Integral inequalities for closed linear Weingarten submanifolds in the product spaces.

An integral inequality for closed linear Weingarten 𝑚-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces 𝑀𝑛(𝑐) × â„, 𝑛 > 𝑚 ≥ 4, where 𝑀𝑛(𝑐) is a space form of constant sectional curvature 𝑐 ∈ {-1, 1}, is proved. As an application is shown that the sharpness in this inequality is attained in the totally umbilical hypersurfaces, and in a certain family of standard product of the form 𝕊1(√1 - 𝑟2) × ð•Šð‘š-1(𝑟) with 0 < 𝑟 < 1 when 𝑐 = 1. In the case where 𝑐 = -1, is obtained an integral inequality whose sharpness is attained only in the totally umbilical hypersurfaces. When 𝑚 = 2 and 𝑚 = 3, an integral inequality is also obtained with equality happening in the totally umbilical hypersurfaces.