Modern Research of Affine and Projective Geometry and its Applications

在本研究中，我们学习了经典微分几何、积分系统理论和信息几何。经典微分几何(1)刻画了余维2的极小仿射超曲面和极小中心仿射浸入。此外，我们还给出了构造余维2的自对偶极小中心仿射曲面的显式方法。(2)研究了Ricci曲率对称确定的射影平坦仿射联络流形，给出了这类流形的射影发展映射的内射性及其像的凸性的基本结果。(3)对于四维球面的共形平坦超曲面，我们定义了一个新的共形不变量。利用这个不变量，我们刻画了经典的例子，并构造了新的例子。(4)发展了关于三维齐次空间中曲线曲面的一个非常具体和全面的理论。可积系统我们研究了出现在经典微分几何中的各种可积系统。我们得到了三维可解李群中的极小曲面和三维双曲空间中的平坦曲面的表示公式。我们还发展了三维黎曼空间型和洛伦兹空间型中具有调和逆平均曲率的(类空)曲面的综合理论。信息几何与统计流形(1)我们定义了复统计流形，并从仿射微分几何和信息几何，特别是量子估计理论的角度对其进行了研究。(2)作为特殊Kahler流形的推广，我们定义了具有相容复结构的统计流形，并研究了它们的基本性质。(3)在(-1)-共形平坦统计流形上，给出了构造Volonoi图的一种显式方法。

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.