We will study geometric properties of Sasakian and K"ahler
quotients. In the presence of Lie group symmetries, we construct a reduction procedure for symplectic and K"ahler manifolds using the ray preimages of the momentum map $J$. More precisely, instead of taking as in point reduction (Weinstein-Marsden reduce spaces, usually denoted $M_mu$), the preimage of a momentum value $mu$, we take the preimage of $RR^+mu$, the positive ray of $mu$. We have three reasons to develop this construction. One is geometric:
the construction of canonical K"ahler reduced spaces
corresponding to a non zero momentum. By canonical we mean that the reduced K"ahler structure is the projection of the initial Kähler structure. And the Weinstein-Marsden reduction is not always canonical for K"ahler manifolds. The second reason is anapplication to the study of conformal Hamiltonian systems. They are mechanical, non-autonomus systems with friction whose integral curves preserve, in the case of symmetries, the ray pre-images of the momentum map, and not the momentum preimages used in the Marsden-Weinstein quotient. Finally, the third reason consists of finding necessary and sufficient conditions for the ray reduced spaces of K"ahler (Sasakian)-Einstein manifolds to be also K"ahler (Sasakian)-Einstein. During this talk we will mostly concentrate on these conditions which enable us to construct new K"ahler (Sasakian)-Einstein manifolds.
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