Abstract
In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.
Highlights
A Kahler manifold Ñ is a complex manifold of complex dimension n and real dimension 2n, which is equipped with a complex structure J defined J : T Ñ → Ñ, where T Ñ is the tangent space of Ñ, satisfying J = 0, i.e., J is parallel with respect to the Levi-Civita connection ∇̃ of Ñ relations J 2 = − Id and ∇and a Riemanian metric G that is compatible with J, i.e., G ( JX, JY ) = G ( X, Y ) for all tangent X, Y on Ñ.The pair (J, G) is called Kahler structure
The step is to introduce new tensors on real hypersurfaces in non-flat complex space forms related to the ∗ -Ricci tensor, since there are results concerning notions and tensors related to the Ricci tensor
It is worthwhile to study if there are non-Hopf real hypersurfaces of dimension greater than three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor, the ∗ -Weyl curvature tensor could be defined on real hypersurfaces in other symmetric
Summary
A Kahler manifold Ñ is a complex manifold of complex dimension n and real dimension 2n, which is equipped with . A complex structure J defined J : T Ñ → Ñ, where T Ñ is the tangent space of Ñ, satisfying J = 0, i.e., J is parallel with respect to the Levi-Civita connection ∇. The pair (J, G) is called Kahler structure. A Kahler manifold of constant holomorphic sectional curvature c is called complex space form. Complete and connected complex space forms depending on the value of holomorphic sectional curvature c are analytically isometric to complex projective space C Pn if c > 0, to complex hyperbolic space C H n if c < 0 or to complex Euclidean space.
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