Barbara Opozda
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
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Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to examine to examine the geodesic completeness of $\mathcal{M}$.
On completeness of foliated structures, and null Killing fields
本文研究带叶状结构的紧致流形中叶的完备性问题，刻画了紧致洛伦兹流形全测地类光叶状结构的完备性，给出幺模仿射类光几何及松弛情形的完备性结果，并提供存在零Killing场时紧致洛伦兹流形非完备的3维例子。
Geodesic Connectedness on Statistical Manifolds with Divisible Cubic Forms
研究统计流形上具有可除三次形式的仿射连接的测地连通性，建立类似Hopf-Rinow定理的结论，并推导出统计流形的Cartan-Hadamard型定理。
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces.
This note proves the geodesic completeness of any compact manifold endowed with a linear connection such that the closure of its holonomy group is compact.
In this paper, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated to Bott connections, canonical connections and Kobayashi-Nomizu connections are Codazzi tensors associated to these connections. We also classify three-dimensional Lorentzian Lie group with the quasi-statistical structure associated to Bott connections, canonical connections and Kobayashi-Nomizu connections.
We show that, for an affine submersion $π: \mathbf{M}\longrightarrow \mathbf{B}$ with horizontal distribution, $\mathbf{B}$ is a statistical manifold with the metric and connection induced from the statistical manifold $\mathbf{M}$. The concept of conformal submersion with horizontal distribution is introduced, which is a generalization of affine submersion with horizontal distribution. Then proved a necessary and sufficient condition for $(\mathbf{M}, \nabla, g_M)$ to become a statistical manifold for a conformal submersion with horizontal distribution. A necessary and sufficient condition is…
In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita connection is flat. We show that no connected semisimple Lie group admits a left invariant flat affine connection. We characterize flat pseudo-Riemannian Lie groups. For a flat left-invariant pseudo-metric on a Lie group, we show the equivalence between the completeness of the Levi-Civita connection and unimodularity of the group. We emphasize the case of flat left i…
We formulate a correspondence between affine and projective special Kähler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special Kähler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special Kähler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative $α'$-corrections in heterotic and type-II string compactifications with $N=2$ supersymmetry. Also affine special Kähler manifolds with quadratic prepotential are mapped to one…
Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption is dropped, the manifold is not necessarily obtained as the quotient of the Euclidean space through a properly discontinuous group of affine transformations. In fact the universal cover may no longer be the Euclidean space. The main result of this paper states that all compact aff…
Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)
本文研究了(2,2)签名的平坦伪黎曼流形的完备性，证明了唯一由离散子群除尽的域是自身，为非欧几里得和洛伦兹情况下的闭合流形提供了首个完备性定理。
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen's inequalities for non integrable distributions of real space forms endowed with a semi-symmetric metric connection and a semi-symmetric non-metric connection. We give so…
The associativity of the multiplication on a Frobenius manifold is equivalent to the WDVV equation of a symmetric cubic form in flat coordinates. Frobenius manifold could be regarded a very special type of statistical manifold. There is a natural commutative product on each tangent space of a statistical manifold. We show that it is associative, hence making it into a manifold with Frobenius algebra structure, if and only if the sectional $K$-curvature vanishes. In other words, WDVV equation is equivalent to zero sectional $K$-curvature. This gives a curvature interpretation for WDVV equation.
A Riemannian Characterization of Compact Affine Manifolds with Parallel Volume
本文通过构造黎曼度量代表，证明具有平行体积的仿射流形必然完备且黎曼平坦，推广了 Markus 猜想，揭示了此类流形的内在刚性。
On completeness of certain locally symmetric pseudo-Riemannian manifolds of signature $(2,2)$
该文研究标号$(2,n)$的紧致局部对称伪黎曼流形的测地完备性，通过分析模型空间性质、离散群作用等，证明4维及任意维下的相关完备性，还推进了Markus猜想、完成流形分类并得到平行流等度连续的结果。
The dually flat structure of statistical manifolds can be derived in a non-parametric way from a particular case of affine space defined on a qualified set of probability measures. The statistically natural displacement mapping of the affine space depends on the notion of Fisher's score. The model space must be carefully defined if the state space is not finite. Among various options, we discuss how to use Orlicz-Sobolev spaces with Gaussian weight. Such a fully non-parametric set-up provides tools to discuss intrinsically infinite-dimensional evolution problems.
In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.
A Variation Problem for Mappings between Statistical Manifolds
本文提出统计双和弦映射，通过变分问题自然推导出该类映射，并证明不恰当仿射超球面诱导此类映射的例子。
New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry
本文从子流形理论与Calabi仿射几何视角研究仿射极大型曲面，刻画了已有显式欧氏完备反例的几何特征，并构造新的非二次欧氏完备仿射极大型超曲面，拓展了对应参数的取值范围。
A two-point approach to the inverse problem in information geometry
该研究在两点张量框架下求解一般度量仿射流形的信息几何逆问题，通过同伦约化得到相关对比函数，并将理论应用于特定流形的分析。
This paper studies the geometry of immersions into statistical manifolds. A necessary and sufficient condition is obtained for statistical manifold structures to be dual to each other for a non-degenerate equiaffine immersion. Then we obtain conditions for realizing an n-dimensional statistical manifold in an (n+1)-dimensional statistical manifold and its converse. Centro-affine immersion of codimension two into a dually flat statistical manifold is defined. Also we have shown that statistical manifold realized in a dually flat statistical manifold of codimension two is conformally-projectivel…
We will show that a statistical manifold $(M, g, \nabla)$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $\nabla$ is projectively flat and the curvatures satisfies $R=R^*$, where $R^*$ is the curvature of the dual connection $\nabla^*$. Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $-1$ statistical structures and vice versa.
We prove the Bonnet theorem for statistical manifolds, which states that if a statistical manifold admits tensors satisfying the Gauss--Codazzi--Ricci equations, then it is locally embeddable to a flat statistical manifold (or a Hessian manifold). The proof is based on the notion of statistical embedding to the product of a vector space and its dual space introduced by Lauritzen. As another application of Lauritzen's embedding, we show that a statistical manifold admitting an affine embedding of codimension 1 or 2 is locally embeddable to a flat statistical manifold of the same codimension.
These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian and semi-Riemannian structures on manifolds. Affine connections, geodesics, torsion and curvature, the exponential map, and the Riemannian connection follow quickly. The Taylor series for of the metric in normal coordinates is an unusual feature. The last third covers, first and second variation of energy, completeness, cut points, the Hadamard-Cartan theore…
Statistical manifold with degenerate metric
针对实际应用中统计流形度量退化导致原有Codazzi结构不适用的问题，本文提出拟Codazzi结构，明确其几何定位与特例，推广了相关几何结构间的对应关系。
Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric Spaces
本文提出一种在仿射连接空间中具有三级精度的平行运输方案，该方案在仿射对称空间中精确，且在黎曼对称流形中几乎全局精确。
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