Submanifold Differential Operators in $\Cal D$-Module Theory II: Generalized Weierstrass and Frenet-Serret Relations as Dirac Equations

Details Submanifold Differential Operators in $\Cal D$-Module Theory II: Generalized Weierstrass and Frenet-Serret Relations as Dirac Equations Submanifold Differential Operators in $\Cal D$-Module Theory II: Generalized Weierstrass and Frenet-Serret Relations as Dirac Equations

1999-10-09

arxiv.org/abs/math/9910052v3

Mathematics

Differential Geometry

AMS-Tex Use 22 pages

Scientific paper

This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case and completely represent the submanifolds. For example, the analytic index of Dirac operator of a space curve is identified with its writhing number. As another example, the operator determinants of the Dirac operators are closely related to invariances of the immersed objects, such as Euler-Bernoulli and Willmore functionals for a space curve and a conformal surface respectively. In this article, we will give mathematical construction of the Dirac operator by means of $\Cal D$-module and reformulate my recent results mathematically.

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