A Generalization Of Minkowski Formulas In Warped Product Space And Some Applications

Minkowski formula is a classical result in differential geometry and has many impor-tant applications.It is well known that the Minkowski type formula holds for hypersurface if the background space is space form(Euclidean space,round sphere,hyperbolic space).And warped product manifold is a definition that contains all the space form,so we want to research the properties of hypersurface in warped product manifold under some condi-tions.We prove a Minkowski type formula for hypersurface in some warped product spaces in this paper,and as a corollary we find that area and volume upper bounds of k-convex or positive(k+1)th mean curvature hypersurface can be controlled by the integral of weighted kth mean curvature under some geometry conditions.Besides,combining the formula we get and the Heintze-Karcher type inequality in warped product space,we also get some Alexandrov type rigid results for star-shaped hypersurface in warped product space by using the formula under some conditions.