Major research activity
The main research subject of our laboratory is differential geometry, in particular, the geometry of statistical manifolds (also called Codazzi manifolds).
A statistical manifold is a manifold provided with a pair of torsion-free affine connection and a Riemannian metric which satisfies a certain compatibility condition, called the Codazzi condition. Any Riemannian manifold (with Levi-Civita connection) is a special kind of statistical manifolds, however, statistical manifolds form a wider class that properly includes Riemannian manifolds.
Statistical manifolds play an important role in three quite different research fields: affine hypersurface theory (AHT), geometry of Hessian manifolds, and information geometry. Hence, the study of statistical manifolds is deeply concerned with these fields, and the obtained results may expect to be applied to each of them. In our research, for instance, we have shown that the usual geometric structure dealt with in information geometry can be appreciated through AHT of finite or infinite dimension. Using this view-point, moreover, some notion and several results in information geometry are reconstructed and generalized in a geometric way. Also, by using AHT, we have given the explicit examples and the classification of Hessian manifolds satisfying a certain special property, which are valuable in both purely theoretic and practical aspects.
Major relevant publications
- A. Fujioka and T. Kurose, Multi-Hamiltonian structures on spaces of closed equicentroaffine plane curves associated to higher KdV flows, Symmetry, Integrability and Geometry: Methods and Applications, 10(2014), 048, 11 pages (DOI: 10.3842/SIGMA.2014.048).
- H. Furuhata and T. Kurose, Hessian manifolds of nonpositive constant Hessian sectional curvature, Tohoku Mathematical Journal, 65(2013), 31-42.
- A. Fujioka and T. Kurose, Hamiltonian formalism for the higher KdV flows on the space of closed complex equicentroaffine curves, International Journal of Geometric Methods in Modern Physics, 7(2010), 165-175.
- A. Fujioka and T. Kurose, Geometry of the space of closed curves in the complex hyperbola, Kyushu Journal of Mathematics, 63(2009), 161-165.
- A. Fujioka and T. Kurose, Motions of curves in the complex hyperbola and the Burgers hierarchy, Osaka Journal of Mathematics, 45(2008), 1057-1065.