What do Ou and Wood mean by Harmonic and Horizontally Weakly Conformal?Let PHI be a quadratic map from Rm to Rn. Write PHI as PHI(X) = ( X* A1 X, X* A2 X, ..., X* An X ) where X is a vector in Rm, X* is the transpose of X, and the Ai (i=1,...,n) are symmetric mxm real matrices. Then: PHI is HARMONIC if and only if all the Ai are traceless, i.e.: tr(Ai) = 0 (i=1,...,n) and PHI is HORIZONTALLY WEAKLY CONFORMAL if and only if Ai Aj + Aj Ai = 0 (i,j=1,...,n; i=/=j) and Ai Ai = Aj Aj (i,j=1,...,n)
Let PHI be an orthogonal multiplication on Rn: PHI: Rn x Rn ---} Rn Then PHI is a Harmonic Morphism if and only if n = 1,2,4,8.
Let PHI be a Hopf construction map: PHI: Rn x Rn ---} R(n+1) Then PHI is a Harmonic Morphism if and only if n = 1,2,4,8.
Let PHI be a Clifford System map: PHI: Rm x Rm = R(2m) ---} R(n+1) represented by 2mx2m symmetric matrices Ai (i=0,1,...,n) such that Ai Aj + Aj Ai = 2dij ID (i,j=1,...,n) where dij is the Kronecker delta and ID is the 2mx2m identity matrix. Then PHI is a Harmonic Morphism, and the Clifford System is irreducible for the following m and n: m n 1 1 2 2 4 3 4 4 8 5 8 6 8 7 8 8 ... ... 16m n+8
Sigmundur Gudmundsson and Stefano Montaldo have a WWW site, THE ATLAS OF HARMONIC MORPHISMS, and Sigmundur Gudmundsson is also editor of THE HARMONIC MORPHISMS BIBLIOGRAPHY.
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