By considering the projectivized spectrum of the Jacobi operator, we introduce the concept of projective Osserman manifold in both the affine and in the pseudo-Riemannian settings. If M is an affine projective Osserman manifold, then the modified Riemannian extension metric on the cotangent bundle is both spacelike and timelike projective Osserman. Since any rank 1 symmetric space is affine projective Osserman, this provides additional information concerning the cotangent bundle of a rank 1 Riemannian symmetric space with the modified Riemannian extension metric. We construct other examples of affine projective Osserman manifolds where the Ricci tensor is not symmetric and thus the connection is not the Levi-Civita connection of any metric. If M is an affine projective Osserman manifold of odd dimension, we use methods of algebraic topology to show the Jacobi operator has only one non-zero eigenvalue and that eigenvalue is real.
曲率模型（V，A）是一个实向量空间V，它配备了一个“曲率算子”A（x，y）z，A具有与仿射曲率算子相同的对称性; A（x，y）z=-A（y，x）z和A（x，y）z+A（y，z）x+A（z，x）y=0。如果Jacobi算子J（y）的谱：x->A（x，y）y是射影常数，则这样的模型称为射影仿射Osserman。球面上向量场的亚当定理对这种模型施加了拓扑条件。本文构造了当维数为奇数、维数为2 mod 4和维数为4 mod 8时的射影仿射Osserman曲率模型，其中特征值结构满足Adam定理.
Conformally Osserman manifolds and conformally complex space forms
本文用Weyl曲率张量刻画了局部共形等价于复射影空间或其负曲率对偶的流形.作为这个调查的副产品，我们分类共形复杂的空间型，如果尺寸至少是8。我们还研究了与黎曼流形的Weyl共形曲率张量相关联的Jacobi算子在单位切向量丛上具有常特征值时的情形，并对在2 mod 4全等维度上不共形平坦的流形进行了分类。
Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics
Flat Metrics, Cubic Differentials and Limits of Projective Holonomies
Labourie和作者分别证明了亏格至少为2的定向曲面上的凸实射影结构等价于共形结构加上全纯三次微分U。我们分析的行为实投影结构的共形结构是固定的，三次微分缩放到无穷大。特别地，我们找到了光滑测地线环路周围的完整性关于|U|的2/3幂所给出的平坦度量的渐近特征值。如果所有回路的渐近完整性都可以计算，那么就可以描述凸实射影结构的变形空间中的一个边界点，如Inkang Kim所构造的。证明涉及C. P. Wang提出的双曲仿射2-球面的仿射微分几何，以及类似于Mike Wolf对调和映射方程和Thurston对Teichmuller空间边界的分析的偏微分方程分析。
