scholarly journals Pseudo-Parallel Characteristic Jacobi Operators on Contact Metric 3 Manifolds

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Wenjie Wang ◽  
Xinxin Dai
Keyword(s):  
Lie Groups ◽  
Jacobi Operator ◽  
Jacobi Operators ◽  
Parallel Characteristic ◽  
Metric Structures

We prove that the characteristic Jacobi operator on a contact metric three manifold is semiparallel if and only if it vanishes. We determine Lie groups of dimension three admitting left invariant contact metric structures such that the characteristic Jacobi operators are pseudoparallel.

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

scholarly journals REAL HYPERSURFACES WITH CYCLIC-PARALLEL STRUCTURE JACOBI OPERATORS IN A NONFLAT COMPLEX SPACE FORM

2009 ◽  
Vol 81 (2) ◽  
pp. 260-273 ◽  
Author(s):  
U-HANG KI ◽  
HIROYUKI KURIHARA
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Jacobi Operator ◽  
Complex Space Form ◽  
Parallel Structure ◽  
Real Hypersurfaces ◽  
Structure Jacobi Operator ◽  
Jacobi Operators

AbstractIt is known that there are no real hypersurfaces with parallel structure Jacobi operators in a nonflat complex space form. In this paper, we classify real hypersurfaces in a nonflat complex space form whose structure Jacobi operator is cyclic-parallel.

scholarly journals Osserman pseudo-Riemannian manifolds of signature (2,2)

2001 ◽  
Vol 71 (3) ◽  
pp. 367-396 ◽  
Author(s):  
Novica Blažić ◽  
Neda Bokan ◽  
Zoran Rakić
Keyword(s):  
Riemannian Manifold ◽  
Characteristic Polynomial ◽  
Riemannian Manifolds ◽  
Jacobi Operator ◽  
Tangent Vector ◽  
Jordan Form ◽  
The Other ◽  
Jacobi Operators ◽  
Osserman Manifold ◽  
Rank One

AbstractA pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

Almost isotropic Kähler manifolds

2020 ◽  
Vol 2020 (767) ◽  
pp. 1-16
Author(s):  
Benjamin Schmidt ◽  
Krishnan Shankar ◽  
Ralf Spatzier
Keyword(s):  
Classical Result ◽  
Jacobi Operator ◽  
Unit Vector ◽  
Complete Riemannian Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Jacobi Operators ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Complex Projective

AbstractLet M be a complete Riemannian manifold and suppose {p\in M}. For each unit vector {v\in T_{p}M}, the Jacobi operator, {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, {\mathcal{J}_{v}(w)=R(w,v)v}. Then p is an isotropic point if there exists a constant {\kappa_{p}\in{\mathbb{R}}} such that {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector {v\in T_{p}M}. If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each {p\in M}, there exists a constant {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators {\mathcal{J}_{v}} satisfy {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector {v\in T_{p}M}. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to {{\mathbb{C}}^{n-1}}.

scholarly journals Transversal Jacobi Operators in Almost Contact Manifolds

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 31
Author(s):  
Jong Taek Cho ◽  
Makoto Kimura
Keyword(s):  
Riemannian Manifold ◽  
Hyperbolic Space ◽  
Jacobi Operator ◽  
Real Hypersurfaces ◽  
Contact Manifolds ◽  
Jacobi Operators ◽  
Ruled Real Hypersurfaces ◽  
Contact Distribution ◽  
Contact Riemannian Manifold

Along a transversal geodesic γ whose tangent belongs to the contact distribution D, we define the transversal Jacobi operator Rγ=R(·,γ˙)γ˙ on an almost contact Riemannian manifold M. Then, using the transversal Jacobi operator Rγ, we give a new characterization of the Sasakian sphere. In the second part, we characterize the complete ruled real hypersurfaces in complex hyperbolic space.

scholarly journals Some Results on D-Homothetic Deformation On Almost Paracontact Metric Manifolds

2020 ◽  
Author(s):  
Mehmet Solgun
Keyword(s):  
Vector Field ◽  
Characteristic Vector ◽  
Parallel Characteristic ◽  
Metric Structures ◽  
Characteristic Vector Field

In this work, we apply the notion of D-homothetic deformation on an almost paracontact metric manifolds and show that the structure after the deformation is also almost paracontact metric structure. Also, we state the classes of almost paracontact metric structures having parallel characteristic vector field and get some results about D- homothetic deformations on these classes.

𝔇⊥-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection

2020 ◽  
Vol 20 (2) ◽  
pp. 163-168
Author(s):  
Eunmi Pak ◽  
Young Jin Suh
Keyword(s):  
Projective Space ◽  
Real Hypersurface ◽  
Jacobi Operator ◽  
Real Hypersurfaces ◽  
Quaternionic Projective Space ◽  
Totally Geodesic ◽  
Open Part ◽  
Jacobi Operators ◽  
Hopf Hypersurfaces ◽  
Normal Jacobi Operator

AbstractWe study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G2(ℂm+2) and prove that a real hypersurface in G2(ℂm+2) with generalized Tanaka–Webster 𝔇⊥-parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍPn in G2(ℂm+2), where m = 2n.

scholarly journals Oscillation Theory for the Density of States of High Dimensional Random Operators

2017 ◽  
Vol 2019 (15) ◽  
pp. 4579-4602
Author(s):  
Julian Groß mann ◽  
Hermann Schulz-Baldes ◽  
Carlos Villegas-Blas
Keyword(s):  
Density Of States ◽  
Von Neumann Algebra ◽  
Jacobi Operator ◽  
Oscillation Theory ◽  
Integrated Density Of States ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Jacobi Operators ◽  
Sturm Liouville ◽  
Finite Trace

Abstract Sturm–Liouville oscillation theory is studied for Jacobi operators with block entries given by covariant operators on an infinite dimensional Hilbert space. It is shown that the integrated density of states of the Jacobi operator is approximated by the winding of the Prüfer phase w.r.t. the trace per unit volume. This rotation number can be interpreted as a spectral flow in a von Neumann algebra with finite trace.

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