Generalized Hsiung–Minkowski formulae on manifolds with density

In this work, using the weighted symmetric functions $$\sigma _{k}^{\infty }$$ σ k ∞ and the weighted Newton transformations $$T_{k}^{\infty }$$ T k ∞ introduced by Case (Alias et al. Proc Edinb Math Soc 46(02):465–488, 2003), we derive some generalized integral formulae for close hypersurfaces in weighted manifolds. We also give some examples and applications of these formulae.