scholarly journals METRIC COMPATIBLE OR NON-COMPATIBLE FINSLER–RICCI FLOWS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250041 ◽  
Author(s):  
SERGIU I. VACARU
Keyword(s):  
Ricci Flow ◽  
Evolution Equations ◽  
Finsler Geometry ◽  
Flow Theory ◽  
Finsler Metric ◽  
Geometric Evolution ◽  
Ricci Flows ◽  
Canonical Metric ◽  
Self Consistent ◽  
Math Phys

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature, etc. In a series of works, we studied (non)-commutative metric compatible Finsler and non-holonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The aim of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived non-holonomic Hamilton evolution equations, when metric non-compatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric non-compatible connections.

scholarly journals Dynamical equations and Lagrange–Ricci flow evolution on prolongation Lie algebroids

2019 ◽  
Vol 97 (2) ◽  
pp. 145-154
Author(s):  
Laurenţiu Bubuianu ◽  
Sergiu I. Vacaru
Keyword(s):  
Lie Algebroids ◽  
Gravity Models ◽  
Geometric Mechanics ◽  
Dynamical Equations ◽  
Geometric Evolution ◽  
Ricci Flows ◽  
Hamilton Equations ◽  
Flow Evolution ◽  
Statistical Functionals ◽  
Math Phys

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems (S. Vacaru. J. Math. Phys. 49, 043504 (2008); Ibid. Rep. Math. Phys. 63, 95 (2009)) is extended to include geometric mechanics and gravity models on Lie algebroids. We prove that such evolution scenarios of geometric mechanics and analogous gravity can be modeled as gradient flows characterized by generalized Perelman functionals if an equivalent geometrization of Lagrange mechanics (J. Kern. Arch. Math. (Basel), 25, 438 (1974)) is considered. The Hamilton equations on Lie algebroids describing Lagrange–Ricci flows are derived. Finally, we show that geometric evolution models on Lie algebroids are described by effective thermodynamical values derived from statistical functionals on prolongation Lie algebroids.

scholarly journals The Β-Change by Finsler Metric of C-Reducible Finsler Spaces in Finsler Geometry

2019 ◽  
Vol 33 (1) ◽  
pp. 1-10
Author(s):  
Khageswar Mandal
Keyword(s):  
Finsler Space ◽  
Homogeneous Function ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Spaces

 This paper considered about the β-Change of Finsler metric L given by L*= f(L, β), where f is any positively homogeneous function of degree one in L and β and obtained the β-Change by Finsler metric of C-reducible Finsler spaces. Also further obtained the condition that a C-reducible Finsler space is transformed to a C-reducible Finsler space by a β-change of Finsler metric.

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