CONFORMAL GEOMETRY OF GENERALIZED METRIC SPACES.
CONFORMAL GEOMETRY OF GENERALIZED METRIC SPACES.
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广义度量空间的共形几何。
DOI：
10.1073/pnas.15.4.376
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发表时间：
1929
影响因子：
11.1
通讯作者：
M. S. Knebelman
中科院分区：
文献类型：
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作者：
M. S. Knebelman
1.设V，是一个n维广义度量空间，其中有一个绝对标量微分不变量f（x，dx），f在dx，.中是二次正齐次的，dxT。如果。i= f1bdx '且gij= f12fij。j向量t相对于元素dx的长度根据定义为t2= g（xdx）t1，（1.1）使用每个重复索引的求和约定。我定义两个度量f（x，dx）和f '（x，dx）为共形的，如果一个度量中任意向量的长度与另一个度量中该向量的长度成比例。设'p（x，dx）为比例因子;则从（1.1）式我们得到gij =' pgij，因此得到gij。k= Vgqj。k+（0. kgVj= Vgjk。i+“。igjk因此
1. Let V,, be an n-dimensional generalized metric space-one to which there is assigned an absolute scalar differential invariant f (x, dx), f being positively homogeneous of degree two in dxl,..., dxT. Iff. i= flbdx'and gij=/2fij. J the length of a vector t relative to the element dx is by definition t2= g (xdx) t1,(1.1) the summation convention for every repeated index being used. I define two metrics f (x, dx) and f'(x, dx) as conformal if the length of an arbitrary vector in the one is proportional to the length of this vector in the other. Let'p (x, dx) be the factor of proportionality; then from (1.1) we obtain gij='pgij and therefore gij. k= Vgqj. k+(o. kgVj= Vgjk. i+'. igjk Hence