Erratum to optimal inequalities for embedded space-times

Abstract

Because of the fact that the existence at a point of a semi-Riemannian manifold of an infimum (or supremum) of the sectional curvature of non-degenerate planes implies that all sectional curvatures are constant at this point [2], Definition 1 of [1] has to be replaced by the following. Definition 1.1. For a given set of mutually orthogonal plane sections {Lj} with dimensions (n1, . . . , nk) such that n1 + . . . + nk ≤ m, the amended scalar curvature Λ(n1, . . . , nk) in the semi-Riemannian case is given by Λ(n1, . . . , nk) = τ − {σ(L1) + . . .+ σ(Lk) | Lj a non-null plane section, Li ⊥ Lj} . With this new definition, Theorem 1 of [1] has to be replaced by the following. Theorem 1.1. Let a m-dimensional Riemannian or Lorentzian manifold (M, g) be locally and isometrically embedded in a (m + 1)-dimensional semi-Riemannian manifold (N , g) with diagonalisable Ricci tensor S (i.e., there exists an orthonormal basis {~ea} of N such that S = ∑m+1 a=1 λa~ea⊗~ea). Then, for every k ≥ 0 and every set (n1, . . . , nk) such that n1 < m and n1 + . . .+ nk ≤ m, we have ‖H‖ ≥ c(n1, . . . , nk)Λ(n1, . . . , nk)− 1 2 c(n1, . . . , nk) { m ∑