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Статті в журналах з теми "Jacobi manifolds"

Автор: Grafiati

Опубліковано: 10 березня 2023

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1

GATSE, Servais Cyr. "Jacobi Manifolds, Contact Manifolds and Contactomorphism." Journal of Mathematics Research 13, no. 4 (July 29, 2021): 85. http://dx.doi.org/10.5539/jmr.v13n4p85.

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Анотація:

Let M be a smooth manifold and let D(M) be the module of first order differential operators on M. In this work, we give a link between Jacobi manifolds and Contact manifolds. We also generalize the notion of contactomorphism on M and thus, we characterize the Contact diffeomorphisms.

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2

Chen, Bang-Yen, Sharief Deshmukh, and Amira A. Ishan. "On Jacobi-Type Vector Fields on Riemannian Manifolds." Mathematics 7, no. 12 (November 21, 2019): 1139. http://dx.doi.org/10.3390/math7121139.

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Анотація:

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

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3

Okassa, Eugène. "Symplectic Lie–Rinehart–Jacobi Algebras and Contact Manifolds." Canadian Mathematical Bulletin 54, no. 4 (December 1, 2011): 716–25. http://dx.doi.org/10.4153/cmb-2011-033-6.

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4

Blažić, Novica, Neda Bokan, and Zoran Rakić. "Osserman pseudo-Riemannian manifolds of signature (2,2)." Journal of the Australian Mathematical Society 71, no. 3 (December 2001): 367–96. http://dx.doi.org/10.1017/s1446788700003001.

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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.

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5

Vitagliano, Luca, and Aïssa Wade. "Holomorphic Jacobi manifolds." International Journal of Mathematics 31, no. 03 (February 14, 2020): 2050024. http://dx.doi.org/10.1142/s0129167x2050024x.

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Анотація:

In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi–Nijenhuis structures.

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6

Haji-Badali, Ali, and Amirhesam Zaeim. "Commutative curvature operators over four-dimensional homogeneous manifolds." International Journal of Geometric Methods in Modern Physics 12, no. 10 (October 25, 2015): 1550123. http://dx.doi.org/10.1142/s0219887815501236.

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Анотація:

Four-dimensional pseudo-Riemannian homogeneous spaces whose isotropy is non-trivial with commuting curvature operators have been studied. The only example of homogeneous Einstein four-manifold which is curvature-Ricci commuting but not semi-symmetric has been presented. Non-trivial examples of semi-symmetric homogeneous four-manifolds which are not locally symmetric, also Jacobi–Jacobi commuting manifolds which are not flat have been presented.

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7

Bascone, Francesco, Franco Pezzella, and Patrizia Vitale. "Topological and Dynamical Aspects of Jacobi Sigma Models." Symmetry 13, no. 7 (July 5, 2021): 1205. http://dx.doi.org/10.3390/sym13071205.

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Анотація:

The geometric properties of sigma models with target space a Jacobi manifold are investigated. In their basic formulation, these are topological field theories—recently introduced by the authors—which share and generalise relevant features of Poisson sigma models, such as gauge invariance under diffeomorphisms and finite dimension of the reduced phase space. After reviewing the main novelties and peculiarities of these models, we perform a detailed analysis of constraints and ensuing gauge symmetries in the Hamiltonian approach. Contact manifolds as well as locally conformal symplectic manifolds are discussed, as main instances of Jacobi manifolds.

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8

Ibáñez, Raul, Manuel de León, Juan C. Marrero, and Edith Padrón. "Nambu-Jacobi and generalized Jacobi manifolds." Journal of Physics A: Mathematical and General 31, no. 4 (January 30, 1998): 1267–86. http://dx.doi.org/10.1088/0305-4470/31/4/015.

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9

Gilkey, Peter, and Veselin Videv. "Jacobi–Jacobi Commuting Models and Manifolds." Journal of Geometry 92, no. 1-2 (January 24, 2009): 60–68. http://dx.doi.org/10.1007/s00022-008-2061-9.

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10

Cariñena, J. F., X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda, and N. Román-Roy. "Structural aspects of Hamilton–Jacobi theory." International Journal of Geometric Methods in Modern Physics 13, no. 02 (January 26, 2016): 1650017. http://dx.doi.org/10.1142/s0219887816500171.

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Анотація:

In our previous papers [J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton–Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1417–1458; Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 431–454] we showed that the Hamilton–Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton–Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (slicing vector fields) on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton–Jacobi theory, by considering special cases like fibered manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.

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11

Marrero, Juan C., Juan Monterde, and Edith Padron. "Jacobi—Nijenhuis manifolds and compatible Jacobi structures." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 329, no. 9 (January 1999): 797–802. http://dx.doi.org/10.1016/s0764-4442(99)90010-1.

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12

VAISMAN, IZU. "COUPLING POISSON AND JACOBI STRUCTURES ON FOLIATED MANIFOLDS." International Journal of Geometric Methods in Modern Physics 01, no. 05 (October 2004): 607–37. http://dx.doi.org/10.1142/s0219887804000307.

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Let M be a differentiable manifold endowed with a foliation ℱ. A Poisson structure P on M is ℱ-coupling if ♯P(ann(Tℱ)) is a normal bundle of the foliation. This notion extends Sternberg's coupling symplectic form of a particle in a Yang–Mills field [11]. In the present paper we extend Vorobiev's theory of coupling Poisson structures [16] from fiber bundles to foliated manifolds and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. We then discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.

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13

Ibort, A., M. de León, and G. Marmo. "Reduction of Jacobi manifolds." Journal of Physics A: Mathematical and General 30, no. 8 (April 21, 1997): 2783–98. http://dx.doi.org/10.1088/0305-4470/30/8/022.

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14

OKASSA, EUGÈNE. "ON LIE–RINEHART–JACOBI ALGEBRAS." Journal of Algebra and Its Applications 07, no. 06 (December 2008): 749–72. http://dx.doi.org/10.1142/s0219498808003107.

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Анотація:

We show that Jacobi algebras (Poisson algebras respectively) can be defined only as Lie–Rinehart–Jacobi algebras (as Lie–Rinehart–Poisson algebras respectively). Also we show that contact manifolds, locally conformal symplectic manifolds (symplectic manifolds respectively) can be defined only as symplectic Lie–Rinehart–Jacobi algebras (only as symplectic Lie–Rinehart–Poisson algebras respectively). We define symplectic Lie algebroids.

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15

GILKEY, P., and S. NIKČEVIĆ. "PSEUDO-RIEMANNIAN JACOBI–VIDEV MANIFOLDS." International Journal of Geometric Methods in Modern Physics 04, no. 05 (August 2007): 727–38. http://dx.doi.org/10.1142/s0219887807002272.

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Анотація:

We exhibit several families of Jacobi–Videv pseudo-Riemannian manifolds which are not Einstein. We also exhibit Jacobi–Videv algebraic curvature tensors where the Ricci operator defines an almost complex structure.

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16

Costa, J. M. Nunes da, and F. Petalidou. "Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids." Journal of Physics A: Mathematical and General 39, no. 33 (August 2, 2006): 10449–75. http://dx.doi.org/10.1088/0305-4470/39/33/014.

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17

Cioroianu, Eugen-Mihaita, and Cornelia Vizman. "Jacobi structures with background." International Journal of Geometric Methods in Modern Physics 17, no. 04 (March 2020): 2050063. http://dx.doi.org/10.1142/s0219887820500632.

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Анотація:

Combining the twisted Jacobi structure [Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen. 39(33) (2006) 10449–10475] with that of a Poisson structure with a 3-form background [Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001) 145–154], alias twisted Poisson, we propose and analyze a new structure, called Jacobi structure with background. The background is a pair consisting of a [Formula: see text]-form and a [Formula: see text]-form. We describe their characteristic leaves. For twisted contact dual pairs, we show the correspondence of characteristic leaves of the two Jacobi manifolds with background.

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18

Munteanu, Marian Ioan, and Ana Irina Nistor. "Magnetic Jacobi Fields in 3-Dimensional Cosymplectic Manifolds." Mathematics 9, no. 24 (December 13, 2021): 3220. http://dx.doi.org/10.3390/math9243220.

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Анотація:

We classify the magnetic Jacobi fields in cosymplectic manifolds of dimension 3, enriching the results in the study of magnetic Jacobi fields derived from uniform magnetic fields. In particular, we give examples of Jacobi magnetic fields in the Euclidean space E3 and we conclude with the description of magnetic Jacobi fields in the product spaces S2×R and H2×R.

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19

Schmidt, Benjamin, Krishnan Shankar, and Ralf Spatzier. "Almost isotropic Kähler manifolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 767 (October 1, 2020): 1–16. http://dx.doi.org/10.1515/crelle-2019-0030.

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Анотація:

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}}.

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20

Nunes da Costa, J. M., and Fani Petalidou. "Reduction of Jacobi–Nijenhuis manifolds." Journal of Geometry and Physics 41, no. 3 (March 2002): 181–95. http://dx.doi.org/10.1016/s0393-0440(01)00054-7.

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21

Pirhadi, Vahid, and Asadollah Razavi. "Integrability of Transitive Jacobi Manifolds." Iranian Journal of Science and Technology, Transactions A: Science 43, no. 4 (August 13, 2018): 1657–64. http://dx.doi.org/10.1007/s40995-018-0609-6.

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22

Beltran, J. V. "Star calculus on Jacobi manifolds." Differential Geometry and its Applications 16, no. 2 (March 2002): 181–98. http://dx.doi.org/10.1016/s0926-2245(02)00059-1.

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23

Brozos-Vázquez, Miguel, and Peter Gilkey. "Manifolds with commuting Jacobi operators." Journal of Geometry 86, no. 1-2 (April 2007): 21–30. http://dx.doi.org/10.1007/s00022-006-1898-z.

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24

Kal'nitskii, V. S. "Jacobi Algebras on Flat Manifolds." Journal of Mathematical Sciences 131, no. 1 (November 2005): 5345–50. http://dx.doi.org/10.1007/s10958-005-0406-6.

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25

Yoldas, Halil. "Some Results on Cosymplectic Manifolds Admitting Certain Vector Fields." Journal of Geometry and Symmetry in Physics 60 (2021): 83–94. http://dx.doi.org/10.7546/jgsp-60-2021-83-94.

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Анотація:

The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.

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26

Fitzpatrick, Sean. "On the Geometry of Almost -Manifolds." ISRN Geometry 2011 (December 13, 2011): 1–12. http://dx.doi.org/10.5402/2011/879042.

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An -structure on a manifold is an endomorphism field satisfying . We call an f-structure regular if the distribution is involutive and regular, in the sense of Palais. We show that when a regular f-structure on a compact manifold M is an almost -structure, it determines a torus fibration of M over a symplectic manifold. When rank , this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with -structure or -structure, we do not assume that the f-structure is normal. We also show that given an almost -structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.

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27

BLAŽIĆ, N., and P. GILKEY. "CONFORMALLY OSSERMAN MANIFOLDS AND CONFORMALLY COMPLEX SPACE FORMS." International Journal of Geometric Methods in Modern Physics 01, no. 01n02 (April 2004): 97–106. http://dx.doi.org/10.1142/s021988780400006x.

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We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.

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28

Petalidou, Fani, and J. M. Nunes da Costa. "Local structure of Jacobi–Nijenhuis manifolds." Journal of Geometry and Physics 45, no. 3-4 (March 2003): 323–67. http://dx.doi.org/10.1016/s0393-0440(01)00074-2.

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29

Chinea, Domingo, Juan C. Marrero, and Manuel de León. "Prequantizable Poisson manifolds and Jacobi structures." Journal of Physics A: Mathematical and General 29, no. 19 (October 7, 1996): 6313–24. http://dx.doi.org/10.1088/0305-4470/29/19/016.

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30

Costa, J. M. Nunes da. "Compatible Jacobi manifolds: geometry and reduction." Journal of Physics A: Mathematical and General 31, no. 3 (January 23, 1998): 1025–33. http://dx.doi.org/10.1088/0305-4470/31/3/013.

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31

Obukhov, Valeriy V. "Algebra of the Symmetry Operators of the Klein–Gordon–Fock Equation for the Case When Groups of Motions G3 Act Transitively on Null Subsurfaces of Spacetime." Symmetry 14, no. 2 (February 9, 2022): 346. http://dx.doi.org/10.3390/sym14020346.

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Анотація:

The algebras of the symmetry operators for the Hamilton–Jacobi and Klein–Gordon–Fock equations are found for a charged test particle, moving in an external electromagnetic field in a spacetime manifold on the isotropic (null) hypersurface, of which a three-parameter groups of motions acts transitively. We have found all admissible electromagnetic fields for which such algebras exist. We have proved that an admissible field does not deform the algebra of symmetry operators for the free Hamilton–Jacobi and Klein–Gordon–Fock equations. The results complete the classification of admissible electromagnetic fields, in which the Hamilton–Jacobi and Klein–Gordon–Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of the r-parametric groups of motions of spacetime manifolds if (r≤4).

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32

Cho, Jong Taek, and Makoto Kimura. "Transversal Jacobi Operators in Almost Contact Manifolds." Mathematics 9, no. 1 (December 24, 2020): 31. http://dx.doi.org/10.3390/math9010031.

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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.

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33

Bruce, Andrew James. "Odd Jacobi Manifolds and Loday-Poisson Brackets." Journal of Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/630749.

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We construct a nonskew symmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.

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34

Wang, Wenjie, and Xinxin Dai. "Pseudo-Parallel Characteristic Jacobi Operators on Contact Metric 3 Manifolds." Journal of Mathematics 2021 (July 21, 2021): 1–6. http://dx.doi.org/10.1155/2021/6148940.

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Анотація:

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.

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35

BONOME, Agustin, Regina CASTRO, Eduardo GARCÍA-RÍO, Luis HERVELLA, and Ramón VÁZQUEZ-LORENZO. "Pseudo-Riemannian manifolds with simple Jacobi operators." Journal of the Mathematical Society of Japan 54, no. 4 (October 2002): 847–75. http://dx.doi.org/10.2969/jmsj/1191591994.

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36

Chinea, Domingo, Juan C. Marrero, and Manuel de León. "A canonical differential complex for Jacobi manifolds." Michigan Mathematical Journal 45, no. 3 (December 1998): 547–79. http://dx.doi.org/10.1307/mmj/1030132300.

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37

de León, Manuel, Juan C. Marrero, and Edith Padrón. "On the geometric quantization of Jacobi manifolds." Journal of Mathematical Physics 38, no. 12 (December 1997): 6185–213. http://dx.doi.org/10.1063/1.532207.

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38

Brozos-Vázquez, M., and P. Gilkey. "Pseudo-riemannian manifolds with commuting jacobi operators." Rendiconti del Circolo Matematico di Palermo 55, no. 2 (June 2006): 163–74. http://dx.doi.org/10.1007/bf02874699.

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39

Lê, Hong Vân, Yong-Geun Oh, Alfonso G. Tortorella, and Luca Vitagliano. "Deformations of coisotropic submanifolds in Jacobi manifolds." Journal of Symplectic Geometry 16, no. 4 (2018): 1051–116. http://dx.doi.org/10.4310/jsg.2018.v16.n4.a7.

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40

de León, Manuel, David Martín de Diego, and Miguel Vaquero. "A Hamilton-Jacobi theory on Poisson manifolds." Journal of Geometric Mechanics 6, no. 1 (2014): 121–40. http://dx.doi.org/10.3934/jgm.2014.6.121.

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41

Vitagliano, Luca, and Aïssa Wade. "Holomorphic Jacobi manifolds and holomorphic contact groupoids." Mathematische Zeitschrift 294, no. 3-4 (May 20, 2019): 1181–225. http://dx.doi.org/10.1007/s00209-019-02320-x.

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42

Zapata-Carratalá, Carlos. "Jacobi geometry and Hamiltonian mechanics: The unit-free approach." International Journal of Geometric Methods in Modern Physics 17, no. 12 (September 24, 2020): 2030005. http://dx.doi.org/10.1142/s0219887820300056.

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Анотація:

We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function [Formula: see text]. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

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43

Andrejic, Vladica. "Quasi-special Osserman manifolds." Filomat 28, no. 3 (2014): 623–33. http://dx.doi.org/10.2298/fil1403623a.

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In this paper we deal with a pseudo-Riemannian Osserman curvature tensor whose reduced Jacobi operator is diagonalizable with exactly two distinct eigenvalues. The main result gives new insight into the theory of the duality principle for pseudo-Riemannian Osserman manifolds. We concern with special Osserman curvature tensor and propose new ways to exclude some additional duality principle conditions from its definition.

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44

Andrejic, Vladica. "On Lorentzian spaces of constant sectional curvature." Publications de l'Institut Math?matique (Belgrade) 103, no. 117 (2018): 7–15. http://dx.doi.org/10.2298/pim1817007a.

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We investigate Osserman-like conditions for Lorentzian curvature tensors that imply constant sectional curvature. It is known that Osserman (moreover zwei-stein) Lorentzian manifolds have constant sectional curvature. We prove that some generalizations of the Rakic duality principle (Lorentzian totally Jacobi-dual or four-dimensional Lorentzian Jacobi-dual) imply constant sectional curvature. Moreover, any four-dimensional Jacobi-dual algebraic curvature tensor such that the Jacobi operator for some nonnull vector is diagonalizable, is Osserman. Additionally, any Lorentzian algebraic curvature tensor such that the reduced Jacobi operator for all nonnull vectors has a single eigenvalue has a constant sectional curvature.

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45

Rovenski, Vladimir, and Dhriti Sundar Patra. "Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons." Fractal and Fractional 7, no. 2 (February 4, 2023): 156. http://dx.doi.org/10.3390/fractalfract7020156.

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This article presents some results of a geometric classification of Sasakian manifolds (SM) that admit an almost ∗-Ricci soliton (RS) structure (g,ω,X). First, we show that a complete SM equipped with an almost ∗-RS with ω≠ const is a unit sphere. Then we prove that if an SM has an almost ∗-RS structure, whose potential vector is a Jacobi vector field on the integral curves of the characteristic vector field, then the manifold is a null or positive SM. Finally, we characterize those SM represented as almost ∗-RS, which are ∗-RS, ∗-Einstein or ∗-Ricci flat.

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46

Berndt, Jürgen, Friedbert Prüfer, and Lieven Vanhecke. "Symmetric-like Riemannian manifolds and geodesic symmetries." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 2 (1995): 265–82. http://dx.doi.org/10.1017/s0308210500028031.

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47

Hounkonnou, Mahouton Norbert, Mahougnon Justin Landalidji, and Melanija Mitrović. "Einstein Field Equation, Recursion Operators, Noether and Master Symmetries in Conformable Poisson Manifolds." Universe 8, no. 4 (April 17, 2022): 247. http://dx.doi.org/10.3390/universe8040247.

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We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Christoffel symbols, components of Riemann and Ricci tensors, Ricci constant and components of Einstein tensor. We highlight the existence of a hierarchy of bi-Hamiltonian structures in both the manifolds, and compute a family of recursion operators and master symmetries generating the constants of motion.

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48

Rezaei-Aghdam, A., and M. Sephid. "Jacobi–Lie symmetry and Jacobi–Lie T-dual sigma models on group manifolds." Nuclear Physics B 926 (January 2018): 602–13. http://dx.doi.org/10.1016/j.nuclphysb.2017.12.003.

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49

Esen, Oğul, Manuel de León, Cristina Sardón, and Marcin Zajşc. "Hamilton–Jacobi formalism on locally conformally symplectic manifolds." Journal of Mathematical Physics 62, no. 3 (March 1, 2021): 033506. http://dx.doi.org/10.1063/5.0021790.

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50

JAVALOYES, MIGUEL ANGEL, and BRUNO LEARTH SOARES. "Geodesics and Jacobi fields of pseudo-Finsler manifolds." Publicationes Mathematicae Debrecen 87, no. 1-2 (June 1, 2015): 57–78. http://dx.doi.org/10.5486/pmd.2015.7028.

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