| Summary: | The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface <i>M</i> in a non-flat complex space form. For any nonnull constant <i>k</i> and any vector field <i>X</i> tangent to <i>M</i> the k-th Cho operator <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>F</mi> <mi>X</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msubsup> </semantics> </math> </inline-formula> is defined and is related to both connections. If <i>X</i> belongs to the maximal holomorphic distribution <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">D</mi> </semantics> </math> </inline-formula> on <i>M</i>, the corresponding operator does not depend on <i>k</i> and is denoted by <inline-formula> <math display="inline"> <semantics> <msub> <mi>F</mi> <mi>X</mi> </msub> </semantics> </math> </inline-formula> and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>F</mi> <mi>X</mi> </msub> <mi>S</mi> <mo>=</mo> <mi>S</mi> <msub> <mi>F</mi> <mi>X</mi> </msub> </mrow> </semantics> </math> </inline-formula>, where <i>S</i> denotes the Ricci tensor of <i>M</i> and a further condition is satisfied, are classified.
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