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In this paper we define the concept of quaternionic submersion, we study its fundamental properties and give an example.

PrefaceImmersions and submersions, which are special tools in Differential Geometry, also play a fundamental role in Riemannian Geometry, expecially when the involved manifolds carry an additional structure (of contact, Hermitian, quaternionic type, etc.). Even if submersions are, in a certain sense, a counterpart of immersions, the corresponding theories are quite different, also from a historical point of view.The theory of isometric immersions, started with the work of Gauss on surfaces in the Euclidean 3-space, is classical and widely explained in many books, whereas the theory of Riemannian submersion goes back to four decades ago, when B. O'Neill and A. Gray, independently, formulated the basis of such theory, which has hugely been developed in the last two decades. Nowadays several works are still in progress. For instance, a new point of view on Riemannian submersions appears in a paper by H. Karcher in 1999.Obviously the content of this book is not exhaustive, anyway, the results presented are enough to solve problems concerning many areas, like Concerning the content of each chapter, which we are going to outline, we only remark that Chap. 8 is rather independent of the others, with the exception of Chap. 1.In Chap. 1, where the basic tools of the mentioned theory are given, the main properties of the invariant tensors introduced by B. O'Neill are stated, together with explicit formulas relating the curvatures of the total space, the fibres and the base space. We also describe classical examples, like Hopf fibrations and generalized Hopf fibrations.Chapter 2 essentially concerns with Riemannian submersions having totally geodesic fibres. In particular we state the classification theorem, due to R. Escobales and A. Ranjan, on the Riemannian submersions with totally geodesic fibres and the standard rn-sphere as total space. For any n 2 1, the Hopf projections provide examples of the above submersions and the representation theory of Clifford algebras is applied to obtain the uniqueness result, up to equivalence. Then, combining a theorem of R. Escobales with a result of J. Ucci, we classify the Riemannian submersions from PZn+l (C) onto Pn(Q) with complete, complex and totally geodesic fibres.Almost Hermitian submersions are investigated into details in Chap. 3. In particular, we discuss the transference of geometric properties from the total space to the fibres and to the base space. We also describe some results, due to D. L. Johnson, in order to illustrate the relationship between the existence of Kahler submersions and of holomorphic connections on principal bundles over Kahler manifolds. Furthermore, we examine almost Hermitian submersions having integrable horizontal distribution, as it happens for Kahler, almost Kahler and nearly Kahler submersions. Considering locally conformal Kahler submersions, we explain the results of J. C. Marrero and J. Rocha and give a large class of examples, involving generalized Hopf manifolds, also known as Vaisman manifolds. Finally, we discuss almost complex conform...

The authors consider the spontaneous compactification induced by a scalar sector in the form of a non-linear sigma model. A very general class of solutions is given by Riemannian submersions from the extra dimensional space onto the space in which the scalar fields take values. An explicit example is constructed taking for the extra dimensional space a generalised Hopf manifold. A massless gauge field is associated with a vertical Killing vector of the Lee type.

In this note we introduce the concept of (?,?’)-holomorphic map between two almost quaternionic Hermitian manifolds. We prove that a (?,?’)-holomorphic map between two quaternionic Kähler manifolds with a certain property is a harmonic map and give some conditions for the stability of such a map

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Abstract. In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M × R, where M is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.

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