By Agnew R.P.
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Proof. 1), we deduce that g is parallel with respect to D if and only if it is D⊥ –parallel with respect to D, that is, (DQ X g)(QY, QZ) = 0, ∀ X, Y, Z ∈ Γ (T M ). 5 Intrinsic and Induced Linear Connections on Semi–Riemannian... 37) = g(∇QY Q X, QZ) + g(QY, ∇QZ Q X). 35) we obtain the equivalence of (i) and (ii). 37) is equivalent to 0 = g(Q X, ∇QY QZ + ∇QZ QY ) = 2g(Q X, hs (QY, QZ)), which completes the proof of the theorem. So far we have obtained characterizations of two important classes of distributions on (M, g).
9) respectively. 10) + g([X, Y ], Z) − g([Y, Z], X) + g([Z, X], Y ), for any X, Y, Z ∈ Γ (T M ). 10) is the well known Levi–Civita connection which was considered as a miracle of semi–Riemannian geometry (cf. O’Neill [O83], p. 60). 10). 12) where g cd are the entries of the inverse matrix of [gcd ]. 1 the local coefﬁcients for the linear connection D. Next we consider an (n+p)–dimensional semi–Riemannian manifold (M, g) and suppose that (D, g) is a semi–Riemannian n–distribution on M . Then (D⊥ , g) is a semi–Riemannian p–distribution on M .
6. The curvature tensors R and R∗ of Levi–Civita and Vr˘ anceanu connections are related by R(X, Y )Z = R∗ (X, Y )Z + (∇∗X h )(QZ, Q Y ) −(∇∗Y h )(QZ, Q X) + (∇X h )(Y, Q Z) −(∇Y h )(X, Q Z) + (∇∗X h)(Q Z, QY ) −(∇∗Y h)(Q Z, QX) + (∇⊥ X h)(Y, QZ) −(∇⊥ Y h)(X, QZ) + h (h (QZ, Q Y ), Q X) −h (h (QZ, Q X), Q Y ) + h (X, h(Y, QZ)) −h (Y, h(X, QZ)) + h (QZ, Q T ∗ (X, Y )) +h (T ◦ (X, Y ), Q Z) + h(h(Q Z, QY ), QX) −h(h(Q Z, QX), QY ) + h(X, h (Y, Q Z)) −h(Y, h (X, Q Z)) + h(Q Z, QT ∗ (X, Y )) +h(T ◦ (X, Y ), QZ), for any X, Y, Z ∈ Γ (T M ).