Abstract
In this paper we consider the notion of ξ-invariant or (equation omitted)-invariant real hypersurfaces in a complex two-plane Grassmannian <TEX>$G_2$</TEX>( <TEX>$C^{m+2}$</TEX>) and prove that there do not exist such kinds of real hypersurfaces in <TEX>$G_2$</TEX>( <TEX>$C^{m+2}$</TEX>) with parallel second fundamental tensor on a distribution ζ defined by ζ = ξ U(equation omitted), where(equation omitted) = Span ｛ξ<TEX>$_1$</TEX>, ξ<TEX>$_2$</TEX>, ξ<TEX>$_3$</TEX>｝.X>｝.
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