BIRS Workshop Lecture Videos
Sasaki-Einstein structures and their compactification Gover, Rod
Sasaki geometry is often viewed as an odd dimensional analogue of Kaehler geometry. In particular a Riemannian or pseudo-Riemannian manifold is Sasakian if its standard metric cone is Kaehler or, respectively, pseudo-Kaehler. We show that there is a natural link between Sasaki geometry and projective differential geometry. The situation is particularly elegant for Sasaki-Einstein geometries and in this setting we use projective geometry to provide the resolution of these geometries into â less rigidâ components. This is analogous to usual picture of a Kaehler structure: a symplectic manifold equipped also with a compatible complex structure etc. However the treatment of Sasaki geometry this way is locally more interesting and involves the projective Cartan or tractor connection. This enables us to describe a natural notion of compactification for complete non-compact pseudo-Riemannian Sasakian geometries. For such compactifications the boundary at infinity is a conformal manifold with a Fefferman space structureâ so it fibres over a CR manifold. This is nicely compatible with the compactification of the Kaehler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. This is joint work with Katharina Neusser and Travis Willse.
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