scholarly journals SUB-RIEMANNIAN RICCI CURVATURES AND UNIVERSAL DIAMETER BOUNDS FOR 3-SASAKIAN MANIFOLDS

2017 ◽  
Vol 18 (4) ◽  
pp. 783-827 ◽  
Author(s):  
Luca Rizzi ◽  
Pavel Silveira
Keyword(s):  
Comparison Theorems ◽  
Riemannian Structure ◽  
Similar Statement ◽  
Sasakian Manifolds ◽  
Sasakian Structure ◽  
Hopf Fibrations ◽  
Image Position

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension $4d+3$, with $d>1$, has sub-Riemannian diameter bounded by $\unicode[STIX]{x1D70B}$. When $d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on $\mathbb{S}^{4d+3}$ of the quaternionic Hopf fibrations: $$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$ whose exact sub-Riemannian diameter is $\unicode[STIX]{x1D70B}$, for all $d\geqslant 1$.

scholarly journals Invariance of basic Hodge numbers under deformations of Sasakian manifolds

2020 ◽  
Author(s):  
Paweł Raźny
Keyword(s):  
Elliptic Operators ◽  
Holomorphic Foliation ◽  
Sasakian Manifolds ◽  
Contact Manifolds ◽  
Continuity Theorem ◽  
Riemannian Foliations ◽  
Holomorphic Structure ◽  
Sasakian Structure ◽  
Hodge Numbers ◽  
Small Deformations

Abstract We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the $$\partial {\bar{\partial }}$$ ∂ ∂ ¯ -lemma and being transversely Kähler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

Multiparameter Sturm theory

1984 ◽  
Vol 99 (1-2) ◽  
pp. 173-184 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Elliptic Partial Differential Equation ◽  
Comparison Theorems ◽  
Unified Theory ◽  
Focal Points ◽  
The Matrix ◽  
Sturm Liouville ◽  
Left And Right ◽  
Image Position ◽  
Completeness Of Eigenfunctions ◽  
Sturm Theory

SynopsisLet Sturm–Liouville problemswith continuous coefficients and appropriate boundary conditions, be coupled by the eigenvalue λ = (λ1, … λk). When k = 1, there are various oscillation, perturbation and comparison theorems concerning existence and continuous or monotonic dependence of eigenvalues, eigenfunctions and their zeros (i.e. focal points).We attempt a unified theory for such results, valid for general fc, under conditions known as "left" and “right” definiteness. A representative result may be stated loosely as follows: if LD holds then (elementwise) monotonic dependence of p, q and the matrix [ars] forces monotonic dependence of λ. LD is a generalisation of the “polar” case for k = 1, and was originally conceived for a quite different purpose, viz. completeness of eigenfunctions via elliptic partial differential equation theory.

scholarly journals Comparison theorems for the square integrability of solutions of (r(t)y')'+q(t)y=f(t, y)

1972 ◽  
Vol 13 (1) ◽  
pp. 75-79 ◽  
Author(s):  
John S. Bradley
Keyword(s):  
Comparison Theorems ◽  
Half Line ◽  
Local Integrability ◽  
All Solutions ◽  
Image Position ◽  
Square Integrability

Bellman [1], [2, p. 116] proved that, if all solutions of the equationare in L2, ∞) and b(t) is bounded, then all solutions ofare also in L2(a, ∞). The purpose of this paper is to present conditions on the function f that guarantee that all solutions ofbe in the class L2(a, ∞) whenever all solutions of the equationhave this property. It is assumed that r(t) >0, r and qare continuous on a half line (a, ∞) and f is continuous. Actually the continuity assumptions may be weakened to local integrability and L2 (a, ∞) may be replaced by Lp(a, ∞) for any p > 1.

Some Comparison Theorems for Conjugate And σ-Points

1976 ◽  
Vol 28 (6) ◽  
pp. 1172-1179
Author(s):  
Walter Leighton
Keyword(s):  
Differential Equations ◽  
Comparison Theorems ◽  
Image Position

Section 1 of this paper is concerned with the effect on conjugate and σ-points of various perturbations of q(x) for differential equations of the form

scholarly journals On Lightlike Geometry of Para-Sasakian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Bilal Eftal Acet ◽  
Selcen Yüksel Perktaş ◽  
Erol Kılıç
Keyword(s):  
Vector Field ◽  
Characteristic Vector ◽  
Sasakian Manifolds ◽  
Lightlike Hypersurface ◽  
Sasakian Structure ◽  
Integrability Conditions ◽  
Integrable Distribution ◽  
Lightlike Hypersurfaces ◽  
Characteristic Vector Field

We study lightlike hypersurfaces of para-Sasakian manifolds tangent to the characteristic vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively, and give examples. Integrability conditions for the distributions on a screen semi-invariant lightlike hypersurface of para-Sasakian manifolds are investigated. We obtain a para-Sasakian structure on the leaves of an integrable distribution of a screen semi-invariant lightlike hypersurface.

scholarly journals Indefinite Almost Paracontact Metric Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Mukut Mani Tripathi ◽  
Erol Kılıç ◽  
Selcen Yüksel Perktaş ◽  
Sadık Keleş
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Constant Sectional Curvature ◽  
Sasakian Manifolds ◽  
Sasakian Structure

We introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an (ε)-para Sasakian structure. We show that, for an (ε)-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) (ε)-para Sasakian manifoldMnis locally isometric to a pseudohyperbolic spaceHνn(1)(resp., pseudosphereSνn(1)). At last, it is proved that for an (ε)-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

scholarly journals Hypersurfaces of Lorentzian para-Sasakian manifolds

2011 ◽  
Vol 109 (1) ◽  
pp. 5 ◽  
Author(s):  
Selcen Yüksel Perktas ◽  
Erol Kiliç ◽  
Sadik Keles
Keyword(s):  
Sufficient Conditions ◽  
Contact Manifold ◽  
Sasakian Manifold ◽  
Product Structure ◽  
Product Manifold ◽  
Sasakian Manifolds ◽  
Contact Manifolds ◽  
Sasakian Structure ◽  
Almost Contact Manifold ◽  
Necessary And Sufficient

In this paper we study the invariant and noninvariant hypersurfaces of $(1,1,1)$ almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an $(1,1,1)$ almost contact manifold admits an almost product structure. We investigate hypersurfaces of affinely cosymplectic and normal $(1,1,1)$ almost contact manifolds. It is proved that a noninvariant hypersurface of a Lorentzian almost paracontact manifold is an almost product metric manifold. Some necessary and sufficient conditions have been given for a noninvariant hypersurface of a Lorentzian para-Sasakian manifold to be locally product manifold. We establish a Lorentzian para-Sasakian structure for an invariant hypersurface of a Lorentzian para-Sasakian manifold. Finally we give some examples for invariant and noninvariant hypersurfaces of a Lorentzian para-Sasakian manifold.

Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5

2015 ◽  
Vol 12 (03) ◽  
pp. 1550027 ◽  
Author(s):  
Jin Hong Kim
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Contact Manifold ◽  
Existence Problem ◽  
Sasakian Manifolds ◽  
Contact Manifolds ◽  
Sasakian Structure ◽  
Simply Connected ◽  
Complex Projective

The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

Some inequalities for (a + b)p and (a + b)p + (a − b)p

2014 ◽  
Vol 98 (541) ◽  
pp. 96-103 ◽  
Author(s):  
G. J. O. Jameson
Keyword(s):  
Similar Statement ◽  
Image Position

We start from two simple identities:For any p > 0 and 0 ≥ b ≥ a, now letCan we formulate statements about Gp(a, b) that generalise (1) and (2)? We cannot hope for equalities, but perhaps we can establish inequalities which somehow reproduce (1) when p = 1 and (2) when p = 2. For (1), this might mean an inequality of the form Apap ≤ Gp (a, b) ≤ Bpap for certain constants Ap and Bp, and for (2) a similar statement with ap replaced by ap + bp. However, these are not the only possibilities, as we shall see.

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