Modern Research of Affine and Projective Geometry and its Applications

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In this research, we studied classical differential geometries, theory of integral systems and information geometry.1. Classical Differential Geometries (1) We characterized minimal affine hypersurfaces and minimal centroaffine immersions of codimension two. Moreover, we gave an explicit method of constructing self-dual minimal centroaffine surfaces of codimension two.(2) We studied manifolds with projectively flat torsion-free affine connection whose Ricci curvature is symmetric and definite, and showed fundamental results on the injectivity of the projective developing maps of such manifolds and the convexity of their image.(3) For conformally flat hypersurfaces of a 4-dimensional sphere, we defined a new conformal invariant. Using the invariant, we characterized the classical examples and constructed new examples.(4) We developed a very concrete and comprehensive theory on curves and surfaces in 3-dimensional homogeneous spaces.2. Integrable Systems We investigated various integrable systems appeared in classical differential geometries. We obtained representation formulae for minimal surfaces in 3-dimensional solvable Lie groups and flat surfaces in a 3-dimensional hyperbolic space. We also developed a comprehensive theory of (spacelike) surfaces with harmonic inverse mean curvature in 3-dimensional Riemannian space forms and Lorentzian space forms.3. Information Geometry and Statistical Manifolds (1) We defined complex statistical manifolds and studied them from the view points of affine differential geometry and of information geometry, especially of quantum estimation theory.(2) As a generalization of special Kahler manifolds, we defined statistical manifolds with compatible complex structure and investigated their fundamental properties.(3) On (-1)-conformally flat statistical manifolds, we gave an explicit method of constructing the Volonoi diagrams.