(m + 1)/2, λ_{2} = c/2 of ranks m(m − 1), 1, (m + 1)(m − 1), respectively. The single eigenvalue λ_{1} corresponds to the Kaehler form ω ∈ Λ^{1,1} M, ω(X, Y) = g(X, J Y) where by J we denote the complex structure of M; λ_{2} is associated with the subspace of primitive two-forms in Λ^{1,1} M; and λ_{0} corresponds to the subspace of two-forms η ∈ Λ^{2} M which satisfy the condition η(J X, J Y) = −η(X, Y). We immediately infer that the eigenvector

ω = ∑_{j=1}^{m} X_{j}^{∗} ∧ J X_{j}^{∗}

(Kaehler form) is indecomposable and generates the one-dimensional eigenspace associated with the eigenvalue λ_{1} , where by {X_{1}^{∗}, …, X_{m}^{∗}, J X_{1}^{∗}, …, J X_{m}^{∗}} we denote the dual (real) basis to the orthonormal adapted basis {X_{1}, …, X_{m}, J X_{1}, …, J X_{m}}.

References

[BGM] M. Berger, P. Gauduchon, E. Mazet, Le Spectre d’une Variete Riemannienne, Lecture Notes in Mathematics 194 (1971).

[B] A. Besse, Einstein Manifolds, Springer, 1987.

[C] S.S. Chern, On curvature and characteristic classes of a Riemann manifold, Abhandlungen Math. Sem. Univ. Hamburg 20 (1956), 117–126.

[G] A. Gray, Tubes, Addison-Wesley Publishing Company, 1990.

[KN] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Interscience Wiley, New York, 1963,1969.

[K] R.S. Kulkarni, On Bianchi identities, Math. Ann. 199 (1972), 175–204.

[W] R.O. Wells, Differential analysis on complex manifolds,, Graduate Texts in Mathematics, Springer, 1979.

Mariusz Frydrych Institute of Mathematics Łódź University ul. S. Banacha 22 90–238 Łódź, Poland E-mail: FRYDRYCH@plunlo51.bitnet

Received 5 November 1992 Revised 10 January 1993