New Methods in Finsler Geometry

Applications of a Finsler non-linear connection

speaker: Takayoshi Ootsuka (Ochanomizu University)

abstract: We will propose a new definition of Finsler non-linear connection which can be applicable to singular Finsler metric \(L(x,dx)\) from the point-Finsler viewpoint. We define a non-linear covariant derivative, \(\nabla dx^\alpha=-dx^\mu \otimes {N^\alpha}_{\mu}(x,dx)\), on the basis of the cotangent bundle, and suppose the \({N^\alpha}_{\mu}\) satisfy, \(\frac{\partial {N^\alpha}_{\mu}}{\partial dx^\nu}-\frac{\partial {N^\alpha}_{\nu}}{\partial dx^\mu}=0, \quad \frac{\partial L}{\partial x^\mu}-{N^\alpha}_{\mu}\frac{\partial L}{\partial dx^\alpha}=0.\) We prove the existence and uniqueness of such a non-linear connection on arbitrary singular Finsler manifold \((M,L)\). Using this non-linear connection, the Euler-Lagrange equations can be written as auto-parallel equations on a certain constraint surface, and we can classify constrained systems, which are described singular Finsler metrics, as first or second class without introducing Hamiltonian structures. Furthermore we can give another definition of Killing symmetry, and construct fluid mechanics and deformed gravity on Finsler manifolds with our non-linear connection.

timetable:
Mon 21 May, 11:40 - 12:10, Aula Dini
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