Abstract
We consider real hypersurfaces M in complex projective space equipped with both the Levi-Civita and generalized Tanaka–Webster connections. Associated with the generalized Tanaka–Webster connection we can define a differential operator of first order. For any nonnull real number k and any symmetric tensor field of type (1,1) B on M, we can define a tensor field of type (1,2) on M, B^{(k)}_T, related to Lie derivative and such a differential operator. We study symmetry and skew symmetry of the tensor A^{(k)}_T associated with the shape operator A of M.
Highlights
We will denote by CP m, m ≥ 2, the complex projective space equipped with the Kahlerian structure (J, g), J being the complex structure and g the Fubini-Study metric with constant holomorphic sectional curvature 4
For any tangent vector field X on M, we write JX = φX + η(X)N, where φX is the tangent component of JX and η(X) = g(X, ξ)
The purpose of the present paper is to study real hypersurfaces M in CP m such that the shape operator satisfies either (1.2) or (1.3)
Summary
We will denote by CP m, m ≥ 2, the complex projective space equipped with the Kahlerian structure (J, g), J being the complex structure and g the Fubini-Study metric with constant holomorphic sectional curvature 4. On M we can define a differential operator of first order associated with the kth generalized Tanaka–Webster connection by L(Xk)Y = ∇ˆ (Xk)Y − ∇ˆ (Yk)X = LX Y + TX(k)Y , for any X, Y tangent to M. Generalizing this we can consider that the tensor BT(k) is symmetric, that is, BT(k)(X, Y ) = BT(k)(Y, X) for any X, Y tangent to M This is equivalent to have the following Codazzi-type condition. We can suppose that BT(k) is skew symmetric, that is, BT(k)(X, Y ) = −BT(k)(Y, X), for any X, Y tangent to M This is equivalent to the following Killing-type condition: Lie Derivatives of the Shape Operator of a Real Hypersurface.
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