Conformal fields and the stability of leaves with constant higher order mean curvature

Andrzejewski, K.; Walczak, Paweł

doi:10.1016/j.difgeo.2011.08.001

“…To begin with, let us note that one of the interesting developments in geometry of foliations during the last decades was the rise of integral formulas for closed foliated manifolds. These formulas are of some interest, for example in several geometric situations they provide obstructions to the existence of foliations with all the leaves enjoying a given geometric property (see, [4,6,9,25,27] and bibliographies therein). Such formulas have also applications in different areas of differential geometry and analysis on manifolds (see, for example, [7,12,14,26]).…”

Section: Introductionmentioning

confidence: 99%

Generalized Newton transformation and its applications to extrinsic geometry

Andrzejewski

¹

,

Kozłowski

²

,

Niedziałomski

³

2016

Asian Journal of Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…To begin with, let us note that one of the interesting developments in geometry of foliations during the last decades was the rise of integral formulas for closed foliated manifolds. These formulas are of some interest, for example in several geometric situations they provide obstructions to the existence of foliations with all the leaves enjoying a given geometric property (see, [4,6,9,25,27] and bibliographies therein). Such formulas have also applications in different areas of differential geometry and analysis on manifolds (see, for example, [7,12,14,26]).…”

Section: Introductionmentioning

confidence: 99%