Abstract
In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P = ϕ A - A ϕ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field P satisfies geometric conditions are classified.
Highlights
A real hypersurface is a submanifold of a Riemannian manifold with a real co-dimensional one.Among the Riemannian manifolds, it is of great interest in the area of Differential Geometry to study real hypersurfaces in complex space forms
Complete and connected complex space forms are analytically isometric to complex projective space C Pn if c > 0, to complex Euclidean space Cn if c = 0, or to complex hyperbolic space C H n if c < 0
On M, relation λ = ν holds and this results in the structure tensor φ commuting with the shape operator A, i.e., Aφ = φA and, because of Theorem 3 M, is locally congruent to a real hypersurface of type (A), and this completes the proof of Theorem 2
Summary
A real hypersurface is a submanifold of a Riemannian manifold with a real co-dimensional one. Are there real hypersurfaces in non-flat complex space forms whose derivatives with respect to different connections coincide?. Are there real hypersurfaces in non-flat complex space forms whose differential operator L(k) coincides with derivatives with respect to different connections?. In [12], the problem of classifying three-dimensional real hypersurfaces in non-flat complex space forms M2 (c), for which the operator L(k) applied to the shape operator coincides with the covariant derivative of it, has been studied, i.e., LX A = ∇ X A, for any vector field X tangent to M. It is interesting to study real hypersurfaces in non-flat complex spaces forms, whose tensor field P satisfies certain geometric conditions. This paper is organized as follows: in Section 2, basic relations and theorems concerning real hypersurfaces in non-flat complex space forms are presented.
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