Gradient Einstein-type spacetimes and modified gravity

Abstract In this work, at first, we investigate gradient Einstein-type spacetimes. After that, we study the properties of a gradient Einstein-type spacetime admitting a parallel vector field. In the next parts, we present a systematic study about gradient Einstein-type spacetime obeying f ( R , T ) and $$f(R,L_m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> gravity theories. Finally, in the cosmological context we find some remarkable results.