Generalized Newton transformation and its applications to extrinsic geometry

Abstract

Abstract. In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures. IntroductionAnalyzing the study of Riemannian geometry we see that its basic concepts are related with some operators, such as shape, Ricci, Schouten operator, etc. and functions constructed of them, such as mean curvature, scalar curvature, Gauss-Kronecker curvature, etc. The most natural and useful functions are the ones derived from algebraic invariants of these operators e.g. by taking trace, determinant and in general the r-th symmetric functions σ r . However, the case r > 1 is strongly nonlinear and therefore more complicated. The powerful tool to deal with this problem is the Newton transformation T r of an endomorphism A (strictly related with the Newton's identities) which, in a sense, enables a linearization of σ r , (r + 1)σ r+1 = tr (AT r ).Although this operator appeared in geometry many years ago (see, e.g., [21,29]), there is a continues increase of applications of this operator in different areas of geometry in the last years (see, among others, [1,2,3,8,10,17,18,23,24,25,28]).All these results cause a natural question, what happen if we have a family of operators i.e. how to define the Newton transformation for a family of endomorphisms. A partial answer to this question can be found in the literature (operator T r and the scalar S r for even r [5,15]), nevertheless, we expect that this case is much more subtle. This is because in the case of family of operators we should obtain more natural functions as in the case of one and consequently more information about geometry. In order to do this, for any multi-index u 2000 Mathematics Subject Classification. 53C12; 53C65.

On total mean curvatures of foliated half-lightlike submanifolds in semi-Riemannian manifolds

“…Moreover, T r are also self-adjoint and commutes with A W . Furthermore, the following algebraic properties of T r are well-known (see [1], [12] and references therein for details).…”

“…This is partly due to the non-linearity of S r for r > 1, and hence very complicated to study. A great deal of research on higher order mean curvatures S r in Riemannian geometry has been done with numerous applications, for instance see [1] and [12]. This gap has motivated our introduction of lightlike geometry of S r for r > 1.…”

Higher order mean curvatures of SAC half-lightlike submanifolds of indefinite almost contact manifolds

“…It is easy to see that the tensor T * r is unique if and only if the null hypersurface M is totally geodesic. For more details on Newton transformations and their properties, we refer the reader to [2], [15] and many more references therein.…”

“…Then, the following algebraic properties of T * r are well-known (see [1], [2], [15] and references therein for details). In line with (3.12), we can see that the SAC half-lightlike submanifold given in Example 2.1 is r-minimal with 1 ≤ r ≤ 6.…”

Geometric configuration of Riemannian submanifolds of arbitrary codimension

“…where (T u ) u stands for the family of the generalized Newton transformations introduced in [4] associated to the matrix A = (A 1 , ..., A q ); (A α ) α∈{1,...,q} is a system of matrices of the shape operators corresponding to a normal basis to the manifold M n andσ u =σ u (A 1 | Σ , ..., A q | Σ ) are the coefficients of the Newton polynomial P A : R q −→ R defined by…”