New Conditions on Normal Jacobi Operator of Real Hypersurfaces in the Complex Quadric

Abstract

On a real hypersurface M of a complex quadric we have an almost contact metric structure induced by the Kählerian structure of the ambient space. Therefore, on M we have the Levi-Civita connection $\nabla$ and, for any non-null real number k, the so called kth generalized Tanaka Webster connection ${\hat{\nabla }}^{(k)}$. We introduce the notions of $({\hat{\nabla }}^{(k)},\nabla )$-Codazzi and $({\hat{\nabla }}^{(k)},\nabla )$-Killing normal Jacobi operator on such a real hypersurface and classify Hopf real hypersurface in a complex quadric whose normal Jacobi operators satisfy any of both conditions.