Generalized Hsiung–Minkowski formulae on manifolds with density

In this work, using the weighted symmetric functions σk∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{k}^{\infty }$$\end{document} and the weighted Newton transformations Tk∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{k}^{\infty }$$\end{document} 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.


Introduction
The classical integral identities of Minkowski type (see [6,17,19,24] where Y is a conformal vector field. i.e. there exists a smooth function φ, such as and H r denotes the r th mean curvatures of M n . Minkowski formulae for hypersurfaces were first obtained by Hsiung [17] in the Euclidian space (generalizing Minkowski result for r = 0) and later by Bivens [11] in the Euclidean sphere and hyperbolic space. These results were generalized by Alencar and Colares [3] by using the (r + 1)-mean curvature linearized operator L r of the hypersurface.
It is interesting to know if the formulae (1) and (2) can be extended in other cases, and applied to generalize the aforementioned results.
In this work, using the weighted symmetric functions σ ∞ k and the weighted Newton transformations T ∞ k introduced by Case [12], we obtain some integral formulae on weighted manifolds. These formulae generalize (1) and (2). We also give some special cases and applications of these formulae.
Recall that a weighted manifold is a triplet (M n , , , dv f = e − f dv), where M n is a complete n-dimensional Riemannian manifold, dv is the standard volume element of M n and f : M n −→ R is a smooth function.

Preliminaries
In this section we collect some basic facts and definitions about manifolds with density which are needed in this article. We also give the definitions and some properties of the weighted symmetric functions and the weighted Newton transformations. For more details see [4,[12][13][14][15]21,25]. It is well known that A is a linear self-adjoint operator and at each point p ∈ M n , it has real eigenvalues μ 1 , ..., μ n (the principal curvatures).
. We have the following properties of σ ∞ k and T ∞ k (see [12] for the proof). Proposition 2.2 [12] For μ 0 , μ 1 ∈ R and μ ∈ R n , we have In particular, For k ≥ 1 we have : with equality if and only if : is nothing but the definition of the (normalized) weighted mean curvature of the hypersurface M n studied by Gromov [16]. The variations of a functional whose integrant is the r th weighted curvature on the hypersurface of a closed Riemannian manifold was given in [10].
The rest of this section will be devoted to computing the divergence of the weighted Newton transformation T ∞ k . For this purpose recall that the divergence of the weighted Newton transformations is defined by : and {e 1 , ..., e n } is a local orthonormal frame of the tangent space of M n .

Lemma 2.3
The weighted divergence of the weighted Newton transformations T ∞ k are inductively given by the following formula For the proof see [1].

Main results
In this section we will derive some general integral formulae for close oriented hypersurface M n in a weighted manifold M n+1 . Our idea here is to compute the weighted divergence div Suppose now the existence of a closed conformal vector field Y on M n+1 ; that is to say there exists a for every vector fields V, W over M n+1 .
If {e 1 , ..., e n } is an orthonormal basis of T p M n that diagonalizes A, then On the other hand, we have This gives And in virtue of formula (2.2) we have Integrating the two sides of this latter equality and applying the divergence theorem, we obtain for 1 ≤ k ≤ n − 1, Consequently, we have the following proposition: This formula generalizes the kth Minkowski formula for the non weighted case [6].
If M n+1 has constant sectional curvature, then by Corollary (1), we obtain:

Proposition 3.3 Under the hypothesis of Proposition 3.1, if M n+1 has constant sectional curvature, then
Formula (3.2) becomes simple when M n+1 has constant sectional curvature and Y is a Killing vector field, that is φ = 0. In that case we have :

Examples and applications
Example 4.1 Suppose that the Killing vector field Y never vanishes. If the distribution : has constant rank n, and it is integrable, then it determines a codimension 1 Riemannian foliation Since Y is a Killing vector field, we have : Hence, it is easy to see by (5) that each leaf of the foliation F(Y ) satisfies : Taking k = 0 in (3.1) and applying the divergence theorem,we obtain for every Killing vector filed Y : If the mean curvature σ ∞ 1 is constant, multiplying by the constant σ ∞ 1 , the last equation allows us to write : On the other hand, for k = 1, (3.3) gives : So that subtracting these two formulae we obtain that : It is not difficult to prove that : If M n+1 = R n+1 , denoting by be the compact domain whose boundary is x (M n ), and N the global vector fields normal to M n . We have div Y = (n + 1) and div By applying the weighted version of the divergence theorem, we have : If M n has constant strictly positive weighted mean curvature, we can choose Y as unit vector field and we obtain : This result was also obtained by [7, Corollary 1.3] using a different argument.
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