Mathematics – Differential Geometry
Scientific paper
2009-09-10
Mathematics
Differential Geometry
Version 5: Minor corrections, mostly in section 7. Changed contact information
Scientific paper
An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetric. AH structures simultaneously generalize Weyl structures and abstract the geometric structure determined on a non-degenerate co-oriented hypersurface in flat affine space by its second fundamental form together with either the projective structure induced by the affine normal or that induced by the conormal Gauss map. There are proposed notions of Einstein equations for AH structures which for Weyl structures specialize to the usual Einstein Weyl equations and such that the AH structure induced on a non-degenerate co-oriented affine hypersurface is Einstein if and only if the hypersurface is an affine hypersphere. It is shown that a convex flat projective structure admits a metric with which it generates an Einstein AH structure, and examples are constructed on mean curvature zero Lagrangian submanifolds of certain para-K\"ahler manifolds. The rough classification of Riemannian Einstein Weyl structures by properties of the scalar curvature is extended to this setting. Known estimates on the growth of the cubic form of an affine hypersphere are partly generalized. The Riemannian Einstein equations are reformulated in terms of a given background metric as an algebraically constrained elliptic system for a cubic tensor. From certain commutative nonassociative algebras there are constructed examples of exact Riemannian signature Einstein AH structures with self-conjugate curvature but which are not Weyl and are neither projectively nor conjugate projectively flat.
No associations
LandOfFree
Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-131361