On almost cosymplectic generalized (<i>k, μ</i>)<i>ʹ</i>-spaces

Abstract We introduce and study almost cosymplectic manifolds whose characteristic vector field ξ belongs to a generalized ( k, μ ) ʹ -nullity distribution, which involves the tensor hʹ = φ ◦ h . If hʹ ≠ 0, we prove that the smooth functions k and μ are uniquely determined by k &lt; 0 and μ = ξ (ln λ ), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>=</m:mo> <m:msqrt> <m:mo>−</m:mo> <m:mi>k</m:mi> </m:msqrt> <m:mo>.</m:mo> </m:math> $\lambda=\sqrt{-k}.$ Moreover, we show that the spectrum of hʹ is { 0 , λ,−λ} and that the considered manifolds cannot be Ricci symmetric. Under certain conditions, we further prove that such manifolds are locally a warped product. Examples are also provided.