Doubly warped product Hermitian manifold with constant holomorphic sectional curvature

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>M</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M_{1},g) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>h</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M_{2},h) be two Hermitian manifolds. Then the doubly warped product (abbreviated as DWP) Hermitian manifold of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>M</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M_{1},g) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>h</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (M_{2},h) is the product manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> M_{1}\times M_{2} endowed with the warped product Hermitian metric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msubsup> <m:mi>f</m:mi> <m:mn>2</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msubsup> <m:mi>f</m:mi> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>h</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> G=f^{2}_{2}g+f^{2}_{1}h , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>f</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> f_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>f</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> f_{2} are positive smooth functions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>M</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> M_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> M_{2} , respectively. In this paper, the formula of altered Chern scalar curvature of DWP-Hermitian manifold is derived. The necessary and sufficient conditions for a DWP-Hermitian manifold to be Kähler-like is obtained. It is also proved that a compact Kähler-like DWP-Hermitian manifold is Chern flat when it has constant holomorphic sectional curvature or constant holomorphic bisectional curvature.