Spin(7) $\operatorname{Spin}(7)$-structure equation and the vector elliptic Liouville equation

Abstract

Abstract The mapping between Belavin–Polyakov (BP) equation for the evolution of a unit tangent vector T∈S2 $T\in \mathbb{S}^{2}$ of a space curve in R3 $\mathbb{R}^{3}$ and the elliptic Liouville equation has been shown by Balakrishnan (see Phys. Lett. A 204:243–246, 1995). In the present work, this result is effectively extended by mapping the BP equation for the unit tangent T∈S6 $T\in \mathbb{S}^{6}$ of a space curve in R7 $\mathbb{R}^{7}$ to the vector elliptic Liouville equation. To show this correspondence, Spin(7) $\operatorname{Spin}(7)$-frame field on the curve is used.

Keywords

• Spin ( 7 ) $\operatorname{Spin}(7)$ -structure equation
• Octonions
• Almost complex structure
• The vector elliptic Liouville equation