On tangent bundles of Walker four-manifolds

Abstract The aim of this study is to explore the complete lifts of almost Norden structures on tangent bundles of Walker four-manifolds. Furthermore, we examine the integrability conditions of the complete lifts <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> </m:math> {J}^{C} of the proper almost complex structure and identify the conditions for the isotropic Kähler-Norden property of the structure <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> </m:math> {J}^{C} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>T</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>J</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(T\left({M}_{4}),{J}^{C},{g}^{C}) . Finally, we investigate the holomorphic conditions of the Norden (Kähler-Norden) metrics <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> </m:math> {g}^{C} and discuss Quasi-Kähler-Norden metrics <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> </m:math> {g}^{C} on the tangent bundle <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>T</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>φ</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(T\left({M}_{4}),{\varphi }^{C},{g}^{C}) of Walker four-manifolds.