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Sep 4, 2017 at 15:08 history edited Robert Bryant CC BY-SA 3.0
Fixed some awkward language and some bad notation.
Sep 4, 2017 at 8:47 vote accept C.F.G
Sep 4, 2017 at 8:28 history edited Robert Bryant CC BY-SA 3.0
Fixed a typo
Sep 4, 2017 at 8:27 comment added Robert Bryant @C.F.G: Oh, you are right, that's a typo. I'll fix it.
Sep 4, 2017 at 5:54 comment added C.F.G Is $(2)$ correct? my assumption is $g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j)$ (see the question).
Sep 4, 2017 at 5:48 comment added C.F.G I am really impressed by your answer. Thank you very much indeed. This question comes from the study of conformally flat almost Hermitian manifolds satisfying certain condition. I must say that I used Kulkarni result which states: A Riemannian manifold $(M,g)$ of dim$\geq 4$ is conformally flat iff for every orthonormal four-tuple $\{e_1,e_2,e_3,e_4\}$ we have $K(e_1,e_2)+K(e_3,e_4)=K(e_1,e_4)+K(e_2,e_3)$. Thank you again.
Sep 4, 2017 at 1:10 history answered Robert Bryant CC BY-SA 3.0