Summary
Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(M,J,g)$ be a K\"ahler--Norden manifold. Using the notions of the horizontal and vertical lifts, a class of almost complex structures $\widetilde J$ is defined on the tangent bundle $T\!M$, and necessary and sufficient conditions for such a structure to be integrable (complex) are described. Next, a class of pseudo-Riemannian metrics $\widetilde g$ of Norden type is defined on $T\!M$, for which $\widetilde J$ is an antiisometry. Thus, the pair $(\widetilde J,\widetilde g)$ becomes an almost complex structure with Norden metric on $T\!M$. It is checked whether the structure $(\widetilde J,\widetilde g)$ is K\"ahler--Norden itself.
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Olszak, Z. On almost complex structures on tangent bundles. Period Math Hung 51, 59–74 (2005). https://doi.org/10.1007/s10998-005-0030-8
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DOI: https://doi.org/10.1007/s10998-005-0030-8