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

Structures on Manifolds

We characterize almost contact metric manifolds which are CR-integrable almost Kenmotsu, through the existence of a suitable linear connection. We classify almost Kenmotsu manifolds satisfying a certain nullity condition, we give examples and completely describe the three dimensional case.

Almost Kenmotsu Manifolds and Nullity Distributions

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

Riemannian Submersions from Quaternionic 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

We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field $\xi$, vanishes. Furthermore, assuming that for a $(2n+1)$-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies $R_{XY}\xi =0$ for any $X, Y$ orthogonal to $\xi$, we prove that the manifold is locally isometric to the Riemannian product of an $(n+1)$-dimensional manifold of constant curvature $-4$ and a flat $n$-dimensional manifold. We give an example of such a manifold.

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Almost Kenmotsu manifolds and local symmetry

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.

Anna Maria Pastore

Papers

Riemannian Submersions and Related Topics

Almost Kenmotsu manifolds and local symmetry

Almost Kenmotsu Manifolds and Nullity Distributions

Harmonic maps on contact metric manifolds

Maps Interchanging f-Structures and their Harmonicity