Finsler geometry of topological singularities for multi-valued fields: Applications to continuum theory of defects

Takahiro Yajima, Hiroyuki Nagahama

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

Topological singularity in a continuum theory of defects and a quantum field theory is studied from a viewpoint of differential geometry. The integrability conditions of singularity (Clairaut-Schwarz-Young theorem) are expressed by a torsion tensor and a curvature tensor when a Finslerian intrinsic parallelism holds for the multi-valued function. In the context of the quantum field theory, the singularity called an extended object is expressed by the torsion when the intrinsic parallelism is related to the spontaneous breakdown of symmetry. In the continuum theory of defects, the path-dependency of point and line defects within a crystal is interpreted by the non-vanishing condition of torsion tensor in a non-Riemannian space osculated from the Finsler space, and the domain is not simply connected. On the other hand, for the rotational singularity, an energy integral (J-integral) around a disclination field is path-independent when a nonlinear connection is single-valued. This means that the topological expression for the sole defect (Gauss-Bonnet theorem with genus g=1) is understood by the integrability of nonlinear connection. (Figure presented.).

Original languageEnglish
Pages (from-to)845-851
Number of pages7
JournalAnnalen der Physik
Volume528
Issue number11-12
DOIs
Publication statusPublished - 2016 Dec 1

Keywords

  • Finsler geometry
  • continuum theory of defects
  • extended object
  • generalized connection structure
  • multi-valued field
  • topological singularity

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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