Holomorphicity and the Walczak formula on Sasakian manifolds

Brînzănescu, Vasile; Slobodeanu, Radu

doi:10.1016/j.geomphys.2006.02.011

Cited by 18 publications

(19 citation statements)

References 16 publications

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“…Assume u ∈ N(q) and |u| = s + 1, s ≥ 0. We show the first identity of (14). A proof of the second one is analogous.…”

Section: Properties Of Generalized Newton Transformationmentioning

confidence: 81%

“…On the other hand, we have a recurrence formula (14) for generalized Newton transformation. We derive the recurrence formula for the divergence of T * u and finally the explicit formula for div…”

Section: Integral Formulasmentioning

confidence: 99%

“…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%

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Generalized Newton transformation and its applications to extrinsic geometry

Andrzejewski

Kozłowski

Niedziałomski

2016

Asian Journal of Mathematics

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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.

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“…Assume u ∈ N(q) and |u| = s + 1, s ≥ 0. We show the first identity of (14). A proof of the second one is analogous.…”

Section: Properties Of Generalized Newton Transformationmentioning

confidence: 81%

Section: Integral Formulasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Newton transformation and its applications to extrinsic geometry

Andrzejewski

Kozłowski

Niedziałomski

2016

Asian Journal of Mathematics

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“…The next result shows the invariance of the above defined holomorphicity and its proof is exactly as in [4]:…”

Section: Let Phol(m ) Be the Set Of All Paracontact-holomorphic Vectomentioning

confidence: 69%

“…Compared with the huge literature in (metric) contact geometry, it seems that new studies are necessary in almost paracontact geometry; a very interesting paper connecting these fields is [5]. The present work is another step in this direction, more precisely from the point of view of some subjects of [4].…”

Section: Introductionmentioning

confidence: 88%

Invariant distributions and holomorphic vector fields in paracontact geometry

Crâşmăreanu¹,

Laurian-Ioan²

2015

Turk J Math

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Having as a model the metric contact case of V. Brînzȃnescu; R. Slobodeanu, we study two similar subjects in the paracontact (metric) geometry: a) distributions that are invariant with respect to the structure endomorphism φ ; b) the class of vector fields of holomorphic type. As examples we consider both the 3 -dimensional case and the general dimensional case through a Heisenberg-type structure inspired also by contact geometry.

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