Characteristic classes of oriented 6-dimensional submanifolds in the octonians
The aim of this paper is to study the characteristic classes of oriented 6-dimensional submanifolds \((M^ 6,I)\) in the space \(\mathbb{R}^ 8\) with the immersion \(I\). \(M^ 6 = (M^ 6,J, \langle; \rangle)\) is an almost Hermitian manifold. Using the structure equations of \((M^ 6,I)\) due to \textit{R. Bryant} [J. Differ. Geom. 17, 185-232 (1982; Zbl 0526.53055)] and a theorem of \textit{C. T. C. Wall} [Invent. Math. 1, 355-374 (1966; Zbl 0149.206)], the author proves the theorem: ``Let \(M^ 6 = (M^ 5, J, \langle; \rangle)\) be a 6-dimensional almost Hermitian submanifold immersed in the octonians \(O\). Then whe have the following relations: (1) \(c_ 1 (T^{1,0}) = - c_ 1 (\nu^{1,0}) = - e(\nu)\); (2) \(c_ 2 (T^{1,0}) = c_ 1 (T^{1,0})^ 2\); (3) \(p_ 1 (TM^ 6) + c_ 1 (T^{1,0})^ 2 = 0\), where \(p_ 1(TM^ 6)\) is the 1-st Pontryagin class of the tangent bundle \(TM^ 6\) of \(M^ 6\), \(c_ i(T^{1,0})\) is the \(i\)-th Chern class of the bundle \(T^{1,0} = \{ v \in TM^ 6 \otimes C/Jv = \sqrt {-1} v\}\), \(e(\nu)\) is the Euler class of the normal bundle \(\nu\) and \(c_ 1 (\nu^{1,0})\) is the 1-st Chern class of the bundle \(\nu^{1,0} = \{v \in \nu \otimes C/Jv = \sqrt {-1} v\}\), respectively''. A corollary of this theorem is a generalization of a result due to \textit{E. Calabi} [Trans. Am. Math. Soc. 87, 407-438 (1958; Zbl 0080.376)]. Finally, some applications and examples are presented.
characteristic classes of oriented 6-dimensional submanifolds in \(\mathbb{R}^ 8\)
scientific article; zbMATH DE number 409644
scientific article