The axiom of spheres in quaternionic geometry (original) (raw)

On the axioms of planes in quaternionic geometry

Annali di Matematica Pura ed Applicata, 1982

The axioms o/planes in Riemanniau geometry and Kaehlerian geomet~'y have been largely studied. In this paper we study axioms/or three kinds o] planes in Quaternionic geometry: the axiom o] g~aternionic 4.planes, the axiom o] hal].quaternionic planes and the axiom o] totally real planes. We also give a characterization o] quaternion.space-yorms in terms o] the constancy o] the totally real sectional curvatures. 1.-Introduction.

Differential geometry of quaternionic manifolds

Annales scientifiques de l'École normale supérieure

http://www.numdam.org/item?id=ASENS\_1986\_4\_19\_1\_31\_0 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1986, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

A direct approach to quaternionic manifolds

Mathematische Nachrichten, 2016

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on H n , in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.

Quaternion Kaehlerian Manifolds Isometrically Immersed in Euclidean Space

Proceedings of the American Mathematical Society, 1983

Let M be a complete 4u-dimensional quaternion Kaehlerian manifold isometrically immersed in the (4« + d (-dimensional Euclidean space. In this note we prove that if d < n, then M is a Riemannian product Q"' X P, where Q'" is the m-dimensional quaternion Euclidean space ( m > n -d ) and P is a Ricci flat quaternion Kaehlerian manifold.

Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions

1993

Geometria differenziale.-Quaternionic-like structures on a manifold: Note L 1-integrability and integrability conditions. Nota(*) di DMITRI V. ALEKSEEVSKY e STE-FANO MARCHIAFAVA, presentata dal Socio E. Martinelli. ABSTRACT.-This Note will be followed by a Note II in these Rendiconti and successively by a wider and more detailed memoir to appear next. Here six quaternionic-like structures on a manifold M (almost quaternionic, hypercomplex, unimodular quaternionic, unimodular hypercomplex, Hermitian quaternionic, Hermitian hypercomplex) are defined and interrelations between them are studied in the framework of general theory of G-structures. Special connections are associated to these structures. 1-integrability and integrability conditions are derived. Decompositions of appropriate spaces of curvature tensors are given. In Note II the automorphism groups of these quaternionic-like structures will be considered.

On Ricci curvature of a quaternion CR-submanifold in a quaternion space form

B. Y. Chen [Kodai Math. J. 4, 399–417 (1981; Zbl 0481.53046)] established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Recently Ximin Liu [Arch. Math. (Brno), 38, 297–305 (2002; Zbl 1090.53052)] obtained results on Ricci curvature of a totally real submanifold in a quaternion projective space extending the results of Chen. In this article, we wish to estimate the Ricci curvature of a quaternion CR-submanifold in a quaternion space form.

Quaternionic structures on a manifold and subordinated structures

Annali di Matematica Pura ed Applicata, 1996

We study differential geometry of an (almost) quaternionic structure Q and of an (almost) hypercomplex structure H on a manifold M 4~ , as well as other quaternioniclike structures (Q, vol), (H, vol), (Q, g), (H, g) which are obtained by adding to Q or H a volume form or a compatible (i.e. Hermitian) metric g. A unified way to do this is provided by the theory of G-structures. In fact these quaternionic-like structures may be identified as G-structures with one of the groups