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. We characterize proper biharmonic anti-invariant surfaces in3-dimensional generalized (κ,µ)-manifolds with constant mean curvatureby means of the scalar curvature of the ambient space and the meancurvature. In addition, we give a method for constructing innity manyexamples of proper biharmonic submanifolds in a certain 3-dimensionalgeneralized (κ,µ)-manifold. Moreover, we determine 3-dimensional gen-eralized (κ,µ)-manifolds which admit a certain kind of proper biharmonicfoliation. 1. IntroductionThe notion of biharmonic submanifolds is a natural extension of the notionof minimal submanifolds from a variational point of view. Non-minimal bihar-monic submanifolds are said to be proper. Considerable advancement has beenmade in the study of proper biharmonic submanifolds in manifolds with specialmetric properties (e.g., real space forms, complex space forms, Sasakian spaceforms, conformally at spaces, etc.) since the beginning of this century.When the dimension of the ambient space is three, it seems worthwhileand interesting to construct and classify proper biharmonic submanifolds incontact metric 3-manifolds. Some classication results for proper biharmonicsubmanifolds in Sasakian 3-manifolds have been obtained (see, for example,[3], [5], [6]).As an extension of the notion of a Sasakian manifold, Koufogiorgos andTsichlias [7] introduced the notion of a generalized (κ,µ)-manifold for two realfunctions κ (≤ 1) and µ. If κ ≡ 1, then it is a Sasakian manifold. Markellosand Papantoniou [10] studied proper biharmonic Legendre curves and anti-invariant surfaces of 3-dimensional non-Sasakian generalized (κ,µ)-manifolds.However, they provided no examples of such submanifolds. As far as the authorknows, there have been no examples of proper biharmonic submanifolds in non-Sasakian contact metric 3-manifolds.

A SHORT NOTE ON BIHARMONIC SUBMANIFOLDS IN 3-DIMENSIONAL GENERALIZED (𝜅, 𝜇)-MANIFOLDS