Integral Formulas for a Foliation with a Unit Normal Vector Field

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation F and a unit vector field N orthogonal to F, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of F with respect to N and the curvature tensor of the induced connection on the distribution D=TF⊕span(N), and this decomposition of D can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation.

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.

Horizontal Newton operators and high-order Minkowski formula

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

Transversality versus ellipticity on pseudo-Riemannian manifolds

In this paper, we establish sufficient conditions for transversality of two hypersurfaces of a given pseudo-Riemannian manifold. These conditions are the ellipticity of some operators obtained by means of Newton transformations.

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.