Real hypersurfaces in the complex quadric with commuting and parallel Ricci tensor

Abstract

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.

Real Hypersurfaces of Complex Quadric in Terms of Star-Ricci Tensor

“…In addition, if the real hypersurface M admits commuting Ricci tensor, i.e. Ric • φ = φ • Ric, Suh also proved the followings: 13]). Let M be a real hypersurface of the complex quadric Q m , m ≥ 3, with commuting Ricci tensor.…”

“…Sξ = αξ, where α = g(Sξ, ξ) is a smooth function, then M is said to be a Hopf hypersurface. For the real Hopf hypersurfaces of complex quadric many characterizations were obtained by Suh (see [9,10,11,12,13] etc.). In particular, we note that Suh in [9] introduced parallel Ricci tensor, i.e.∇Ric = 0, for the real hypersurfaces in Q m and gave a complete classification for this case.…”

“…Then the unit normal vector field N of M is either U-principal or U-isotropic. 13]). There exist no Hopf real hypersurfaces in the complex quadric Q m , m ≥ 4, with commuting and parallel Ricci tensor.…”

“…Here the parallel vector bundle U means that ( ∇ X A)Y = q(X)AY for all X, Y ∈ T z Q m , z ∈ Q m , where ∇ and q denote a connection and a certain 1-form on T z Q m , respectively. Theorem 1.1 ( [13]). Let M be a real hypersurface of the complex quadric Q m , m ≥ 3, with commuting Ricci tensor.…”

Totally geodesic real hypersurfaces in complex quadric

“…We fix AX = λX and AY = γY , then we get Suppose that the Ricci tensor is recurrent on M . Then we have [15] (∇ Y Ric)(X) = ∇ Y Ric(X) − Ric(∇ Y X) β(Y )Ric(X) = q(Y )g(JAN, X) − g(AX, N ){q(Y )JAN } T + q(Y )g(JAξ, X)Aξ +η(AX){q(Y )JAξ} T .…”