The Sectional Curvature Of Gauss Distribution Manifold And Its Compatibility With Fisher Metric

Information Geometry is a subject that applies concepts and methods of differ-ential geometry to statistics problems.In 1945,Rao first introduced the concept of geodesic in differential geometry into statistics.Since then statisticians keep trying to introduce various ideas in differential geometry to study statistics and finally give birth to the branch information geometry.Usually a statistics model is parameterized by several real values,each parameter corresponding to a probability distribution which can be viewed as a point in this model.Naturally it can be viewed as a manifold.On the other hand,for any such model,there are matrices called Fisher information ma-trices corresponding to each point in this model.These matrices are positive definite and therefore can be defined as the Riemannian metric in this manifold.After equiped with these structures,methods of differential geometry,especially of Riemannian geometry can be applied into a statistics model to invest its properties.In this thesis,we mainly try to study the geometric properties Under a conformal transformation the Fisher information matrix is the same as the pull back of the metric.The discrete distribution manifold also has negative constant sectional curvature.A natural question is that whether these are other statistic model with constant sectional curvature.