Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. Associated with $L$ one has \textit{le carr\'e du champ} $\Gamma$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $\Gamma^Z$ and we assume that $\Gamma$ and $\Gamma^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequality from Riemannian geometry, see Definition \ref{D:cdi}. In our main results we prove that, using solely \eqref{cdi}, one can develop a theory which parallels the celebrated works of Yau, and Li-Yau on complete manifolds with Ricci bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section \ref{S:appendix} we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality \eqref{cdi}. Such classes include all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is bounded from below, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is bounded from below.

Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries

We introduce anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a Langrangian Riemannian submersion, a special anti-invariant Riemannian submersion, to be totally geodesic. Moreover, we obtain decomposition theorems for the total manifold of such submersions.

Anti-invariant Riemannian submersions from almost Hermitian manifolds

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

Riemannian Submersions from Quaternionic Manifolds

A para-Kahler manifold can be defined as a pseudo-Riemannian manifold with a parallel skew-symmetric para-complex structure , that is, a parallel field of skew-symmetric endomorphisms with or, equivalently, as a symplectic manifold with a bi-Lagrangian structure , that is, two complementary integrable Lagrangian distributions. A homogeneous manifold of a semisimple Lie group admits an invariant para-Kahler structure if and only if it is a covering of the adjoint orbit of a semisimple element . A description is given of all invariant para-Kahler structures on such a homogeneous manifold. With the use of a para-complex analogue of basic formulae of Kahler geometry it is proved that any invariant para-complex structure on defines a unique para-Kahler Einstein structure with given non-zero scalar curvature. An explicit formula for the Einstein metric is given. A survey of recent results on para-complex geometry is included.

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Homogeneous para-Kähler Einstein manifolds

We introduce slant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a slant submersion to be totally geodesic. Moreover, we obtain a decomposition theorem for the total manifold of such submersions.

Slant submersions from almost Hermitian manifolds

This book provides the first-ever systematic introduction to the theory of Riemannian submersions, which was initiated by Barrett O'Neill and Alfred Gray less than four decades ago. The authors focus their attention on classification theorems when the total space and the fibres have nice geometric properties. Particular emphasis is placed on the interrelation with almost Hermitian, almost contact and quaternionic geometry. Examples clarifying and motivating the theory are included in every chapter. Recent results on semi-Riemannian submersions are also explained. Finally, the authors point out the close connection of the subject with some areas of physics.

Riemannian Submersions and Related Topics

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.

Kaluza-Klein theory with scalar fields and generalised Hopf manifolds

In this paper, we study some harmonic or 03C6-pluriharmonic maps on contact metric manifolds.

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Harmonic maps on contact metric manifolds

We study some remarkable classes of metric f-structures on differentiable manifolds (namely, almost Hermitian, almost contact, almost S-structures and K-structures). We state and prove the necessary condition(s) for the existence of maps commuting such structures. The paper contains several new results, of geometric significance, on CR-integrable manifolds and the harmonicity of such maps.

Stere Ianus

Papers

Riemannian Submersions and Related Topics

Riemannian Submersions from Quaternionic Manifolds

Kaluza-Klein theory with scalar fields and generalised Hopf manifolds

Harmonic maps on contact metric manifolds

Maps Interchanging f-Structures and their Harmonicity