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Is there a formula for calculating the mean curvature tensor using Ricci tensor, Gauss curvature tensor and the second order derivative of the second fundamental form? More precisely, I found some formula in the paper of S. Funabashi, Totally complex submanifolds of a quaternionic Kaehlerian manifold, Kodai Math. J. 2 (1979), 314-336. on page 327, just below the formula (4.1). Thank you!

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That formula that you refer to is an application of the Ricci identities aka the commutation formulae for the covariant derivatives applied to the Codazzi equations (3.7) using the symmetries of the second fundamental form established in Lemma 3.1. –  Yuri Vyatkin Jun 20 '12 at 6:34

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It should not be difficult to obtain such a formula from the Gauss equation by taking covariant derivatives and appropriate contractions. In fact, the twice contracted Gauss equation relates the scalar curvatures, the Ricci tensor, the square of the mean curvature and the square of the length of the second fundamental form. See eg. p. 47 in "The large scale structure of the space-time" by S.W. Hawking and G.F.R. Ellis.

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