Travel Grant for the International Conference on "Finsler Extension of Relativity Theory"

经典广义相对论在很大程度上是建立在黎曼几何向洛伦兹几何（更一般地说，半黎曼几何）的扩展之上的。事实上，与时间无关的问题本质上基本上是黎曼问题。例如，黑洞的分类是基于场方程的与时间无关的解；正质量定理的（第一个）证明也是基于黎曼几何。目前研究人员感兴趣的是将经典和相对论物理空间（如黎曼几何）的主要几何和物理结构严格推广和推广到芬斯勒空间的问题。相对论建立在黎曼几何基础上的问题之一是“伤亡”的困难。这是因为黎曼几何中的测地线是可逆的。如果一个时空允许CTC（封闭类时曲线），那么一个人可以回到过去，各种（致命的）悖论都可能发生。相反，存在芬斯勒度量，其中测地线是不可逆的，因此，为处理相对论中的“伤亡”提供了更自然的设置。有许多问题，本质上更技术性，涉及到在相对论中有用的芬斯勒几何。其中包括：(a)度量函数为依赖于四阶变量的对称多项式的空间（在芬斯勒几何中称为切尔诺夫空间），(b)度量函数为依赖于四阶变量的对称多项式的空间（在芬斯勒几何中称为伯瓦尔-摩尔空间），(c)多线性对称形式为标量积的芬斯勒扩展，(d)芬斯勒空间与超复数之间的联系，(e)扩展的可能性，在芬斯勒的背景下，洛伦兹分裂定理，最初是由S. T. Yau作为Cheeger和Gromov的黎曼分裂定理的自然推广而推测出来的。

Classical general relativity is built in large part upon the extension of Riemannian Geometry to Lorentzian Geometry (more generally, semi-Riemannian Geometry). In fact, time independent questions are basically Riemannian in nature. For examples the classification of black holes is based on time independent solutions of the field equations; the (first) proof of the positive mass theorem is also based on Riemannian Geometry. At present researchers are interested in issues involved in the rigorous extension and generalization of the principal geometric and physical structures associated with the spaces of classical and relativistic physics (e.g. of Riemannian geometry) to Finsler spaces. One of the problems of basing Relativity on Riemannian Geometry is the difficulty with "casualty". This is due to the fact that geodesics in Riemannian Geometry are reversible. If a space-time admits a CTC (closed time-like curve) then one could travel back in time and all kinds of (murderous) paradoxes could occur. In contrast, there exist Finsler metrics for which the geodesics are irreversible hence, provides a more natural setting for dealing with "casualty" in Relativity. There are many issues, more technical in nature, involving Finsler Geometry that are useful in Relativity. Among these are: (a) spaces whose metric functions are symmetric polynomials depending on four variables of the third order (known as Chernov spaces in Finsler Geometry), (b) spaces whose metric functions are symmetric polynomials depending on four variables of the fourth order (known as Berwald-Moore spaces in Finsler Geometry), (c) multi-linear symmetric forms as Finsler extension of scalar product, (d) connection between Finsler spaces and hypercomplex numbers, (e) possibility of extending, in the Finsler setting, the Lorentzian splitting theorem, originally conjectured by S. T. Yau as the natural extension of the Riemannian splitting theorem of Cheeger and Gromov.