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Proof of Proposition 2.3. Let V2 X(TM )b e given by ^g(Vv;X )= g(c(X);Kv )+ g(v;rd(X)K), for every X 2 TvTM .T hus we have ^g(Vv; Av) = 1, at any v 2 CK(M ), and so Dv = SpanfVv; Avg is a Lorentzian vector subspace of TvTM (recall that ^g(A; A )=0o nCKM ). One easily checks that X 2 TvCKM if and only if ^g(Av;X )= ^g(Vv;X) = 0. Therefore, (CKM;^g )i s aL orentzian manifold. The proof for spacelike bres remains valid. Finally, (Tv(CKM )p)? = fX 2 TvTM : c(X )= g(v;rd(X)K)vg; and so restricted to CKM is a semi-Riemannian submersion.

Erratum to ``A Berger-Green type inequality for compact Lorentzian manifolds"