• Sasakian manifolds
• Ricci solitons
• Legendrian submanifolds
• Kaehler manifolds
• Extremal metrics
• deformations

The thesis deals with four different problems in Sasakian and Kaehler geometry.

We extend to the Sasakian setting a result of Tian and Zhu about the decomposition of the Lie algebra of holomorphic vector fields on a Kaehler manifold in the presence of a Kaehler-Ricci soliton. We apply deformations of Sasakian structures to a Sasaki-Ricci soliton to obtain a stability result concerning generalized Sasaki-Ricci solitons, generalizing a Kaehler result of Li and also He and Song by relaxing some of their assumptions.

Then, dealing with Kaehler metrics but the argument can work also in the Sasakian ones, we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.

Then we put ourselves in the Sasaki-Einstein case and, given minimal Legendrian submanifold L of a Sasaki-Einstein manifold we construct two families of eigenfunctions of the Laplacian of L and we give a lower bound for the dimension of the corresponding eigenspace. Moreover, in the case the lower bound is attained, we prove that L is totally geodesic and a rigidity result about the ambient manifold. This is a generalization of a result for the standard Sasakian sphere done by Le and Wang.

Finally we consider a space of Sasakian metrics on a given Sasakian manifold. We find that the Ebin metric restricted to the space of type II Sasaki deformations is twice the sum of the Calabi metric and the gradient metric, the natural Sasakian analogues of two known metrics that can be defined on the space of Kaehler metrics. We solve the short time existence of geodesics for this sum metric. By the same techniques we are able also to answer a question by Calabi about the existence of geodesics of the gradient metric for the space of Kaehler potentials for any initial position and velocity.Namely we prove the existence of short time geodesics for the gradient metric as well.