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

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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.