Matsutani, Shigeki Submanifold Dirac operators with torsion. (English) Zbl 1210.53003 Balkan J. Geom. Appl. 9, No. 1, 69-82 (2004). The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. Following the algebraic construction, the author introduces another submanifold Dirac operator which has a gauge field associated with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in \(\mathbb E^4\). The authors provide the theorems which are connected with the generalized Weierstrass relation. For a space curve in three dimensional Euclidean space \(\mathbb E^3\), Takagi and Tanzawa found a submanifold Schrödinger operator with a gauge field, \[ S:=-(\partial_s-\sqrt{-1}a)^2-\frac{1}{4}k^2,\tag{1} \] where \(k\) is a curvature of the curve. Using the algebraic construction, the author generalizes the Takagi-Tanzawa Schrödinger operator (1) to the submanifold Dirac operator over a surface immersed in \(\mathbb E^4\). Reviewer: Serguey M. Pokas (Odessa) Cited in 1 Document MSC: 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 51N20 Euclidean analytic geometry 81T20 Quantum field theory on curved space or space-time backgrounds 53A05 Surfaces in Euclidean and related spaces Keywords:Takagi-Tanzawa Schrödinger operator; Dirac system; Clifford module; Hermit conjugate operator; preHilbert space; Dirac particle; submanifold Schrödinger operator; submanifold Dirac equations; gauge field; Frenet-Serret torsion; generalized Weierstrass relation PDFBibTeX XMLCite \textit{S. Matsutani}, Balkan J. Geom. Appl. 9, No. 1, 69--82 (2004; Zbl 1210.53003) Full Text: arXiv EuDML