Mathematics – Differential Geometry
Scientific paper
2008-11-11
Mathematics
Differential Geometry
Scientific paper
We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold $M$ as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle $N=TM$ carries a canonical structure of (affine) special K\"ahler manifold. This gives an intrinsic description of the $r$-map as the map $M\mapsto N=TM$. On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when $M$ is any Hessian manifold.
Alekseevsky Dmitri V.
Cortés Vicente
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