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Open Problems in Affine Differential Geometry and Related Topics
Abstract
The following problems were raised in the workshop “Affine Differential Geometry and Related Topics” at Graduate School of Information Sciences, Tohoku University at December 16-18, 1996.
... If this is the case, we say that the Riemannian manifold (M, g) admits a Hessian structure. a e-mail: fujiwara@math.sci.osaka-u.ac.jp (corresponding author) This problem was raised in [2, p. 102] and was recalled in [3,4]. Determining whether a manifold admits a global dually flat structure was studied in [5]. ...
The notion of dually flatness is of central importance in information geometry. Nevertheless, little is known about dually flat structures on quantum statistical manifolds except that the Bogoliubov metric admits a global dually flat structure on a quantum state space S ( C d ) for any d ≥ 2 . In this paper, we show that every monotone metric on a two-level quantum state space S ( C 2 ) admits a local dually flat structure.
... As we shall see, results which are easy to prove from the perspective of affine connections can be harder to understand from the perspective of Hessian geometry and vice versa. The issue of determining whether a metric g is a Hessian metric was raised in [8,1] in the language of g-dually flat connections. They posed the following basic questions: Problem 1. ...
We prove that, in dimensions greater than 2, the generic metric is not a Hessian metric and find a curvature condition on Hessian metrics in dimensions greater than 3. In particular we prove that the forms used to define the Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By contrast all analytic Riemannian 2-metrics are Hessian metrics.
In this paper we prove that any smooth surfaces can be locally isometrically embedded into as Lagrangian surfaces. As a byproduct we obtain that any smooth surfaces are Hessian surfaces.
We prove that, in dimensions greater than 2, the generic metric is not a
Hessian metric and find a curvature condition on Hessian metrics in dimensions
greater than 3. In particular we prove that the forms used to define the
Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By
contrast all analytic Riemannian 2-metrics are Hessian metrics.
Let (M;h;r) be a statistical manifold of dimension n(‚ 2). Then we show that (M; h; r) is of constant curvature if and only if the tangent bundle TM over M with complete lift statistical structure (hC; rC) is conformally-projectively ∞at.
We consider the problem of prescribing scalar curvature on the standard 2-sphere S
2. It is proved that any positive smooth function on S
2 is the scalar curvature of some pointwise conformal metric, if an associated map has non-zero degree. As a result we improve some previous important results and give some completely new ones.
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