Mohammed Abdelmalek
Some geometric properties of Newton transformations and applicartions
Abstract.
Let Mⁿ⁺¹ be an (n+1)-dimensional oriented weighted manifold, and ψ:Mⁿ→Mⁿ⁺¹ be an hypersurface isometrically immersed. It's Weingarten (or shape) operator A is defined by alm :
AX=-(▽_{X}N)^{⊺}.
Where ▽ is the Levi-Civita connections on Mⁿ⁺¹, N is the vector field normal to Mⁿ in Mⁿ⁺¹ and X is a tangent vector field X∈ϰ(Mⁿ).
It is well known that A is a linear self adjoint operator and at each point p∈Mⁿ, its eigenvalues μ₁,...,μ_{n} are the principal curvatures of Mⁿ.
For μ=(μ₁,...,μ_{n}) the elementary symmetric polynomial σ_{k}^{∞}:ℝⁿ→ℝ are defined by r :
σ_{k}^{∞}(μ)=∑μ_{i₁}...μ_{i_{n}}.
Associate to A, we can define the weighted Newton transformations T_{k} by <cite>r</cite> :
{
T₀=I,
T_{k}=σ_{k}I-AT_{k-1} for k≥1.
This is equivalent to
T_{k}=∑(-1)^{i}σ_{k-i}.A^{i}.
In this work, we study the Newton transformations T_{k}.We give some algebraic properties for these operators. We use them to prove some geometric results for hypersurfaces embedded in space forms. We also give a relation between the transversality of two given hypersurfaces and the ellipticity of T_{k}. These last result allows us to give a prove of Alexandrov sphere theorem for constant higher order mean curvature hypersurfaces embedded in space forms
