NON-EXISTENCE IN PARTICULAR CASES
 
        


Global
1. No non-constant harmonic morphisms, satisfying a non-degeneracy 
   condition, from $S^4, CP^2$ to surfaces,
   [Th. 7.6, 8.7: Wood 1992] 

2. No non-constant harmonic morphisms $S^{2n}\to N^{2n-1}$ for $k\ge2$,
   [Y. X. Dong 1997]


3. No non-constant harmonic morphisms $(M^{n+1},g)\to (N^n,h)$ if
   (1) $M=S^{n+1}$, $n$ odd  and $n\ge 4$,
   (2) $M=CP^k$,    $k$ even and $2k-1\ge 4$,
   (3) $M=HP^l$,    $l$ even and $4k-1\ge 4
   and $g$ any riemannaina metric on $M$.
   [Dong-Cheng 1996]


Local
1. No non-constant horizontally homothetic harmonic morphisms from 
   $U\subset M$ to $N$ for $(M,N)=(S^m,R^n),(S^m,H^n),(R^m,H^n)$,
   [Cor. 2.7: Gudmundsson 1990] 



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