一类Riemann张量指标表达式的标准型完全分类及其在微分几何中的应用

A complete classification of canonical forms of a class of Riemann tensor indexed expressions and its applications in differential geometry

本文研究了n维微分几何中Riemann张量指标表达式的标准型完全分类问题,通过引入指标结构图的概念,证明了规范类型单项式都是标准型,并且构成次数不大于5的Sakai类型单项式的正交基底,由此得到Sakai类型单项式的标准型完全分类,这是次数大于3时标准型完全分类问题的第一个结果.同时给出了相应标准化算法,通过比较说明了该算法比现有算法更加简便,最后应用于自动推导和证明微分几何中关于Riemann张量的一些公式.

This paper is concerned with the problem of complete classification of canonical forms of Riemann tensor expressions that obey Einstein summation convention. By the idea of index-structure-figuration, we prove that the Riemann tensor monomials, whose index-structure-figuration is composed of canonical index-circles are already in their canonical forms, and are the orthogonal invariants of Sakai-type Riemann tensor monomials of degree of no more than 5, and accordingly we obtain a complete classification of the monomials. This is the first result in literature with respect to the degree of more than 3. We also present a normalization algorithm, which is compared with the exiting algorithm and showed to be simpler and faster. Finally, we apply the algorithm to automatically deriving and proving some formulas involving Riemann tensors in differential geometry.