Riemannian geometry as determined by the volumes of small geodesic balls
Riemannian geometry as determined by the volumes of small geodesic balls
复制标题
由小测地线球的体积确定的黎曼几何
DOI：
10.1007/bf02395060
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发表时间：
1979
期刊：
影响因子：
3.7
通讯作者：
L. Vanhecke
中科院分区：
文献类型：
--
作者：
A. Gray;L. Vanhecke
(这里0 ~-~单位球的体积在R”。eo最简单的表达式是w = (1) (89 "/~ where ({rn)！——f（89 + 1）。首先我们要谈几点。1. 我们攻击猜想(I)的方法是使用Vm(r)的幂级数展开。第3节将详细讨论这一扩展；然而，关于它的一般事实如下：(a)该系列的第一项是玉米；(b)当k为奇数时，r n+~的系数消失；(c) rn +~的系数可以用曲率表示。不幸的是，非零系数以一种相当复杂的方式依赖于曲率，这使得猜想(1)的求解成为一个有趣的问题。2. 据我们所知，Vm(r)的幂级数展开式是由Bertrand-Diguet-Puiseux于1848年提出的。参见[14，第209页]。在这些论文中，对RS中的曲面计算了Fro(r)展开的前两项：
(Here o~-~the volume of the unit ball in R". The simplest expression for eo is w = (1]( 89 "/~ where ({rn)! ---F( 89 + 1).) First we make several remarks. 1. Our method for attacking the conjecture (I) will be to use the power series expansion for Vm(r). This expansion will be considered in detail in section 3; however, the general facts about it are the following: (a) the first term in the series is corn; (b) the coefficient of r n+~ vanishes provided k is odd; (c) the coefficients of r n+~ for/r even can be expressed in terms of curvature. Unfortunately the nonzero coefficients depend on curvature in a rather complicated way, and this is what makes the resolution of the conjecture (I) an interesting problem. 2. To our knowledge the power series expansion for Vm(r) was first considered in 1848 by Bertrand-Diguet-Puiseux [6]. See also [14, p. 209]. In these papers the first two terms of the expansion for Fro(r) are computed for surfaces in RS: