Abstract. We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Q m = SO m+2 /SO m SO 2 . It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in Q m = SO m+2 /SO m SO 2 with commuting Ricci tensor.

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In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians G 2 (C m+2 ). By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in G 2 (C m+2 ).

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