Geodesics on faces of calibrations of degree two. (Q1608086)

This paper continues the article [\textit{A. N. Glushakov} and \textit{S. E. Kozlov}, Geometry of the sphere of calibration of degree two, ibid. 110, No. 4, 2776--2782 (2002); translation from Zap. Nauchn. Semin. POMI 261, 43--54 (1999; Zbl 1039.53057)]. In the space of \(p\)-forms \(\Lambda^p(\mathbb R^n)\) there is defined a Minkowski norm (the comass norm) \(| | \phi| | ^* = \sup\{\langle\phi,\omega\rangle \mid \omega \in G^+_{p,n} \subset \Lambda_p(\mathbb R^n)\}\), where \(G^+_{p,n}\) is the Grassmann manifold of unit simple \(p\)-vectors. The norm in \(\Lambda_p(R^n)\) which is conjugated to the comass norm is called the mass norm. Let \(S^*_{2,n}\) (\(S_{2,n}\)) be the unit sphere of the comass (mass) norm in \(\Lambda^2(\mathbb R^n)\) (\(\Lambda_2(\mathbb R^n)\)). The authors prove that the faces of the \(S^*_{2,n}\) (\(S_{2,n}\)) are totally geodesic submanifolds in the manifold \(ext\) of extreme points of the sphere, and that \(ext(S_{2,n}) = G^+_{2,n}\). Then they apply these results to studying intrinsic geometry of complex projective space, in particular, prove Wong's theorem [see \textit{Y.-C. Wang}, Proc. Natl. Acad. Sci. USA 60, 75--79 (1968; Zbl 0169.23904)].