Differential Geometry of Spray and Finsler Spaces
Differential Geometry of Spray and Finsler Spaces
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DOI：
10.1007/978-94-015-9727-2
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发表时间：
2001-03
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影响因子：
--
通讯作者：
Z. Shen
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作者：
Z. Shen
在这本书中，我们研究了喷雾和芬斯勒指标。粗略地说，流形上的喷雾由相容的二阶常微分方程组组成。流形上的Finsler度量是切空间中的一族范数，这些范数随基点光滑变化。每个Finsler度规通过其测地线方程组来确定喷雾。因此，Finsler空间可以看作是特殊的喷雾空间。另一方面，每个Finsler度量通过极小曲线的长度来定义距离函数。因此，Finsler空间可视为正则度量空间。黎曼空间是特殊的正则度量空间。1854年，B.Riemann在他的开创性的哈密尔顿定理中引入了黎曼空间的黎曼曲率。此后，这些特殊的正则度量空间的几何就以他的名字命名。Riemann也提到了一般正则度量空间，但他认为在一般情况下没有什么新鲜事。事实上，从技术上讲，处理一般正则度量空间要困难得多。半个多世纪以来，直到P·芬斯勒在1918年完成了他的开创性工作，在这个方向上一直没有实质性的进展。Finsler研究了一般正则度量空间中曲线曲面的变分问题。一些难题被他解决了。从那时起，这种正则度量空间被称为Finsler空间。然而，芬斯勒并没有更进一步地引入正则度量空间的曲率。毕业后不久，他就把研究方向转向了集理论。
In this book we study sprays and Finsler metrics. Roughly speaking, a spray on a manifold consists of compatible systems of second-order ordinary differential equations. A Finsler metric on a manifold is a family of norms in tangent spaces, which vary smoothly with the base point. Every Finsler metric determines a spray by its systems of geodesic equations. Thus, Finsler spaces can be viewed as special spray spaces. On the other hand, every Finsler metric defines a distance function by the length of minimial curves. Thus Finsler spaces can be viewed as regular metric spaces. Riemannian spaces are special regular metric spaces. In 1854, B. Riemann introduced the Riemann curvature for Riemannian spaces in his ground-breaking Habilitationsvortrag. Thereafter the geometry of these special regular metric spaces is named after him. Riemann also mentioned general regular metric spaces, but he thought that there were nothing new in the general case. In fact, it is technically much more difficult to deal with general regular metric spaces. For more than half century, there had been no essential progress in this direction until P. Finsler did his pioneering work in 1918. Finsler studied the variational problems of curves and surfaces in general regular metric spaces. Some difficult problems were solved by him. Since then, such regular metric spaces are called Finsler spaces. Finsler, however, did not go any further to introduce curvatures for regular metric spaces. He switched his research direction to set theory shortly after his graduation.