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.   

Tony Smith's Home Page

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