38 citations · 123 across the 28 of their papers we have counts for
22 papers · 1 filter
Conelike radiant structures
Daniel J. F. Fox
Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulatio…
Curvature equations coupling symmetric tensors with a metric
Daniel J. F. Fox
There are described hierarchies of equations coupling a metric with a trace-free tensor having prescribed symmetries and in the kernel of certain generalized gradients. These speci…
Left-symmetric algebras and homogeneous improper affine spheres
Daniel J. F. Fox
The nonzero level sets in -dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the func…
Remarks on symplectic sectional curvature
Daniel J. F. Fox
In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines t…
Infinitesimal affine automorphisms of symplectic connections
Daniel J. F. Fox
Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is giv…
Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero
Daniel J. F. Fox
Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of -dimensional nondegenerate hypersurface…
Critical symplectic connections on surfaces
Daniel J. F. Fox
The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms o…
Conservation laws for surfaces of constant mean curvature in 3-dimensional space forms
Daniel Fox, Joe S. Wang
The exterior differential system for constant mean curvature (CMC) surfaces in a 3-dimensional space form is an elliptic Monge-Ampere system defined on the unit tangent bundle. We…
Functions dividing their Hessian determinants and affine spheres
Daniel J. F. Fox
The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the funct…
Ricci flows on surfaces related to the Einstein Weyl and Abelian vortex equations
Daniel J. F. Fox
There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of…
Killing fields and Conservation Laws for rank-1 Toda field equations
Daniel Fox
We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomo…
A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone
Daniel J. F. Fox
This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-…
Einstein-like geometric structures on surfaces
Daniel J. F. Fox
An AH (affine hypersurface) structure is a pair comprising a projective equivalence class of torsion-free connections and a conformal structure satisfying a compatibility condition…
Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations
Daniel J. F. Fox
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 st…
Higher-order conservation laws for the non-linear Poisson equation via the characteristic cohomology
Daniel Fox, Oliver Goertsches
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation as elements of the characte…
Generators for Rational Loop Groups and Geometric Applications
Neil Donaldson, Daniel Fox, Oliver Goertsches
Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a…
Cayley cones ruled by 2-planes: desingularization and implications of the twistor fibration
Daniel Fox
Cayley cones in the octonions that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well…
Coassociative cones that are ruled by 2-planes
Daniel Fox
It is shown that coassociative cones in R^7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R^7. The ge…
Contact Path Geometries
Daniel J. F. Fox
Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivizati…
Projectively Invariant Star Products
Daniel J. F. Fox
It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth, fiberwise-polynomial functions on the cotangent bundle a one-para…
Contact Schwarzian Derivatives
Daniel J. F. Fox
H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construct…
Contact Projective Structures
Daniel J. F. Fox
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to…