Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations

Abstract

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-Kähler 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. 65 6. Riemannian Einstein AH structures with self-conjugate curvature 68 6.1. Symmetric tensor algebra 68 6.2. Differential operators on trace-free symmetric tensors 70 6.3. Curvature operator on trace-free symmetric tensors 73 6.4. Weitzenböck type identities for trace-free symmetric tensors 76 6.5. Alternative formulations of the Riemannian Einstein AH equations 80 6.6. Growth of the cubic torsion 85 6.7. Infinitesimal automorphisms of convex flat projective structures 91 7. Einstein AH structures from commutative nonassociative algebras 95 7.1. Partial differential equations for homogeneous cubic polynomials 95 7.2. Einstein commutative Codazzi algebras 101 8. The Einstein AH structure on a mean curvature zero Lagrangian submanifold of a para-Kähler space form 113 9. Analogy with conformal structures 119 9.1. Möbius structures, the Codazzi Laplacian, and an analogue of the Yamabe problem 120 9.2. Generalizing the Bach tensor 123 References 125

Killing and conformal Killing tensors

“…in connection with geometric inverse problems, integrable systems and Einstein-Weyl geometry, cf. [5], [8], [9], [13], [15], [23], [26].…”

The commutative nonassociative algebra of metric curvature tensors

“…In [39,52] it is shown that the cubic isoparametric polynomials satisfy (1.6). That the cubic isoparametric polynomials solve (1.5) was observed in section 7 of [18] and [19], and is also shown in Equation (6.4.15) (see also Proposition 6.11.1 and Corollary 6.11.2) of [39]. In [55] there are identified two classes of Hsiang algebras, called exceptional and mutant, and it is shown that a Hsiang algebra satisfies (1.5) if and only if it is exceptional or mutant.…”

“…In one particularly interesting class of examples considered previously by the author in [18] in a context related to affine spheres, some special cases of which are considered in [39] (see further remarks below), the algebra is nonunital (this can always be arranged by considering the retraction along a unit); exact, meaning that its (left) multiplication representation L • : A → End(A) defined by L • (x) = x • y satisfies tr L • (x) = 0 for all x ∈ A (this condition is called harmonic in [39]); and Killing Einstein, meaning that the bilinear form h is a nonzero multiple of the bilinear form τ (x, y) = tr L • (x)L • (y). The last two conditions, exact and Killing Einstein, are equivalent to the requirements that the associated cubic polynomial P be h-harmonic and satisfy…”

Einstein-like geometric structures on surfaces

“…In [30] the author defined a class of geometric structures, called AH (affine hypersurface) structures, which generalize both Weyl structures and the structures induced on a co-oriented non-degenerate immersed hypersurface in flat affine space by its second fundamental form, affine normal and co-normal Gauß map, and defined for these AH structures equations, called Einstein, specializing on the one hand to the usual Einstein Weyl equations and, on the other hand, to the equations for an affine hypersphere. In the present paper these equations are solved on compact orientable surfaces and some aspects of their geometry in this case are described.…”

“…The specialization to surfaces of the notion of Einstein AH structure defined in [30] says that an AH structure on a surface is Einstein if there vanishes the trace-free symmetric part of the Ricci curvature of its aligned representative ∇, while the scalar trace of its Ricci curvature satisfies the condition (6.1), generalizing the constancy of the scalar curvature of an Einstein metric, and taken by D. Calderbank in [13,14,15] as the definition of a two-dimensional Einstein Weyl structure. Calderbank's point of view in the two-dimensional case was motivating for the definition of Einstein AH equations in general.…”