scholarly journals Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation

2018 ◽  
Vol 38 (2) ◽  
pp. 201 ◽  
Author(s):  
Mitsuo Kato ◽  
Toshiyuki Mano ◽  
Jiro Sekiguchi
Keyword(s):  
Vector Fields ◽  
Potential Vector ◽  
Flat Structure ◽  
Painlevé Vi Equation ◽  
Algebraic Solutions

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

Rotation of potential vector fields

1976 ◽  
Vol 20 (2) ◽  
pp. 700-704
Author(s):  
V. S. Klimov
Keyword(s):  
Vector Fields ◽  
Potential Vector

CONTACT GEOMETRY AND RICCI SOLITONS

2010 ◽  
Vol 07 (06) ◽  
pp. 951-960 ◽  
Author(s):  
JONG TAEK CHO ◽  
RAMESH SHARMA
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Base Manifold ◽  
Gradient Ricci Soliton ◽  
Unit Tangent Bundle ◽  
Potential Vector Field

We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to En+1 × Sn(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.

scholarly journals Algebraic solutions of plane vector fields

2009 ◽  
Vol 213 (1) ◽  
pp. 144-153 ◽  
Author(s):  
S.C. Coutinho ◽  
L. Menasché Schechter
Keyword(s):  
Vector Fields ◽  
Algebraic Solutions

scholarly journals Flat Structure on the Space of Isomonodromic Deformations

2020 ◽  
Author(s):  
Mitsuo Kato ◽  
◽  
Toshiyuki Mano ◽  
Jiro Sekiguchi ◽  
◽  
...  
Keyword(s):  
Special Kind ◽  
Frobenius Manifold ◽  
Topological Field Theory ◽  
Wdvv Equation ◽  
Isomonodromic Deformations ◽  
Flat Structure ◽  
Complex Reflection Groups ◽  
Topological Field ◽  
Linear Differential ◽  
Algebraic Solutions

Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

Sign in / Sign up

Export Citation Format

Share Document