A counterexample to a conjecture on flat bilinear forms

Marcos Dajczer & Luis A. Florit

Abstract. We provide a counterexample to a conjecture on the dimension of the nullity of a flat symmetric bilinear form.

The algebraic theory of flat bilinear forms was developed by J. D. Moore after the seminal work of E. Cartan on exteriorly orthogonal quadratic forms as a tool to treat the “rigidity problem” for submanifolds; see [5] and references therein. An ℝ-bilinear form β: 𝕍^{n} × 𝕍^{n} → 𝕎^{p,q} into a vector space endowed with an indefinite inner product of type (p, q) is said to be flat if

⟨β(X, Y), β(Z, W)⟩ − ⟨β(X, W), β(Z, Y)⟩ = 0 for all X, Y, Z, W ∈ 𝕍^{n}.

One main goal of the theory is to estimate the dimension of the nullity space

N(β) = {X ∈ 𝕍^{n} : β(X, Y) = 0 : Y ∈ 𝕍^{n}}

of a given β that is assumed to be onto, that is, 𝕎^{p,q} = span{β(X, Y) : X, Y ∈ 𝕍^{n}}.

In [1] the following result for a symmetric bilinear form was proved.

Theorem 1. Let β: 𝕍^{n} × 𝕍^{n} → 𝕎^{q,q} be a flat symmetric bilinear form. If q ≤ 5 and β is onto then dim N(β) ≥ n − 2q.

A proof of the preceding result for q = 2 is contained in the argument by Cartan in [2]. It was conjectured around 1984 by the first author of this paper that the same estimate holds for arbitrary dimension q. A positive answer to the conjecture would have important consequences. For instance, the isometric and conformal rigidity results in [1] would hold after dropping the restriction on the codimension. Moreover, an extension of the results in [3] and [4] for arbitrary codimension with the same bounds would be possible. However, we give next a counterexample that shows that the conjecture is already false for q = 6 and that there is no linear estimate.

Theorem 2. For a given τ ∈ ℕ with τ ≥ 3 set 2p = τ(τ + 1). Then, there is an onto flat symmetric bilinear form β: 𝕍^{n} × 𝕍^{n} → 𝕎^{p,p} such that dim N(β) = n − 2p − \binom{τ}{3}.

2000 Mathematics Subject Classification. Primary 53B25. Key words and phrases. symmetric flat bilinear form, nullity.

Proof: Denote L = {1, 2, …, τ}, I = (L × L)/S(2) and J = (L × L × L)/S(3), where S(n) is the group of permutations of n elements. Then, #I = p and #J = m := \binom{τ+2}{3}. For a ∈ L, k = [(i, j)] ∈ I and s = [(u, v, w)] ∈ J, we say that a ∈ s if a ∈ {u, v, w}, and define ∗ : L × I → J by a ∗ k = [(a, i, j)]. Then either a ∉ s or there is a unique k ∈ I such that a ∗ k = s.

Let 𝕍^{n} = ℝ^{τ} ⊕ ℝ^{m}, and take bases {y_{1}, …, y_{τ}} and {n_{s} : s ∈ J} of ℝ^{τ} and ℝ^{m}, respectively. Let B_{1} = {e_{r} : r ∈ I} and B_{2} = {ê_{r} : r ∈ I} be two basis of ℝ^{p}, and consider on 𝕎^{p,p} = ℝ^{2p} the metric of type (p, p) given by ⟨e_{r}, e_{s}⟩ = ⟨ê_{r}, ê_{s}⟩ = 0, ⟨e_{r}, ê_{s}⟩ = δ_{r,s} for all r, s ∈ I. The (ordered) union of the bases B_{1} and B_{2} is called a pseudo-orthonormal basis of 𝕎^{p,p}. Define a symmetric bilinear map β as follows:

β(n_{s}, n_{r}) = 0,  r, s ∈ J,
β(y_{i}, y_{j}) = ê_{[(i,j)]}, i, j ∈ L,
β(y_{i}, n_{s}) = 0,  if i ∉ s,
β(y_{i}, n_{s}) = e_{k},  if i ∈ s and i ∗ k = s.

To prove that N(β) = 0, take x = y +n ∈ N(β) with y = ∑_{j=1}^{τ} a_{j}y_{j} and n = ∑_{s∈J} b_{s}n_{s}. Then a_{i} = ⟨β(x, y_{i}), e_{[(i,i)]})⟩ = 0, and b_{s} = ⟨β(x, y_{u}), ê_{[(v,w)]}⟩ = 0 for s = [(u, v, w)]. To see that β is flat just observe that ⟨β(y_{i}, y_{j}), β(y_{t}, n_{s})⟩ = δ_{s,[(i,j,t)]} is symmetric in i, j, t ∈ L.

References

[1] M. do Carmo and M. Dajczer, Conformal Rigidity, Amer. J. Math. 109 (1987), 963–985.

[2] E. Cartan, La déformation des hypersurfaces dans l’espace conforme réel a n ≥ 5 dimensions. Bull. Soc. Math. France 45 (1917), 57–121.

[3] M. Dajczer and L. A. Florit, Compositions of isometric immersions in higher codimension, Manuscripta Math. 105 (2001), 507–517. Erratum, Manuscripta Math. To appear.

[4] M. Dajczer and R. Tojeiro, A rigidity theorem for conformal immersions, Indiana Univ. Math. J. 46 (1997), 491–504.

[5] J. D. Moore, Submanifolds of constant positive curvature I, Duke Math. J. 44 (1977), 449–484.

IMPA – Estrada Dona Castorina, 110

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marcos@impa.br —— luis@impa.br