𝔇⊥-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection

We study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G2(ℂm+2) and prove that a real hypersurface in G2(ℂm+2) with generalized Tanaka–Webster 𝔇⊥-parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍPn in G2(ℂm+2), where m = 2n.

ON THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE – ADDENDUM

The page numbers shown in Appendix B have been updated to reflect the final pagination of the article.

PROJECTIVE STRUCTURES AND -CONNECTIONS

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

ON THE SYMMETRIC SQUARES OF COMPLEX AND QUATERNIONIC PROJECTIVE SPACE

The problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently. We offer a solution for the complex and quaternionic projective spaces$\mathbb{K}$Pn, by utilising their rich geometrical structure. Our description involves generators and relations, and our methods entail ideas from the literature of quantum chemistry, theoretical physics, and combinatorics. We begin with the case$\mathbb{K}$P∞, and then identify the truncation required for passage to finiten. The calculations rely upon a ladder of long exact cohomology sequences, which compares cofibrations associated to the diagonals of the symmetric square and the corresponding Borel construction. These incorporate the one-point compactifications of classic configuration spaces of unordered pairs of points in$\mathbb{K}$Pn, which are identified as Thom spaces by combining Löwdin's symmetric orthogonalisation (and its quaternionic analogue) with a dash ofPingeometry. The relations in the ensuing cohomology rings are conveniently expressed using generalised Fibonacci polynomials. Our conclusions are compatible with those of Gugnin mod torsion and Nakaoka mod 2, and with homological results of Milgram.

On submersion and immersion submanifolds of a quaternionic projective space

We study submanifolds of a quaternionic projective space, it is of great interest how to pull down some formulae deduced for submanifolds of a sphere to those for submanifolds of a quaternionic projective space.

Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator

Regarding the generalized Tanaka-Webster connection, we considered a new notion of $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster $$\mathfrak{D}^ \bot$$-parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.

QR-Submanifolds of(p−1) QR-Dimension in a Quaternionic Projective SpaceQP(n+p)/4under Some Curvature Conditions

The purpose of this paper is to studyn-dimensionalQR-submanifolds of(p−1)QR-dimension in a quaternionic projective spaceQP(n+p)/4and especially to determine such submanifolds under some curvature conditions.